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The Computational Theory of Mind

Could a machine think? Could the mind itself be a thinking machine? The computer revolution transformed discussion of these questions, offering our best prospects yet for machines that emulate reasoning, decision-making, problem solving, perception, linguistic comprehension, and other mental processes. Advances in computing raise the prospect that the mind itself is a computational system—a position known as the computational theory of mind (CTM). Computationalists are researchers who endorse CTM, at least as applied to certain important mental processes. CTM played a central role within cognitive science during the 1960s and 1970s. For many years, it enjoyed orthodox status. More recently, it has come under pressure from various rival paradigms. A key task facing computationalists is to explain what one means when one says that the mind “computes”. A second task is to argue that the mind “computes” in the relevant sense. A third task is to elucidate how computational description relates to other common types of description, especially neurophysiological description (which cites neurophysiological properties of the organism’s brain or body) and intentional description (which cites representational properties of mental states).

1. Turing machines

2. artificial intelligence, 3.1 machine functionalism, 3.2 the representational theory of mind, 4.1 relation between neural networks and classical computation, 4.2 arguments for connectionism, 4.3 systematicity and productivity, 4.4 computational neuroscience, 5.1 computation as formal, 5.2 externalism about mental content, 5.3 content-involving computation, 6.1 information-processing, 6.2 function evaluation, 6.3 structuralism, 6.4 mechanistic theories, 6.5 pluralism, 7.1 triviality arguments, 7.2 gödel’s incompleteness theorem, 7.3 limits of computational modeling, 7.4 temporal arguments, 7.5 embodied cognition, other internet resources, related entries.

The intuitive notions of computation and algorithm are central to mathematics. Roughly speaking, an algorithm is an explicit, step-by-step procedure for answering some question or solving some problem. An algorithm provides routine mechanical instructions dictating how to proceed at each step. Obeying the instructions requires no special ingenuity or creativity. For example, the familiar grade-school algorithms describe how to compute addition, multiplication, and division. Until the early twentieth century, mathematicians relied upon informal notions of computation and algorithm without attempting anything like a formal analysis. Developments in the foundations of mathematics eventually impelled logicians to pursue a more systematic treatment. Alan Turing’s landmark paper “On Computable Numbers, With an Application to the Entscheidungsproblem” (Turing 1936) offered the analysis that has proved most influential.

A Turing machine is an abstract model of an idealized computing device with unlimited time and storage space at its disposal. The device manipulates symbols , much as a human computing agent manipulates pencil marks on paper during arithmetical computation. Turing says very little about the nature of symbols. He assumes that primitive symbols are drawn from a finite alphabet. He also assumes that symbols can be inscribed or erased at “memory locations”. Turing’s model works as follows:

• There are infinitely many memory locations, arrayed in a linear structure. Metaphorically, these memory locations are “cells” on an infinitely long “paper tape”. More literally, the memory locations might be physically realized in various media (e.g., silicon chips).
• There is a central processor, which can access one memory location at a time. Metaphorically, the central processor is a “scanner” that moves along the paper tape one “cell” at a time.
• The central processor can enter into finitely many machine states .
• The central processor can perform four elementary operations: write a symbol at a memory location; erase a symbol from a memory location; access the next memory location in the linear array (“move to the right on the tape”); access the previous memory location in the linear array (“move to the left on the tape”).
• Which elementary operation the central processor performs depends entirely upon two facts: which symbol is currently inscribed at the present memory location; and the scanner’s own current machine state.
• A machine table dictates which elementary operation the central processor performs, given its current machine state and the symbol it is currently accessing. The machine table also dictates how the central processor’s machine state changes given those same factors. Thus, the machine table enshrines a finite set of routine mechanical instructions governing computation.

Turing translates this informal description into a rigorous mathematical model. For more details, see the entry on Turing machines .

Turing motivates his approach by reflecting on idealized human computing agents. Citing finitary limits on our perceptual and cognitive apparatus, he argues that any symbolic algorithm executed by a human can be replicated by a suitable Turing machine. He concludes that the Turing machine formalism, despite its extreme simplicity, is powerful enough to capture all humanly executable mechanical procedures over symbolic configurations. Subsequent discussants have almost universally agreed.

Turing computation is often described as digital rather than analog . What this means is not always so clear, but the basic idea is usually that computation operates over discrete configurations. By comparison, many historically important algorithms operate over continuously variable configurations. For example, Euclidean geometry assigns a large role to ruler-and-compass constructions , which manipulate geometric shapes. For any shape, one can find another that differs to an arbitrarily small extent. Symbolic configurations manipulated by a Turing machine do not differ to arbitrarily small extent. Turing machines operate over discrete strings of elements (digits) drawn from a finite alphabet. One recurring controversy concerns whether the digital paradigm is well-suited to model mental activity or whether an analog paradigm would instead be more fitting (MacLennan 2012; Piccinini and Bahar 2013).

Besides introducing Turing machines, Turing (1936) proved several seminal mathematical results involving them. In particular, he proved the existence of a universal Turing machine (UTM). Roughly speaking, a UTM is a Turing machine that can mimic any other Turing machine. One provides the UTM with a symbolic input that codes the machine table for Turing machine M . The UTM replicates M ’s behavior, executing instructions enshrined by M ’s machine table. In that sense, the UTM is a programmable general purpose computer . To a first approximation, all personal computers are also general purpose: they can mimic any Turing machine, when suitably programmed. The main caveat is that physical computers have finite memory, whereas a Turing machine has unlimited memory. More accurately, then, a personal computer can mimic any Turing machine until it exhausts its limited memory supply .

Turing’s discussion helped lay the foundations for computer science , which seeks to design, build, and understand computing systems. As we know, computer scientists can now build extremely sophisticated computing machines. All these machines implement something resembling Turing computation, although the details differ from Turing’s simplified model.

Rapid progress in computer science prompted many, including Turing, to contemplate whether we could build a computer capable of thought. Artificial Intelligence (AI) aims to construct “thinking machinery”. More precisely, it aims to construct computing machines that execute core mental tasks such as reasoning, decision-making, problem solving, and so on. During the 1950s and 1960s, this goal came to seem increasingly realistic (Haugeland 1985).

Early AI research emphasized logic . Researchers sought to “mechanize” deductive reasoning. A famous example was the Logic Theorist computer program (Newell and Simon 1956), which proved 38 of the first 52 theorems from Principia Mathematica (Whitehead and Russell 1925). In one case, it discovered a simpler proof than Principia ’s.

Early success of this kind stimulated enormous interest inside and outside the academy. Many researchers predicted that intelligent machines were only a few years away. Obviously, these predictions have not been fulfilled. Intelligent robots do not yet walk among us. Even relatively low-level mental processes such as perception vastly exceed the capacities of current computer programs. When confident predictions of thinking machines proved too optimistic, many observers lost interest or concluded that AI was a fool’s errand. Nevertheless, the decades have witnessed gradual progress. One striking success was IBM’s Deep Blue, which defeated chess champion Gary Kasparov in 1997. Another major success was the driverless car Stanley (Thrun, Montemerlo, Dahlkamp, et al. 2006), which completed a 132-mile course in the Mojave Desert, winning the 2005 Defense Advanced Research Projects Agency (DARPA) Grand Challenge. A less flashy success story is the vast improvement in speech recognition algorithms.

One problem that dogged early work in AI is uncertainty . Nearly all reasoning and decision-making operates under conditions of uncertainty. For example, you may need to decide whether to go on a picnic while being uncertain whether it will rain. Bayesian decision theory is the standard mathematical model of inference and decision-making under uncertainty. Uncertainty is codified through probability . Precise rules dictate how to update probabilities in light of new evidence and how to select actions in light of probabilities and utilities. (See the entries Bayes’s theorem and normative theories of rational choice: expected utility for details.) In the 1980s and 1990s, technological and conceptual developments enabled efficient computer programs that implement or approximate Bayesian inference in realistic scenarios. An explosion of Bayesian AI ensued (Thrun, Burgard, and Fox 2006), including the aforementioned advances in speech recognition and driverless vehicles. Tractable algorithms that handle uncertainty are a major achievement of contemporary AI (Murphy 2012), and possibly a harbinger of more impressive future progress.

Some philosophers insist that computers, no matter how sophisticated they become, will at best mimic rather than replicate thought. A computer simulation of the weather does not really rain. A computer simulation of flight does not really fly. Even if a computing system could simulate mental activity, why suspect that it would constitute the genuine article?

Turing (1950) anticipated these worries and tried to defuse them. He proposed a scenario, now called the Turing Test , where one evaluates whether an unseen interlocutor is a computer or a human. A computer passes the Turing test if one cannot determine that it is a computer. Turing proposed that we abandon the question “Could a computer think?” as hopelessly vague, replacing it with the question “Could a computer pass the Turing test?”. Turing’s discussion has received considerable attention, proving especially influential within AI. Ned Block (1981) offers an influential critique. He argues that certain possible machines pass the Turing test even though these machines do not come close to genuine thought or intelligence. See the entry the Turing test for discussion of Block’s objection and other issues surrounding the Turing Test.

For more on AI, see the entry logic and artificial intelligence . For much more detail, see Russell and Norvig (2010).

3. The classical computational theory of mind

Warren McCulloch and Walter Pitts (1943) first suggested that something resembling the Turing machine might provide a good model for the mind. In the 1960s, Turing computation became central to the emerging interdisciplinary initiative cognitive science , which studies the mind by drawing upon psychology, computer science (especially AI), linguistics, philosophy, economics (especially game theory and behavioral economics), anthropology, and neuroscience. The label classical computational theory of mind (which we will abbreviate as CCTM) is now fairly standard. According to CCTM, the mind is a computational system similar in important respects to a Turing machine, and core mental processes (e.g., reasoning, decision-making, and problem solving) are computations similar in important respects to computations executed by a Turing machine. These formulations are imprecise. CCTM is best seen as a family of views, rather than a single well-defined view. [ 1 ]

It is common to describe CCTM as embodying “the computer metaphor”. This description is doubly misleading.

First, CCTM is better formulated by describing the mind as a “computing system” or a “computational system” rather than a “computer”. As David Chalmers (2011) notes, describing a system as a “computer” strongly suggests that the system is programmable . As Chalmers also notes, one need not claim that the mind is programmable simply because one regards it as a Turing-style computational system. (Most Turing machines are not programmable.) Thus, the phrase “computer metaphor” strongly suggests theoretical commitments that are inessential to CCTM. The point here is not just terminological. Critics of CCTM often object that the mind is not a programmable general purpose computer (Churchland, Koch, and Sejnowski 1990). Since classical computationalists need not claim (and usually do not claim) that the mind is a programmable general purpose computer, the objection is misdirected.

Second, CCTM is not intended metaphorically. CCTM does not simply hold that the mind is like a computing system. CCTM holds that the mind literally is a computing system. Of course, the most familiar artificial computing systems are made from silicon chips or similar materials, whereas the human body is made from flesh and blood. But CCTM holds that this difference disguises a more fundamental similarity, which we can capture through a Turing-style computational model. In offering such a model, we prescind from physical details. We attain an abstract computational description that could be physically implemented in diverse ways (e.g., through silicon chips, or neurons, or pulleys and levers). CCTM holds that a suitable abstract computational model offers a literally true description of core mental processes.

It is common to summarize CCTM through the slogan “the mind is a Turing machine”. This slogan is also somewhat misleading, because no one regards Turing’s precise formalism as a plausible model of mental activity. The formalism seems too restrictive in several ways:

• Turing machines execute pure symbolic computation. The inputs and outputs are symbols inscribed in memory locations. In contrast, the mind receives sensory input (e.g., retinal stimulations) and produces motor output (e.g., muscle activations). A complete theory must describe how mental computation interfaces with sensory inputs and motor outputs.
• A Turing machine has infinite discrete memory capacity. Ordinary biological systems have finite memory capacity. A plausible psychological model must replace the infinite memory store with a large but finite memory store
• Modern computers have random access memory : addressable memory locations that the central processor can directly access. Turing machine memory is not addressable. The central processor can access a location only by sequentially accessing intermediate locations. Computation without addressable memory is hopelessly inefficient. For that reason, C.R. Gallistel and Adam King (2009) argue that addressable memory gives a better model of the mind than non-addressable memory.
• A Turing machine has a central processor that operates serially , executing one instruction at a time. Other computational formalisms relax this assumption, allowing multiple processing units that operate in parallel . Classical computationalists can allow parallel computations (Fodor and Pylyshyn 1988; Gallistel and King 2009: 174). See Gandy (1980) and Sieg (2009) for general mathematical treatments that encompass both serial and parallel computation.
• Turing computation is deterministic : total computational state determines subsequent computational state. One might instead allow stochastic computations. In a stochastic model, current state does not dictate a unique next state. Rather, there is a certain probability that the machine will transition from one state to another.

CCTM claims that mental activity is “Turing-style computation”, allowing these and other departures from Turing’s own formalism.

Hilary Putnam (1967) introduced CCTM into philosophy. He contrasted his position with logical behaviorism and type-identity theory . Each position purports to reveal the nature of mental states, including propositional attitudes (e.g., beliefs), sensations (e.g., pains), and emotions (e.g., fear). According to logical behaviorism, mental states are behavioral dispositions. According to type-identity theory, mental states are brain states. Putnam advances an opposing functionalist view, on which mental states are functional states. According to functionalism, a system has a mind when the system has a suitable functional organization . Mental states are states that play appropriate roles in the system’s functional organization. Each mental state is individuated by its interactions with sensory input, motor output, and other mental states.

Functionalism offers notable advantages over logical behaviorism and type-identity theory:

• Behaviorists want to associate each mental state with a characteristic pattern of behavior—a hopeless task, because individual mental states do not usually have characteristic behavioral effects. Behavior almost always results from distinct mental states operating together (e.g., a belief and a desire). Functionalism avoids this difficulty by individuating mental states through characteristic relations not only to sensory input and behavior but also to one another.
• Type-identity theorists want to associate each mental state with a characteristic physical or neurophysiological state. Putnam casts this project into doubt by arguing that mental states are multiply realizable : the same mental state can be realized by diverse physical systems, including not only terrestrial creatures but also hypothetical creatures (e.g., a silicon-based Martian). Functionalism is tailor-made to accommodate multiple realizability. According to functionalism, what matters for mentality is a pattern of organization, which could be physically realized in many different ways. See the entry multiple realizability for further discussion of this argument.

Putnam defends a brand of functionalism now called machine functionalism . He emphasizes probabilistic automata , which are similar to Turing machines except that transitions between computational states are stochastic. He proposes that mental activity implements a probabilistic automaton and that particular mental states are machine states of the automaton’s central processor. The machine table specifies an appropriate functional organization, and it also specifies the role that individual mental states play within that functional organization. In this way, Putnam combines functionalism with CCTM.

Machine functionalism faces several problems. One problem, highlighted by Ned Block and Jerry Fodor (1972), concerns the productivity of thought . A normal human can entertain a potential infinity of propositions. Machine functionalism identifies mental states with machine states of a probabilistic automaton. Since there are only finitely many machine states, there are not enough machine states to pair one-one with possible mental states of a normal human. Of course, an actual human will only ever entertain finitely many propositions. However, Block and Fodor contend that this limitation reflects limits on lifespan and memory, rather than (say) some psychological law that restricts the class of humanly entertainable propositions. A probabilistic automaton is endowed with unlimited time and memory capacity yet even still has only finitely many machine states. Apparently, then, machine functionalism mislocates the finitary limits upon human cognition.

Another problem for machine functionalism, also highlighted by Block and Fodor (1972), concerns the systematicity of thought. An ability to entertain one proposition is correlated with an ability to think other propositions. For example, someone who can entertain the thought that John loves Mary can also entertain the thought that Mary loves John . Thus, there seem to be systematic relations between mental states. A good theory should reflect those systematic relations. Yet machine functionalism identifies mental states with unstructured machines states, which lack the requisite systematic relations to another. For that reason, machine functionalism does not explain systematicity. In response to this objection, machine functionalists might deny that they are obligated to explain systematicity. Nevertheless, the objection suggests that machine functionalism neglects essential features of human mentality. A better theory would explain those features in a principled way.

While the productivity and systematicity objections to machine functionalism are perhaps not decisive, they provide strong impetus to pursue an improved version of CCTM. See Block (1978) for additional problems facing machine functionalism and functionalism more generally.

Fodor (1975, 1981, 1987, 1990, 1994, 2008) advocates a version of CCTM that accommodates systematicity and productivity much more satisfactorily. He shifts attention to the symbols manipulated during Turing-style computation.

An old view, stretching back at least to William of Ockham’s Summa Logicae , holds that thinking occurs in a language of thought (sometimes called Mentalese ). Fodor revives this view. He postulates a system of mental representations, including both primitive representations and complex representations formed from primitive representations. For example, the primitive Mentalese words JOHN, MARY, and LOVES can combine to form the Mentalese sentence JOHN LOVES MARY. Mentalese is compositional : the meaning of a complex Mentalese expression is a function of the meanings of its parts and the way those parts are combined. Propositional attitudes are relations to Mentalese symbols. Fodor calls this view the representational theory of mind ( RTM ). Combining RTM with CCTM, he argues that mental activity involves Turing-style computation over the language of thought. Mental computation stores Mentalese symbols in memory locations, manipulating those symbols in accord with mechanical rules.

A prime virtue of RTM is how readily it accommodates productivity and systematicity:

Productivity : RTM postulates a finite set of primitive Mentalese expressions, combinable into a potential infinity of complex Mentalese expressions. A thinker with access to primitive Mentalese vocabulary and Mentalese compounding devices has the potential to entertain an infinity of Mentalese expressions. She therefore has the potential to instantiate infinitely many propositional attitudes (neglecting limits on time and memory).

Systematicity : According to RTM, there are systematic relations between which propositional attitudes a thinker can entertain. For example, suppose I can think that John loves Mary. According to RTM, my doing so involves my standing in some relation R to a Mentalese sentence JOHN LOVES MARY, composed of Mentalese words JOHN, LOVES, and MARY combined in the right way. If I have this capacity, then I also have the capacity to stand in relation R to the distinct Mentalese sentence MARY LOVES JOHN, thereby thinking that Mary loves John. So the capacity to think that John loves Mary is systematically related to the capacity to think that Mary loves John.

By treating propositional attitudes as relations to complex mental symbols, RTM explains both productivity and systematicity.

CCTM+RTM differs from machine functionalism in several other respects. First, machine functionalism is a theory of mental states in general , while RTM is only a theory of propositional attitudes. Second, proponents of CCTM+RTM need not say that propositional attitudes are individuated functionally. As Fodor (2000: 105, fn. 4) notes, we must distinguish computationalism (mental processes are computational) from functionalism (mental states are functional states). Machine functionalism endorses both doctrines. CCTM+RTM endorses only the first. Unfortunately, many philosophers still mistakenly assume that computationalism entails a functionalist approach to propositional attitudes (see Piccinini 2004 for discussion).

Philosophical discussion of RTM tends to focus mainly on high-level human thought , especially belief and desire. However, CCTM+RTM is applicable to a much wider range of mental states and processes. Many cognitive scientists apply it to non-human animals. For example, Gallistel and King (2009) apply it to certain invertebrate phenomena (e.g., honeybee navigation). Even confining attention to humans, one can apply CCTM+RTM to subpersonal processing . Fodor (1983) argues that perception involves a subpersonal “module” that converts retinal input into Mentalese symbols and then performs computations over those symbols. Thus, talk about a language of thought is potentially misleading, since it suggests a non-existent restriction to higher-level mental activity.

Also potentially misleading is the description of Mentalese as a language , which suggests that all Mentalese symbols resemble expressions in a natural language. Many philosophers, including Fodor, sometimes seem to endorse that position. However, there are possible non-propositional formats for Mentalese symbols. Proponents of CCTM+RTM can adopt a pluralistic line, allowing mental computation to operate over items akin to images, maps, diagrams, or other non-propositional representations (Johnson-Laird 2004: 187; McDermott 2001: 69; Pinker 2005: 7; Sloman 1978: 144–176). The pluralistic line seems especially plausible as applied to subpersonal processes (such as perception) and non-human animals. Michael Rescorla (2009a,b) surveys research on cognitive maps (Tolman 1948; O’Keefe and Nadel 1978; Gallistel 1990), suggesting that some animals may navigate by computing over mental representations more similar to maps than sentences. Elisabeth Camp (2009), citing research on baboon social interaction (Cheney and Seyfarth 2007), argues that baboons may encode social dominance relations through non-sentential tree-structured representations.

CCTM+RTM is schematic. To fill in the schema, one must provide detailed computational models of specific mental processes. A complete model will:

• describe the Mentalese symbols manipulated by the process;
• isolate elementary operations that manipulate the symbols (e.g., inscribing a symbol in a memory location ); and
• delineate mechanical rules governing application of elementary operations.

By providing a detailed computational model, we decompose a complex mental process into a series of elementary operations governed by precise, routine instructions.

CCTM+RTM remains neutral in the traditional debate between physicalism and substance dualism. A Turing-style model proceeds at a very abstract level, not saying whether mental computations are implemented by physical stuff or Cartesian soul-stuff (Block 1983: 522). In practice, all proponents of CCTM+RTM embrace a broadly physicalist outlook. They hold that mental computations are implemented not by soul-stuff but rather by the brain. On this view, Mentalese symbols are realized by neural states, and computational operations over Mentalese symbols are realized by neural processes. Ultimately, physicalist proponents of CCTM+RTM must produce empirically well-confirmed theories that explain how exactly neural activity implements Turing-style computation. As Gallistel and King (2009) emphasize, we do not currently have such theories—though see Zylberberg, Dehaene, Roelfsema, and Sigman (2011) for some speculations.

Fodor (1975) advances CCTM+RTM as a foundation for cognitive science. He discusses mental phenomena such as decision-making, perception, and linguistic processing. In each case, he maintains, our best scientific theories postulate Turing-style computation over mental representations. In fact, he argues that our only viable theories have this form. He concludes that CCTM+RTM is “the only game in town”. Many cognitive scientists argue along similar lines. C.R. Gallistel and Adam King (2009), Philip Johnson-Laird (1988), Allen Newell and Herbert Simon (1976), and Zenon Pylyshyn (1984) all recommend Turing-style computation over mental symbols as the best foundation for scientific theorizing about the mind.

4. Neural networks

In the 1980s, connectionism emerged as a prominent rival to classical computationalism. Connectionists draw inspiration from neurophysiology rather than logic and computer science. They employ computational models, neural networks , that differ significantly from Turing-style models. A neural network is a collection of interconnected nodes. Nodes fall into three categories: input nodes, output nodes, and hidden nodes (which mediate between input and output nodes). Nodes have activation values, given by real numbers. One node can bear a weighted connection to another node, also given by a real number. Activations of input nodes are determined exogenously: these are the inputs to computation. Total input activation of a hidden or output node is a weighted sum of the activations of nodes feeding into it. Activation of a hidden or output node is a function of its total input activation; the particular function varies with the network. During neural network computation, waves of activation propagate from input nodes to output nodes, as determined by weighted connections between nodes.

In a feedforward network , weighted connections flow only in one direction. Recurrent networks have feedback loops, in which connections emanating from hidden units circle back to hidden units. Recurrent networks are less mathematically tractable than feedforward networks. However, they figure crucially in psychological modeling of various phenomena, such as phenomena that involve some kind of memory (Elman 1990).

Weights in a neural network are typically mutable, evolving in accord with a learning algorithm . The literature offers various learning algorithms, but the basic idea is usually to adjust weights so that actual outputs gradually move closer to the target outputs one would expect for the relevant inputs. The backpropagation algorithm is a widely used algorithm of this kind (Rumelhart, Hinton, and Williams 1986).

Connectionism traces back to McCulloch and Pitts (1943), who studied networks of interconnected logic gates (e.g., AND-gates and OR-gates). One can view a network of logic gates as a neural network, with activations confined to two values (0 and 1) and activation functions given by the usual truth-functions. McCulloch and Pitts advanced logic gates as idealized models of individual neurons. Their discussion exerted a profound influence on computer science (von Neumann 1945). Modern digital computers are simply networks of logic gates. Within cognitive science, however, researchers usually focus upon networks whose elements are more “neuron-like” than logic gates. In particular, modern-day connectionists typically emphasize analog neural networks whose nodes take continuous rather than discrete activation values. Some authors even use the phrase “neural network” so that it exclusively denotes such networks.

Neural networks received relatively scant attention from cognitive scientists during the 1960s and 1970s, when Turing-style models dominated. The 1980s witnessed a huge resurgence of interest in neural networks, especially analog neural networks, with the two-volume Parallel Distributed Processing (Rumelhart, McClelland, and the PDP research group, 1986; McClelland, Rumelhart, and the PDP research group, 1987) serving as a manifesto. Researchers constructed connectionist models of diverse phenomena: object recognition, speech perception, sentence comprehension, cognitive development, and so on. Impressed by connectionism, many researchers concluded that CCTM+RTM was no longer “the only game in town”.

In the 2010s, a class of computational models known as deep neural networks became quite popular (Krizhevsky, Sutskever, and Hinton 2012; LeCun, Bengio, and Hinton 2015). These models are neural networks with multiple layers of hidden nodes (sometimes hundreds of such layers). Deep neural networks—trained on large data sets through one or another learning algorithm (usually backpropagation)—have achieved great success in many areas of AI, including object recognition and strategic game-playing. Deep neural networks are now widely deployed in commercial applications, and they are the focus of extensive ongoing investigation within both academia and industry. Researchers have also begun using them to model the mind (e.g. Marblestone, Wayne, and Kording 2016; Kriegeskorte 2015).

For a detailed overview of neural networks, see Haykin (2008). For a user-friendly introduction, with an emphasis on psychological applications, see Marcus (2001). For a philosophically oriented introduction to deep neural networks, see Buckner (2019).

Neural networks have a very different “feel” than classical (i.e., Turing-style) models. Yet classical computation and neural network computation are not mutually exclusive:

• One can implement a neural network in a classical model . Indeed, every neural network ever physically constructed has been implemented on a digital computer.
• One can implement a classical model in a neural network . Modern digital computers implement Turing-style computation in networks of logic gates. Alternatively, one can implement Turing-style computation using an analog recurrent neural network whose nodes take continuous activation values (Graves, Wayne, and Danihelka 2014, Other Internet Resources; Siegelmann and Sontag 1991; Siegelmann and Sontag 1995).

Although some researchers suggest a fundamental opposition between classical computation and neural network computation, it seems more accurate to identify two modeling traditions that overlap in certain cases but not others (cf. Boden 1991; Piccinini 2008b). In this connection, it is also worth noting that classical computationalism and connectionist computationalism have their common origin in the work of McCulloch and Pitts.

Philosophers often say that classical computation involves “rule-governed symbol manipulation” while neural network computation is non-symbolic. The intuitive picture is that “information” in neural networks is globally distributed across the weights and activations, rather than concentrated in localized symbols. However, the notion of “symbol” itself requires explication, so it is often unclear what theorists mean by describing computation as symbolic versus non-symbolic. As mentioned in §1 , the Turing formalism places very few conditions on “symbols”. Regarding primitive symbols, Turing assumes just that there are finitely many of them and that they can be inscribed in read/write memory locations. Neural networks can also manipulate symbols satisfying these two conditions: as just noted, one can implement a Turing-style model in a neural network.

Many discussions of the symbolic/non-symbolic dichotomy employ a more robust notion of “symbol”. On the more robust approach, a symbol is the sort of thing that represents a subject matter. Thus, something is a symbol only if it has semantic or representational properties. If we employ this more robust notion of symbol, then the symbolic/non-symbolic distinction cross-cuts the distinction between Turing-style computation and neural network computation. A Turing machine need not employ symbols in the more robust sense. As far as the Turing formalism goes, symbols manipulated during Turing computation need not have representational properties (Chalmers 2011). Conversely, a neural network can manipulate symbols with representational properties. Indeed, an analog neural network can manipulate symbols that have a combinatorial syntax and semantics (Horgan and Tienson 1996; Marcus 2001).

Following Steven Pinker and Alan Prince (1988), we may distinguish between eliminative connectionism and implementationist connectionism .

Eliminative connectionists advance connectionism as a rival to classical computationalism. They argue that the Turing formalism is irrelevant to psychological explanation. Often, though not always, they seek to revive the associationist tradition in psychology, a tradition that CCTM had forcefully challenged. Often, though not always, they attack the mentalist, nativist linguistics pioneered by Noam Chomsky (1965). Often, though not always, they manifest overt hostility to the very notion of mental representation. But the defining feature of eliminative connectionism is that it uses neural networks as replacements for Turing-style models. Eliminative connectionists view the mind as a computing system of a radically different kind than the Turing machine. A few authors explicitly espouse eliminative connectionism (Churchland 1989; Rumelhart and McClelland 1986; Horgan and Tienson 1996), and many others incline towards it.

Implementationist connectionism is a more ecumenical position. It allows a potentially valuable role for both Turing-style models and neural networks, operating harmoniously at different levels of description (Marcus 2001; Smolensky 1988). A Turing-style model is higher-level, whereas a neural network model is lower-level. The neural network illuminates how the brain implements the Turing-style model, just as a description in terms of logic gates illuminates how a personal computer executes a program in a high-level programming language.

Connectionism excites many researchers because of the analogy between neural networks and the brain. Nodes resemble neurons, while connections between nodes resemble synapses. Connectionist modeling therefore seems more “biologically plausible” than classical modeling. A connectionist model of a psychological phenomenon apparently captures (in an idealized way) how interconnected neurons might generate the phenomenon.

When evaluating the argument from biological plausibility, one should recognize that neural networks vary widely in how closely they match actual brain activity. Many networks that figure prominently in connectionist writings are not so biologically plausible (Bechtel and Abrahamsen 2002: 341–343; Bermúdez 2010: 237–239; Clark 2014: 87–89; Harnish 2002: 359–362). A few examples:

• Real neurons are much more heterogeneous than the interchangeable nodes that figure in typical connectionist networks.
• Real neurons emit discrete spikes (action potentials) as outputs. But the nodes that figure in many prominent neural networks, including the best known deep neural networks, instead have continuous outputs.
• The backpropagation algorithm requires that weights between nodes can vary between excitatory and inhibitory, yet actual synapses cannot so vary (Crick and Asanuma 1986). Moreover, the algorithm assumes target outputs supplied exogenously by modelers who know the desired answer . In that sense, learning is supervised . Very little learning in actual biological systems involves anything resembling supervised training.

On the other hand, some neural networks are more biologically realistic (Buckner and Garson 2019; Illing, Gerstner, and Brea 2019). For instance, there are neural networks that replace backpropagation with more realistic learning algorithms, such as a reinforcement learning algorithm (Pozzi, Bohté, and Roelfsema 2019, Other Internet Resources) or an unsupervised learning algorithm (Krotov and Hopfield 2019). There are also neural networks whose nodes output discrete spikes roughly akin to those emitted by real neurons in the brain (Maass 1996; Buesing, Bill, Nessler, and Maass 2011).

Even when a neural network is not biologically plausible, it may still be more biologically plausible than classical models. Neural networks certainly seem closer than Turing-style models, in both details and spirit, to neurophysiological description. Many cognitive scientists worry that CCTM reflects a misguided attempt at imposing the architecture of digital computers onto the brain. Some doubt that the brain implements anything resembling digital computation, i.e., computation over discrete configurations of digits (Piccinini and Bahar 2013). Others doubt that brains display clean Turing-style separation between central processor and read/write memory (Dayan 2009). Neural networks fare better on both scores: they do not require computation over discrete configurations of digits, and they do not postulate a clean separation between central processor and read/write memory.

Classical computationalists typically reply that it is premature to draw firm conclusions based upon biological plausibility, given how little we understand about the relation between neural, computational, and cognitive levels of description (Gallistel and King 2009; Marcus 2001). Using measurement techniques such as cell recordings and functional magnetic resonance imaging (fMRI), and drawing upon disciplines as diverse as physics, biology, AI, information theory, statistics, graph theory, and dynamical systems theory, neuroscientists have accumulated substantial knowledge about the brain at varying levels of granularity (Zednik 2019). We now know quite a lot about individual neurons, about how neurons interact within neural populations, about the localization of mental activity in cortical regions (e.g. the visual cortex), and about interactions among cortical regions. Yet we still have a tremendous amount to learn about how neural tissue accomplishes the tasks that it surely accomplishes: perception, reasoning, decision-making, language acquisition, and so on. Given our present state of relative ignorance, it would be rash to insist that the brain does not implement anything resembling Turing computation.

Connectionists offer numerous further arguments that we should employ connectionist models instead of, or in addition to, classical models. See the entry connectionism for an overview. For purposes of this entry, we mention two additional arguments.

The first argument emphasizes learning (Bechtel and Abrahamsen 2002: 51). A vast range of cognitive phenomena involve learning from experience. Many connectionist models are explicitly designed to model learning, through backpropagation or some other algorithm that modifies the weights between nodes. By contrast, connectionists often complain that there are no good classical models of learning. Classical computationalists can respond by citing perceived defects of connectionist learning algorithms (e.g., the heavy reliance of backpropagation upon supervised training). Classical computationalists can also cite Bayesian decision theory, which models learning as probabilistic updating. More specifically, classical computationalists can cite the achievements of Bayesian cognitive science , which uses Bayesian decision theory to construct mathematical models of mental activity (Ma 2019). Over the past few decades, Bayesian cognitive science has accrued many explanatory successes. This impressive track record suggests that some mental processes are Bayesian or approximately Bayesian (Rescorla 2020). Moreover, the advances mentioned in §2 show how classical computing systems can execute or at least approximately execute Bayesian updating in various realistic scenarios. These developments provide hope that classical computation can model many important cases of learning.

The second argument emphasizes speed of computation . Neurons are much slower than silicon-based components of digital computers. For this reason, neurons could not execute serial computation quickly enough to match rapid human performance in perception, linguistic comprehension, decision-making, etc. Connectionists maintain that the only viable solution is to replace serial computation with a “massively parallel” computational architecture—precisely what neural networks provide (Feldman and Ballard 1982; Rumelhart 1989). However, this argument is only effective against classical computationalists who insist upon serial processing. As noted in §3 , some Turing-style models involve parallel processing. Many classical computationalists are happy to allow “massively parallel” mental computation, and the argument gains no traction against these researchers. That being said, the argument highlights an important question that any computationalist—whether classical, connectionist, or otherwise—must address: How does a brain built from relatively slow neurons execute sophisticated computations so quickly? Neither classical nor connectionist computationalists have answered this question satisfactorily (Gallistel and King 2009: 174 and 265).

Fodor and Pylyshyn (1988) offer a widely discussed critique of eliminativist connectionism. They argue that systematicity and productivity fail in connectionist models, except when the connectionist model implements a classical model. Hence, connectionism does not furnish a viable alternative to CCTM. At best, it supplies a low-level description that helps bridge the gap between Turing-style computation and neuroscientific description.

This argument has elicited numerous replies and counter-replies. Some argue that neural networks can exhibit systematicity without implementing anything like classical computational architecture (Horgan and Tienson 1996; Chalmers 1990; Smolensky 1991; van Gelder 1990). Some argue that Fodor and Pylyshyn vastly exaggerate systematicity (Johnson 2004) or productivity (Rumelhart and McClelland 1986), especially for non-human animals (Dennett 1991). These issues, and many others raised by Fodor and Pylyshyn’s argument, have been thoroughly investigated in the literature. For further discussion, see Bechtel and Abrahamsen (2002: 156–199), Bermúdez (2005: 244–278), Chalmers (1993), Clark (2014: 84–86), and the encyclopedia entries on the language of thought hypothesis and on connectionism .

Gallistel and King (2009) advance a related but distinct productivity argument. They emphasize productivity of mental computation , as opposed to productivity of mental states . Through detailed empirical case studies, they argue that many non-human animals can extract, store, and retrieve detailed records of the surrounding environment. For example, the Western scrub jay records where it cached food, what kind of food it cached in each location, when it cached the food, and whether it has depleted a given cache (Clayton, Emery, and Dickinson 2006). The jay can access these records and exploit them in diverse computations: computing whether a food item stored in some cache is likely to have decayed; computing a route from one location to another; and so on. The number of possible computations a jay can execute is, for all practical purposes, infinite.

CCTM explains the productivity of mental computation by positing a central processor that stores and retrieves symbols in addressable read/write memory. When needed, the central processor can retrieve arbitrary, unpredicted combinations of symbols from memory. In contrast, Gallistel and King argue, connectionism has difficulty accommodating the productivity of mental computation. Although Gallistel and King do not carefully distinguish between eliminativist and implementationist connectionism, we may summarize their argument as follows:

• Eliminativist connectionism cannot explain how organisms combine stored memories (e.g., cache locations) for computational purposes (e.g., computing a route from one cache to another). There are a virtual infinity of possible combinations that might be useful, with no predicting in advance which pieces of information must be combined in future computations. The only computationally tractable solution is symbol storage in readily accessible read/write memory locations—a solution that eliminativist connectionists reject.
• Implementationist connectionists can postulate symbol storage in read/write memory, as implemented by a neural network . However, the mechanisms that connectionists usually propose for implementing memory are not plausible. Existing proposals are mainly variants upon a single idea: a recurrent neural network that allows reverberating activity to travel around a loop (Elman 1990). There are many reasons why the reverberatory loop model is hopeless as a theory of long-term memory. For example, noise in the nervous system ensures that signals would rapidly degrade in a few minutes. Implementationist connectionists have thus far offered no plausible model of read/write memory. [ 2 ]

Gallistel and King conclude that CCTM is much better suited than either eliminativist or implementationist connectionism to explain a vast range of cognitive phenomena.

Critics attack this new productivity argument from various angles, focusing mainly on the empirical case studies adduced by Gallistel and King. Peter Dayan (2009), John Donahoe (2010), and Christopher Mole (2014) argue that biologically plausible neural network models can accommodate at least some of the case studies. Dayan and Donahoe argue that empirically adequate neural network models can dispense with anything resembling read/write memory. Mole argues that, in certain cases, empirically adequate neural network models can implement the read/write memory mechanisms posited by Gallistel and King. Debate on these fundamental issues seems poised to continue well into the future.

Computational neuroscience describes the nervous system through computational models. Although this research program is grounded in mathematical modeling of individual neurons, the distinctive focus of computational neuroscience is systems of interconnected neurons. Computational neuroscience usually models these systems as neural networks. In that sense, it is a variant, off-shoot, or descendant of connectionism. However, most computational neuroscientists do not self-identify as connectionists. There are several differences between connectionism and computational neuroscience:

• Neural networks employed by computational neuroscientists are much more biologically realistic than those employed by connectionists. The computational neuroscience literature is filled with talk about firing rates, action potentials, tuning curves, etc. These notions play at best a limited role in connectionist research, such as most of the research canvassed in (Rogers and McClelland 2014).
• Computational neuroscience is driven in large measure by knowledge about the brain, and it assigns huge importance to neurophysiological data (e.g., cell recordings). Connectionists place much less emphasis upon such data. Their research is primarily driven by behavioral data (although more recent connectionist writings cite neurophysiological data with somewhat greater frequency).
• Computational neuroscientists usually regard individual nodes in neural networks as idealized descriptions of actual neurons. Connectionists usually instead regard nodes as neuron-like processing units (Rogers and McClelland 2014) while remaining neutral about how exactly these units map onto actual neurophysiological entities.

One might say that computational neuroscience is concerned mainly with neural computation (computation by systems of neurons), whereas connectionism is concerned mainly with abstract computational models inspired by neural computation. But the boundaries between connectionism and computational neuroscience are admittedly somewhat porous. For an overview of computational neuroscience, see Trappenberg (2010) or Miller (2018).

Serious philosophical engagement with neuroscience dates back at least to Patricia Churchland’s Neurophilosophy (1986). As computational neuroscience matured, Churchland became one of its main philosophical champions (Churchland, Koch, and Sejnowski 1990; Churchland and Sejnowski 1992). She was joined by Paul Churchland (1995, 2007) and others (Eliasmith 2013; Eliasmith and Anderson 2003; Piccinini and Bahar 2013; Piccinini and Shagrir 2014). All these authors hold that theorizing about mental computation should begin with the brain, not with Turing machines or other inappropriate tools drawn from logic and computer science. They also hold that neural network modeling should strive for greater biological realism than connectionist models typically attain. Chris Eliasmith (2013) develops this neurocomputational viewpoint through the Neural Engineering Framework , which supplements computational neuroscience with tools drawn from control theory (Brogan 1990). He aims to “reverse engineer” the brain, building large-scale, biologically plausible neural network models of cognitive phenomena.

Computational neuroscience differs in a crucial respect from CCTM and connectionism: it abandons multiply realizability. Computational neuroscientists cite specific neurophysiological properties and processes, so their models do not apply equally well to (say) a sufficiently different silicon-based creature. Thus, computational neuroscience sacrifices a key feature that originally attracted philosophers to CTM. Computational neuroscientists will respond that this sacrifice is worth the resultant insight into neurophysiological underpinnings. But many computationalists worry that, by focusing too much on neural underpinnings, we risk losing sight of the cognitive forest for the neuronal trees. Neurophysiological details are important, but don’t we also need an additional abstract level of computational description that prescinds from such details? Gallistel and King (2009) argue that a myopic fixation upon what we currently know about the brain has led computational neuroscience to shortchange core cognitive phenomena such as navigation, spatial and temporal learning, and so on. Similarly, Edelman (2014) complains that the Neural Engineering Framework substitutes a blizzard of neurophysiological details for satisfying psychological explanations.

Partly in response to such worries, some researchers propose an integrated cognitive computational neuroscience that connects psychological theories with neural implementation mechanisms (Naselaris et al. 2018; Kriegeskorte and Douglas 2018). The basic idea is to use neural network models to illuminate how mental processes are instantiated in the brain, thereby grounding multiply realizable cognitive description in the neurophysiological. A good example is recent work on neural implementation of Bayesian inference (e.g., Pouget et al. 2013; Orhan and Ma 2017; Aitchison and Lengyel 2016). Researchers articulate (multiply realizable) Bayesian models of various mental processes; they construct biologically plausible neural networks that execute or approximately execute the posited Bayesian computations; and they evaluate how well these neural network models fit with neurophysiological data.

Despite the differences between connectionism and computational neuroscience, these two movements raise many similar issues. In particular, the dialectic from §4.4 regarding systematicity and productivity arises in similar form.

5. Computation and representation

Philosophers and cognitive scientists use the term “representation” in diverse ways. Within philosophy, the most dominant usage ties representation to intentionality, i.e., the “aboutness” of mental states. Contemporary philosophers usually elucidate intentionality by invoking representational content . A representational mental state has a content that represents the world as being a certain way, so we can ask whether the world is indeed that way. Thus, representationally contentful mental states are semantically evaluable with respect to properties such as truth, accuracy, fulfillment, and so on. To illustrate:

• Beliefs are the sorts of things that can be true or false. My belief that Emmanuel Macron is French is true if Emmanuel Macron is French, false if he is not.
• Perceptual states are the sorts of things that can be accurate or inaccurate. My perceptual experience as of a red sphere is accurate only if a red sphere is before me.
• Desires are the sorts of things that can fulfilled or thwarted. My desire to eat chocolate is fulfilled if I eat chocolate, thwarted if I do not eat chocolate.

Beliefs have truth-conditions (conditions under which they are true), perceptual states have accuracy-conditions (conditions under which they are accurate), and desires have fulfillment-conditions (conditions under which they are fulfilled).

In ordinary life, we frequently predict and explain behavior by invoking beliefs, desires, and other representationally contentful mental states. We identify these states through their representational properties. When we say “Frank believes that Emmanuel Macron is French”, we specify the condition under which Frank’s belief is true (namely, that Emmanuel Macron is French). When we say “Frank wants to eat chocolate”, we specify the condition under which Frank’s desire is fulfilled (namely, that Frank eats chocolate). So folk psychology assigns a central role to intentional descriptions , i.e., descriptions that identify mental states through their representational properties. Whether scientific psychology should likewise employ intentional descriptions is a contested issue within contemporary philosophy of mind.

Intentional realism is realism regarding representation. At a minimum, this position holds that representational properties are genuine aspects of mentality. Usually, it is also taken to hold that scientific psychology should freely employ intentional descriptions when appropriate. Intentional realism is a popular position, advocated by Tyler Burge (2010a), Jerry Fodor (1987), Christopher Peacocke (1992, 1994), and many others. One prominent argument for intentional realism cites cognitive science practice . The argument maintains that intentional description figures centrally in many core areas of cognitive science, such as perceptual psychology and linguistics. For example, perceptual psychology describes how perceptual activity transforms sensory inputs (e.g., retinal stimulations) into representations of the distal environment (e.g., perceptual representations of distal shapes, sizes, and colors). The science identifies perceptual states by citing representational properties (e.g., representational relations to specific distal shapes, sizes, colors). Assuming a broadly scientific realist perspective, the explanatory achievements of perceptual psychology support a realist posture towards intentionality.

Eliminativism is a strong form of anti-realism about intentionality. Eliminativists dismiss intentional description as vague, context-sensitive, interest-relative, explanatorily superficial, or otherwise problematic. They recommend that scientific psychology jettison representational content. An early example is W.V. Quine’s Word and Object (1960), which seeks to replace intentional psychology with behaviorist stimulus-response psychology. Paul Churchland (1981), another prominent eliminativist, wants to replace intentional psychology with neuroscience.

Between intentional realism and eliminativism lie various intermediate positions. Daniel Dennett (1971, 1987) acknowledges that intentional discourse is predictively useful, but he questions whether mental states really have representational properties. According to Dennett, theorists who employ intentional descriptions are not literally asserting that mental states have representational properties. They are merely adopting the “intentional stance”. Donald Davidson (1980) espouses a neighboring interpretivist position. He emphasizes the central role that intentional ascription plays within ordinary interpretive practice, i.e., our practice of interpreting one another’s mental states and speech acts. At the same time, he questions whether intentional psychology will find a place within mature scientific theorizing. Davidson and Dennett both profess realism about intentional mental states. Nevertheless, both philosophers are customarily read as intentional anti-realists. (In particular, Dennett is frequently read as a kind of instrumentalist about intentionality.) One source of this customary reading involves indeterminacy of interpretation . Suppose that behavioral evidence allows two conflicting interpretations of a thinker’s mental states. Following Quine, Davidson and Dennett both say there is then “no fact of the matter” regarding which interpretation is correct. This diagnosis indicates a less than fully realist attitude towards intentionality.

Debates over intentionality figure prominently in philosophical discussion of CTM. Let us survey some highlights.

Classical computationalists typically assume what one might call the formal-syntactic conception of computation (FSC). The intuitive idea is that computation manipulates symbols in virtue of their formal syntactic properties rather than their semantic properties.

FSC stems from innovations in mathematical logic during the late 19 th and early 20 th centuries, especially seminal contributions by George Boole and Gottlob Frege. In his Begriffsschrift (1879/1967), Frege effected a thoroughgoing formalization of deductive reasoning. To formalize, we specify a formal language whose component linguistic expressions are individuated non-semantically (e.g., by their geometric shapes). We may have some intended interpretation in mind, but elements of the formal language are purely syntactic entities that we can discuss without invoking semantic properties such as reference or truth-conditions. In particular, we can specify inference rules in formal syntactic terms. If we choose our inference rules wisely, then they will cohere with our intended interpretation: they will carry true premises to true conclusions. Through formalization, Frege invested logic with unprecedented rigor. He thereby laid the groundwork for numerous subsequent mathematical and philosophical developments.

Formalization plays a significant foundational role within computer science. We can program a Turing-style computer that manipulates linguistic expressions drawn from a formal language. If we program the computer wisely, then its syntactic machinations will cohere with our intended semantic interpretation. For example, we can program the computer so that it carries true premises only to true conclusions, or so that it updates probabilities as dictated by Bayesian decision theory.

FSC holds that all computation manipulates formal syntactic items, without regard to any semantic properties those items may have. Precise formulations of FSC vary. Computation is said to be “sensitive” to syntax but not semantics, or to have “access” only to syntactic properties, or to operate “in virtue” of syntactic rather than semantic properties, or to be impacted by semantic properties only as “mediated” by syntactic properties. It is not always so clear what these formulations mean or whether they are equivalent to one another. But the intuitive picture is that syntactic properties have causal/explanatory primacy over semantic properties in driving computation forward.

Fodor’s article “Methodological Solipsism Considered as a Research Strategy in Cognitive Psychology” (1980) offers an early statement. Fodor combines FSC with CCTM+RTM. He analogizes Mentalese to formal languages studied by logicians: it contains simple and complex items individuated non-semantically, just as typical formal languages contain simple and complex expressions individuated by their shapes. Mentalese symbols have a semantic interpretation, but this interpretation does not (directly) impact mental computation. A symbol’s formal properties, rather than its semantic properties, determine how computation manipulates the symbol. In that sense, the mind is a “syntactic engine”. Virtually all classical computationalists follow Fodor in endorsing FSC.

Connectionists often deny that neural networks manipulate syntactically structured items. For that reason, many connectionists would hesitate to accept FSC. Nevertheless, most connectionists endorse a generalized formality thesis : computation is insensitive to semantic properties. The generalized formality thesis raises many of the same philosophical issues raised by FSC. We focus here on FSC, which has received the most philosophical discussion.

Fodor combines CCTM+RTM+FSC with intentional realism. He holds that CCTM+RTM+FSC vindicates folk psychology by helping us convert common sense intentional discourse into rigorous science. He motivates his position with a famous abductive argument for CCTM+RTM+FSC (1987: 18–20). Strikingly, mental activity tracks semantic properties in a coherent way. For example, deductive inference carries premises to conclusions that are true if the premises are true. How can we explain this crucial aspect of mental activity? Formalization shows that syntactic manipulations can track semantic properties, and computer science shows how to build physical machines that execute desired syntactic manipulations. If we treat the mind as a syntax-driven machine, then we can explain why mental activity tracks semantic properties in a coherent way. Moreover, our explanation does not posit causal mechanisms radically different from those posited within the physical sciences. We thereby answer the pivotal question: How is rationality mechanically possible ?

Stephen Stich (1983) and Hartry Field (2001) combine CCTM+FSC with eliminativism. They recommend that cognitive science model the mind in formal syntactic terms, eschewing intentionality altogether. They grant that mental states have representational properties, but they ask what explanatory value scientific psychology gains by invoking those properties. Why supplement formal syntactic description with intentional description? If the mind is a syntax-driven machine, then doesn’t representational content drop out as explanatorily irrelevant?

At one point in his career, Putnam (1983: 139–154) combined CCTM+FSC with a Davidson-tinged interpretivism . Cognitive science should proceed along the lines suggested by Stich and Field, delineating purely formal syntactic computational models. Formal syntactic modeling co-exists with ordinary interpretive practice, in which we ascribe intentional contents to one another’s mental states and speech acts. Interpretive practice is governed by holistic and heuristic constraints, which stymie attempts at converting intentional discourse into rigorous science. For Putnam, as for Field and Stich, the scientific action occurs at the formal syntactic level rather than the intentional level.

CTM+FSC comes under attack from various directions. One criticism targets the causal relevance of representational content (Block 1990; Figdor 2009; Kazez 1995). Intuitively speaking, the contents of mental states are causally relevant to mental activity and behavior. For example, my desire to drink water rather than orange juice causes me to walk to the sink rather than the refrigerator. The content of my desire ( that I drink water ) seems to play an important causal role in shaping my behavior. According to Fodor (1990: 137–159), CCTM+RTM+FSC accommodates such intuitions. Formal syntactic activity implements intentional mental activity, thereby ensuring that intentional mental states causally interact in accord with their contents. However, it is not so clear that this analysis secures the causal relevance of content. FSC says that computation is “sensitive” to syntax but not semantics. Depending on how one glosses the key term “sensitive”, it can look like representational content is causally irrelevant, with formal syntax doing all the causal work. Here is an analogy to illustrate the worry. When a car drives along a road, there are stable patterns involving the car’s shadow. Nevertheless, shadow position at one time does not influence shadow position at a later time. Similarly, CCTM+RTM+FSC may explain how mental activity instantiates stable patterns described in intentional terms, but this is not enough to ensure the causal relevance of content. If the mind is a syntax-driven machine, then causal efficacy seems to reside at the syntactic rather the semantic level. Semantics is just “along for the ride”. Apparently, then, CTM+FSC encourages the conclusion that representational properties are causally inert. The conclusion may not trouble eliminativists, but intentional realists usually want to avoid it.

A second criticism dismisses the formal-syntactic picture as speculation ungrounded in scientific practice. Tyler Burge (2010a,b, 2013: 479–480) contends that formal syntactic description of mental activity plays no significant role within large areas of cognitive science, including the study of theoretical reasoning, practical reasoning, and perception. In each case, Burge argues, the science employs intentional description rather than formal syntactic description. For example, perceptual psychology individuates perceptual states not through formal syntactic properties but through representational relations to distal shapes, sizes, colors, and so on. To understand this criticism, we must distinguish formal syntactic description and neurophysiological description . Everyone agrees that a complete scientific psychology will assign prime importance to neurophysiological description. However, neurophysiological description is distinct from formal syntactic description, because formal syntactic description is supposed to be multiply realizable in the neurophysiological. The issue here is whether scientific psychology should supplement intentional descriptions and neurophysiological descriptions with multiply realizable, non-intentional formal syntactic descriptions.

Putnam’s landmark article “The Meaning of ‘Meaning’” (1975: 215–271) introduced the Twin Earth thought experiment , which postulates a world just like our own except that H 2 O is replaced by a qualitatively similar substance XYZ with different chemical composition. Putnam argues that XYZ is not water and that speakers on Twin Earth use the word “water” to refer to XYZ rather than to water. Burge (1982) extends this conclusion from linguistic reference to mental content . He argues that Twin Earthlings instantiate mental states with different contents. For example, if Oscar on Earth thinks that water is thirst-quenching , then his duplicate on Twin Earth thinks a thought with a different content, which we might gloss as that twater is thirst-quenching . Burge concludes that mental content does not supervene upon internal neurophysiology. Mental content is individuated partly by factors outside the thinker’s skin, including causal relations to the environment. This position is externalism about mental content .

Formal syntactic properties of mental states are widely taken to supervene upon internal neurophysiology. For example, Oscar and Twin Oscar instantiate the same formal syntactic manipulations. Assuming content externalism, it follows that there is a huge gulf between ordinary intentional description and formal syntactic description.

Content externalism raises serious questions about the explanatory utility of representational content for scientific psychology:

Argument from Causation (Fodor 1987, 1991): How can mental content exert any causal influence except as manifested within internal neurophysiology? There is no “psychological action at a distance”. Differences in the physical environment impact behavior only by inducing differences in local brain states. So the only causally relevant factors are those that supervene upon internal neurophysiology. Externally individuated content is causally irrelevant .

Argument from Explanation (Stich 1983): Rigorous scientific explanation should not take into account factors outside the subject’s skin. Folk psychology may taxonomize mental states through relations to the external environment, but scientific psychology should taxonomize mental states entirely through factors that supervene upon internal neurophysiology. It should treat Oscar and Twin Oscar as psychological duplicates. [ 3 ]

Some authors pursue the two arguments in conjunction with one another. Both arguments reach the same conclusion: externally individuated mental content finds no legitimate place within causal explanations provided by scientific psychology. Stich (1983) argues along these lines to motivate his formal-syntactic eliminativism.

Many philosophers respond to such worries by promoting content internalism . Whereas content externalists favor wide content (content that does not supervene upon internal neurophysiology), content internalists favor narrow content (content that does so supervene). Narrow content is what remains of mental content when one factors out all external elements. At one point in his career, Fodor (1981, 1987) pursued internalism as a strategy for integrating intentional psychology with CCTM+RTM+FSC. While conceding that wide content should not figure in scientific psychology, he maintained that narrow content should play a central explanatory role.

Radical internalists insist that all content is narrow. A typical analysis holds that Oscar is thinking not about water but about some more general category of substance that subsumes XYZ, so that Oscar and Twin Oscar entertain mental states with the same contents. Tim Crane (1991) and Gabriel Segal (2000) endorse such an analysis. They hold that folk psychology always individuates propositional attitudes narrowly. A less radical internalism recommends that we recognize narrow content in addition to wide content. Folk psychology may sometimes individuate propositional attitudes widely, but we can also delineate a viable notion of narrow content that advances important philosophical or scientific goals. Internalists have proposed various candidate notions of narrow content (Block 1986; Chalmers 2002; Cummins 1989; Fodor 1987; Lewis 1994; Loar 1988; Mendola 2008). See the entry narrow mental content for an overview of prominent candidates.

Externalists complain that existing theories of narrow content are sketchy, implausible, useless for psychological explanation, or otherwise objectionable (Burge 2007; Sawyer 2000; Stalnaker 1999). Externalists also question internalist arguments that scientific psychology requires narrow content:

Argument from Causation : Externalists insist that wide content can be causally relevant. The details vary among externalists, and discussion often becomes intertwined with complex issues surrounding causation, counterfactuals, and the metaphysics of mind. See the entry mental causation for an introductory overview, and see Burge (2007), Rescorla (2014a), and Yablo (1997, 2003) for representative externalist discussion.

Argument from Explanation : Externalists claim that psychological explanation can legitimately taxonomize mental states through factors that outstrip internal neurophysiology (Peacocke 1993; Shea, 2018). Burge observes that non-psychological sciences often individuate explanatory kinds relationally , i.e., through relations to external factors. For example, whether an entity counts as a heart depends (roughly) upon whether its biological function in its normal environment is to pump blood. So physiology individuates organ kinds relationally. Why can’t psychology likewise individuate mental states relationally? For a notable exchange on these issues, see Burge (1986, 1989, 1995) and Fodor (1987, 1991).

Externalists doubt that we have any good reason to replace or supplement wide content with narrow content. They dismiss the search for narrow content as a wild goose chase.

Burge (2007, 2010a) defends externalism by analyzing current cognitive science. He argues that many branches of scientific psychology (especially perceptual psychology) individuate mental content through causal relations to the external environment. He concludes that scientific practice embodies an externalist perspective. By contrast, he maintains, narrow content is a philosophical fantasy ungrounded in current science.

Suppose we abandon the search for narrow content. What are the prospects for combining CTM+FSC with externalist intentional psychology? The most promising option emphasizes levels of explanation . We can say that intentional psychology occupies one level of explanation, while formal-syntactic computational psychology occupies a different level. Fodor advocates this approach in his later work (1994, 2008). He comes to reject narrow content as otiose. He suggests that formal syntactic mechanisms implement externalist psychological laws. Mental computation manipulates Mentalese expressions in accord with their formal syntactic properties, and these formal syntactic manipulations ensure that mental activity instantiates appropriate law-like patterns defined over wide contents.

In light of the internalism/externalism distinction, let us revisit the eliminativist challenge raised in §5.1 : what explanatory value does intentional description add to formal-syntactic description? Internalists can respond that suitable formal syntactic manipulations determine and maybe even constitute narrow contents, so that internalist intentional description is already implicit in suitable formal syntactic description (cf. Field 2001: 75). Perhaps this response vindicates intentional realism, perhaps not. Crucially, though, no such response is available to content externalists. Externalist intentional description is not implicit in formal syntactic description, because one can hold formal syntax fixed while varying wide content. Thus, content externalists who espouse CTM+FSC must say what we gain by supplementing formal-syntactic explanations with intentional explanations. Once we accept that mental computation is sensitive to syntax but not semantics, it is far from clear that any useful explanatory work remains for wide content. Fodor addresses this challenge at various points, offering his most systematic treatment in The Elm and the Expert (1994). See Arjo (1996), Aydede (1998), Aydede and Robbins (2001), Wakefield (2002); Perry (1998), and Wakefield (2002) for criticism. See Rupert (2008) and Schneider (2005) for positions close to Fodor’s. Dretske (1993) and Shea (2018, pp. 197–226) pursue alternative strategies for vindicating the explanatory relevance of wide content.

The perceived gulf between computational description and intentional description animates many writings on CTM. A few philosophers try to bridge the gulf using computational descriptions that individuate computational states in representational terms. These descriptions are content-involving , to use Christopher Peacocke’s (1994) terminology. On the content-involving approach, there is no rigid demarcation between computational and intentional description. In particular, certain scientifically valuable descriptions of mental activity are both computational and intentional. Call this position content-involving computationalism .

Content-involving computationalists need not say that all computational description is intentional. To illustrate, suppose we describe a simple Turing machine that manipulates symbols individuated by their geometric shapes. Then the resulting computational description is not plausibly content-involving. Accordingly, content-involving computationalists do not usually advance content-involving computation as a general theory of computation. They claim only that some important computational descriptions are content-involving.

One can develop content-involving computationalism in an internalist or externalist direction. Internalist content-involving computationalists hold that some computational descriptions identify mental states partly through their narrow contents. Murat Aydede (2005) recommends a position along these lines. Externalist content-involving computationalism holds that certain computational descriptions identify mental states partly through their wide contents. Tyler Burge (2010a: 95–101), Christopher Peacocke (1994, 1999), Michael Rescorla (2012), and Mark Sprevak (2010) espouse this position. Oron Shagrir (2001, forthcoming) advocates a content-involving computationalism that is neutral between internalism and externalism.

Externalist content-involving computationalists typically cite cognitive science practice as a motivating factor. For example, perceptual psychology describes the perceptual system as computing an estimate of some object’s size from retinal stimulations and from an estimate of the object’s depth. Perceptual “estimates” are identified representationally, as representations of specific distal sizes and depths. Quite plausibly, representational relations to specific distal sizes and depths do not supervene on internal neurophysiology. Quite plausibly, then, perceptual psychology type-identifies perceptual computations through wide contents. So externalist content-involving computationalism seems to harmonize well with current cognitive science.

A major challenge facing content-involving computationalism concerns the interface with standard computationalism formalisms, such as the Turing machine. How exactly do content-involving descriptions relate to the computational models found in logic and computer science? Philosophers usually assume that these models offer non-intentional descriptions. If so, that would be a major and perhaps decisive blow to content-involving computationalism.

Arguably, though, many familiar computational formalisms allow a content-involving rather than formal syntactic construal. To illustrate, consider the Turing machine. One can individuate the “symbols” comprising the Turing machine alphabet non-semantically, through factors akin to geometric shape. But does Turing’s formalism require a non-semantic individuative scheme? Arguably, the formalism allows us to individuate symbols partly through their contents. Of course, the machine table for a Turing machine does not explicitly cite semantic properties of symbols (e.g., denotations or truth-conditions). Nevertheless, the machine table can encode mechanical rules that describe how to manipulate symbols, where those symbols are type-identified in content-involving terms. In this way, the machine table dictates transitions among content-involving states without explicitly mentioning semantic properties. Aydede (2005) suggests an internalist version of this view, with symbols type-identified through their narrow contents. [ 4 ] Rescorla (2017a) develops the view in an externalist direction, with symbols type-identified through their wide contents. He argues that some Turing-style models describe computational operations over externalistically individuated Mentalese symbols. [ 5 ]

In principle, one might embrace both externalist content-involving computational description and formal syntactic description. One might say that these two kinds of description occupy distinct levels of explanation. Peacocke suggests such a view. Other content-involving computationalists regard formal syntactic descriptions of the mind more skeptically. For example, Burge questions what explanatory value formal syntactic description contributes to certain areas of scientific psychology (such as perceptual psychology). From this viewpoint, the eliminativist challenge posed in §5.1 has matters backwards. We should not assume that formal syntactic descriptions are explanatorily valuable and then ask what value intentional descriptions contribute. We should instead embrace the externalist intentional descriptions offered by current cognitive science and then ask what value formal syntactic description contributes.

Proponents of formal syntactic description respond by citing implementation mechanisms . Externalist description of mental activity presupposes that suitable causal-historical relations between the mind and the external physical environment are in place. But surely we want a “local” description that ignores external causal-historical relations, a description that reveals underlying causal mechanisms. Fodor (1987, 1994) argues in this way to motivate the formal syntactic picture. For possible externalist responses to the argument from implementation mechanisms, see Burge (2010b), Rescorla (2017b), Shea (2013), and Sprevak (2010). Debate over this argument, and more generally over the relation between computation and representation, seems likely to continue into the indefinite future.

6. Alternative conceptions of computation

The literature offers several alternative conceptions, usually advanced as foundations for CTM. In many cases, these conceptions overlap with one another or with the conceptions considered above.

It is common for cognitive scientists to describe computation as “information-processing”. It is less common for proponents to clarify what they mean by “information” or “processing”. Lacking clarification, the description is little more than an empty slogan.

Claude Shannon introduced a scientifically important notion of “information” in his 1948 article “A Mathematical Theory of Communication”. The intuitive idea is that information measures reduction in uncertainty , where reduced uncertainty manifests as an altered probability distribution over possible states. Shannon codified this idea within a rigorous mathematical framework, laying the foundation for information theory (Cover and Thomas 2006). Shannon information is fundamental to modern engineering. It finds fruitful application within cognitive science, especially cognitive neuroscience. Does it support a convincing analysis of computation as “information-processing”? Consider an old-fashioned tape machine that records messages received over a wireless radio. Using Shannon’s framework, one can measure how much information is carried by some recorded message. There is a sense in which the tape machine “processes” Shannon information whenever we replay a recorded message. Still, the machine does not seem to implement a non-trivial computational model. [ 6 ] Certainly, neither the Turing machine formalism nor the neural network formalism offers much insight into the machine’s operations. Arguably, then, a system can process Shannon information without executing computations in any interesting sense.

Confronted with such examples, one might try to isolate a more demanding notion of “processing”, so that the tape machine does not “process” Shannon information. Alternatively, one might insist that the tape machine executes non-trivial computations. Piccinini and Scarantino (2010) advance a highly general notion of computation—which they dub generic computation —with that consequence.

A second prominent notion of information derives from Paul Grice’s (1989) influential discussion of natural meaning . Natural meaning involves reliable, counterfactual-supporting correlations. For example, tree rings correlate with the age of the tree, and pox correlate with chickenpox. We colloquially describe tree rings as carrying information about tree age, pox as carrying information about chickenpox, and so on. Such descriptions suggest a conception that ties information to reliable, counterfactual-supporting correlations. Fred Dretske (1981) develops this conception into a systematic theory, as do various subsequent philosophers. Does Dretske-style information subserve a plausible analysis of computation as “information-processing”? Consider an old-fashioned bimetallic strip thermostat . Two metals are joined together into a strip. Differential expansion of the metals causes the strip to bend, thereby activating or deactivating a heating unit. Strip state reliably correlates with current ambient temperature, and the thermostat “processes” this information-bearing state when activating or deactivating the heater. Yet the thermostat does not seem to implement any non-trivial computational model. One would not ordinarily regard the thermostat as computing. Arguably, then, a system can process Dretske-style information without executing computations in any interesting sense. Of course, one might try to handle such examples through maneuvers parallel to those from the previous paragraph.

A third prominent notion of information is semantic information , i.e., representational content. [ 7 ] Some philosophers hold that a physical system computes only if the system’s states have representational properties (Dietrich 1989; Fodor 1998: 10; Ladyman 2009; Shagrir 2006; Sprevak 2010). In that sense, information-processing is necessary for computation. As Fodor memorably puts it, “no computation without representation” (1975: 34). However, this position is debatable. Chalmers (2011) and Piccinini (2008a) contend that a Turing machine might execute computations even though symbols manipulated by the machine have no semantic interpretation. The machine’s computations are purely syntactic in nature, lacking anything like semantic properties. On this view, representational content is not necessary for a physical system to count as computational.

It remains unclear whether the slogan “computation is information-processing” provides much insight. Nevertheless, the slogan seems unlikely to disappear from the literature anytime soon. For further discussion of possible connections between computation and information, see Gallistel and King (2009: 1–26), Lizier, Flecker, and Williams (2013), Milkowski (2013), Piccinini and Scarantino (2010), and Sprevak (forthcoming).

In a widely cited passage, the perceptual psychologist David Marr (1982) distinguishes three levels at which one can describe an “information-processing device”:

Computational theory : “[t]he device is characterized as a mapping from one kind of information to another, the abstract properties of this mapping are defined precisely, and its appropriateness and adequacy for the task as hand are demonstrated” (p. 24). Representation and algorithm : “the choice of representation for the input and output and the algorithm to be used to transform one into the other” (pp. 24–25). Hardware implementation : “the details of how the algorithm and representation are realized physically” (p. 25).

Marr’s three levels have attracted intense philosophical scrutiny. For our purposes, the key point is that Marr’s “computational level” describes a mapping from inputs to outputs, without describing intermediate steps. Marr illustrates his approach by providing “computational level” theories of various perceptual processes, such as edge detection.

Marr’s discussion suggests a functional conception of computation , on which computation is a matter of transforming inputs into appropriate outputs. Frances Egan elaborates the functional conception over a series of articles (1991, 1992, 1999, 2003, 2010, 2014, 2019). Like Marr, she treats computational description as description of input-output relations. She also claims that computational models characterize a purely mathematical function: that is, a mapping from mathematical inputs to mathematical outputs. She illustrates by considering a visual mechanism (called “Visua”) that computes an object’s depth from retinal disparity. She imagines a neurophysiological duplicate (“Twin Visua”) embedded so differently in the physical environment that it does not represent depth. Visua and Twin Visua instantiate perceptual states with different representational properties. Nevertheless, Egan says, vision science treats Visua and Twin Visua as computational duplicates . Visua and Twin Visua compute the same mathematical function, even though the computations have different representational import in the two cases. Egan concludes that computational modeling of the mind yields an “abstract mathematical description” consistent with many alternative possible representational descriptions. Intentional attribution is just a heuristic gloss upon underlying computational description.

Chalmers (2012) argues that the functional conception neglects important features of computation. As he notes, computational models usually describe more than just input-output relations. They describe intermediate steps through which inputs are transformed into outputs. These intermediate steps, which Marr consigns to the “algorithmic” level, figure prominently in computational models offered by logicians and computer scientists. Restricting the term “computation” to input-output description does not capture standard computational practice.

An additional worry faces functional theories, such as Egan’s, that exclusively emphasize mathematical inputs and outputs. Critics complain that Egan mistakenly elevates mathematical functions, at the expense of intentional explanations routinely offered by cognitive science (Burge 2005; Rescorla 2015; Silverberg 2006; Sprevak 2010). To illustrate, suppose perceptual psychology describes the perceptual system as estimating that some object’s depth is 5 meters. The perceptual depth-estimate has a representational content: it is accurate only if the object’s depth is 5 meters. We cite the number 5 to identify the depth-estimate. But our choice of this number depends upon our arbitrary choice of measurement units. Critics contend that the content of the depth-estimate, not the arbitrarily chosen number through which we theorists specify that content, is what matters for psychological explanation. Egan’s theory places the number rather than the content at explanatory center stage. According to Egan, computational explanation should describe the visual system as computing a particular mathematical function that carries particular mathematical inputs into particular mathematical outputs . Those particular mathematical inputs and outputs depend upon our arbitrary choice of measurement units, so they arguably lack the explanatory significance that Egan assigns to them.

We should distinguish the functional approach, as pursued by Marr and Egan, from the functional programming paradigm in computer science. The functional programming paradigm models evaluation of a complex function as successive evaluation of simpler functions. To take a simple example, one might evaluate \(f(x,y) = (x^{2}+y)\) by first evaluating the squaring function and then evaluating the addition function. Functional programming differs from the “computational level” descriptions emphasized by Marr, because it specifies intermediate computational stages. The functional programming paradigm stretches back to Alonzo Church’s (1936) lambda calculus , continuing with programming languages such as PCF and LISP. It plays an important role in AI and theoretical computer science. Some authors suggest that it offers special insight into mental computation (Klein 2012; Piantadosi, Tenenbaum, and Goodman 2012). However, many computational formalisms do not conform to the functional paradigm: Turing machines; imperative programming languages, such as C; logic programming languages, such as Prolog; and so on. Even though the functional paradigm describes numerous important computations (possibly including mental computations), it does not plausibly capture computation in general .

Many philosophical discussions embody a structuralist conception of computation : a computational model describes an abstract causal structure, without taking into account particular physical states that instantiate the structure. This conception traces back at least to Putnam’s original treatment (1967). Chalmers (1995, 1996a, 2011, 2012) develops it in detail. He introduces the combinatorial-state automaton (CSA) formalism, which subsumes most familiar models of computation (including Turing machines and neural networks). A CSA provides an abstract description of a physical system’s causal topology : the pattern of causal interaction among the system’s parts, independent of the nature of those parts or the causal mechanisms through which they interact. Computational description specifies a causal topology.

Chalmers deploys structuralism to delineate a very general version of CTM. He assumes the functionalist view that psychological states are individuated by their roles in a pattern of causal organization. Psychological description specifies causal roles, abstracted away from physical states that realize those roles. So psychological properties are organizationally invariant , in that they supervene upon causal topology. Since computational description characterizes a causal topology, satisfying a suitable computational description suffices for instantiating appropriate mental properties. It also follows that psychological description is a species of computational description, so that computational description should play a central role within psychological explanation. Thus, structuralist computation provides a solid foundation for cognitive science. Mentality is grounded in causal patterns, which are precisely what computational models articulate.

Structuralism comes packaged with an attractive account of the implementation relation between abstract computational models and physical systems. Under what conditions does a physical system implement a computational model? Structuralists say that a physical system implements a model just in case the model’s causal structure is “isomorphic” to the model’s formal structure. A computational model describes a physical system by articulating a formal structure that mirrors some relevant causal topology. Chalmers elaborates this intuitive idea, providing detailed necessary and sufficient conditions for physical realization of CSAs. Few if any alternative conceptions of computation can provide so substantive an account of the implementation relation.

We may instructively compare structuralist computationalism with some other theories discussed above:

Machine functionalism . Structuralist computationalism embraces the core idea behind machine functionalism: mental states are functional states describable through a suitable computational formalism. Putnam advances CTM as an empirical hypothesis, and he defends functionalism on that basis. In contrast, Chalmers follows David Lewis (1972) by grounding functionalism in the conceptual analysis of mentalistic discourse. Whereas Putnam defends functionalism by defending computationalism, Chalmers defends computationalism by assuming functionalism.

Classical computationalism, connectionism, and computational neuroscience . Structuralist computationalism emphasizes organizationally invariant descriptions, which are multiply realizable. In that respect, it diverges from computational neuroscience. Structuralism is compatible with both classical and connectionist computationalism, but it differs in spirit from those views. Classicists and connectionists present their rival positions as bold, substantive hypotheses. Chalmers advances structuralist computationalism as a relatively minimalist position unlikely to be disconfirmed.

Intentional realism and eliminativism . Structuralist computationalism is compatible with both positions. CSA description does not explicitly mention semantic properties such as reference, truth-conditions, representational content, and so on. Structuralist computationalists need not assign representational content any important role within scientific psychology. On the other hand, structuralist computationalism does not preclude an important role for representational content.

The formal-syntactic conception of computation . Wide content depends on causal-historical relations to the external environment, relations that outstrip causal topology. Thus, CSA description leaves wide content underdetermined. Narrow content presumably supervenes upon causal topology, but CSA description does not explicitly mention narrow contents. Overall, then, structuralist computationalism prioritizes a level of formal, non-semantic computational description. In that respect, it resembles FSC. On the other hand, structuralist computationalists need not say that computation is “insensitive” to semantic properties, so they need not endorse all aspects of FSC.

Although structuralist computationalism is distinct from CTM+FSC, it raises some similar issues. For example, Rescorla (2012) denies that causal topology plays the central explanatory role within cognitive science that structuralist computationalism dictates. He suggests that externalist intentional description rather than organizationally invariant description enjoys explanatory primacy. Coming from a different direction, computational neuroscientists will recommend that we forego organizationally invariant descriptions and instead employ more neurally specific computational models. In response to such objections, Chalmers (2012) argues that organizationally invariant computational description yields explanatory benefits that neither intentional description nor neurophysiological description replicate: it reveals the underlying mechanisms of cognition (unlike intentional description); and it abstracts away from neural implementation details that are irrelevant for many explanatory purposes.

The mechanistic nature of computation is a recurring theme in logic, philosophy, and cognitive science. Gualtiero Piccinini (2007, 2012, 2015) and Marcin Milkowski (2013) develop this theme into a mechanistic theory of computing systems. A functional mechanism is a system of interconnected components, where each component performs some function within the overall system. Mechanistic explanation proceeds by decomposing the system into parts, describing how the parts are organized into the larger system, and isolating the function performed by each part. A computing system is a functional mechanism of a particular kind. On Piccinini’s account, a computing system is a mechanism whose components are functionally organized to process vehicles in accord with rules. Echoing Putnam’s discussion of multiple realizability, Piccinini demands that the rules be medium-independent , in that they abstract away from the specific physical implementations of the vehicles. Computational explanation decomposes the system into parts and describes how each part helps the system process the relevant vehicles. If the system processes discretely structured vehicles, then the computation is digital. If the system processes continuous vehicles, then the computation is analog. Milkowski’s version of the mechanistic approach is similar. He differs from Piccinini by pursuing an “information-processing” gloss, so that computational mechanisms operate over information-bearing states. Milkowski and Piccinini deploy their respective mechanistic theories to defend computationalism.

Mechanistic computationalists typically individuate computational states non-semantically. They therefore encounter worries about the explanatory role of representational content, similar to worries encountered by FSC and structuralism. In this spirit, Shagrir (2014) complains that mechanistic computationalism does not accommodate cognitive science explanations that are simultaneously computational and representational. The perceived force of this criticism will depend upon one’s sympathy for content-involving computationalism.

We have surveyed various contrasting and sometimes overlapping conceptions of computation: classical computation, connectionist computation, neural computation, formal-syntactic computation, content-involving computation, information-processing computation, functional computation, structuralist computation, and mechanistic computation. Each conception yields a different form of computationalism. Each conception has its own strengths and weaknesses. One might adopt a pluralistic stance that recognizes distinct legitimate conceptions. Rather than elevate one conception above the others, pluralists happily employ whichever conception seems useful in a given explanatory context. Edelman (2008) takes a pluralistic line, as does Chalmers (2012) in his most recent discussion.

The pluralistic line raises some natural questions. Can we provide a general analysis that encompasses all or most types of computation? Do all computations share certain characteristic marks with one another? Are they perhaps instead united by something like family resemblance? Deeper understanding of computation requires us to grapple with these questions.

7. Arguments against computationalism

CTM has attracted numerous objections. In many cases, the objections apply only to specific versions of CTM (such as classical computationalism or connectionist computationalism). Here are a few prominent objections. See also the entry the Chinese room argument for a widely discussed objection to classical computationalism advanced by John Searle (1980).

A recurring worry is that CTM is trivial , because we can describe almost any physical system as executing computations. Searle (1990) claims that a wall implements any computer program, since we can discern some pattern of molecular movements in the wall that is isomorphic to the formal structure of the program. Putnam (1988: 121–125) defends a less extreme but still very strong triviality thesis along the same lines. Triviality arguments play a large role in the philosophical literature. Anti-computationalists deploy triviality arguments against computationalism, while computationalists seek to avoid triviality.

Computationalists usually rebut triviality arguments by insisting that the arguments overlook constraints upon computational implementation, constraints that bar trivializing implementations. The constraints may be counterfactual, causal, semantic, or otherwise, depending on one’s favored theory of computation. For example, David Chalmers (1995, 1996a) and B. Jack Copeland (1996) hold that Putnam’s triviality argument ignores counterfactual conditionals that a physical system must satisfy in order to implement a computational model. Other philosophers say that a physical system must have representational properties to implement a computational model (Fodor 1998: 11–12; Ladyman 2009; Sprevak 2010) or at least to implement a content-involving computational model (Rescorla 2013, 2014b). The details here vary considerably, and computationalists debate amongst themselves exactly which types of computation can avoid which triviality arguments. But most computationalists agree that we can avoid any devastating triviality worries through a sufficiently robust theory of the implementation relation between computational models and physical systems.

Pancomputationalism holds that every physical system implements a computational model. This thesis is plausible, since any physical system arguably implements a sufficiently trivial computational model (e.g., a one-state finite state automaton). As Chalmers (2011) notes, pancomputationalism does not seem worrisome for computationalism. What would be worrisome is the much stronger triviality thesis that almost every physical system implements almost every computational model.

For further discussion of triviality arguments and computational implementation, see Sprevak (2019) and the entry computation in physical systems .

According to some authors, Gödel’s incompleteness theorems show that human mathematical capacities outstrip the capacities of any Turing machine (Nagel and Newman 1958). J.R. Lucas (1961) develops this position into a famous critique of CCTM. Roger Penrose pursues the critique in The Emperor’s New Mind (1989) and subsequent writings. Various philosophers and logicians have answered the critique, arguing that existing formulations suffer from fallacies, question-begging assumptions, and even outright mathematical errors (Bowie 1982; Chalmers 1996b; Feferman 1996; Lewis 1969, 1979; Putnam 1975: 365–366, 1994; Shapiro 2003). There is a wide consensus that this criticism of CCTM lacks any force. It may turn out that certain human mental capacities outstrip Turing-computability, but Gödel’s incompleteness theorems provide no reason to anticipate that outcome.

Could a computer compose the Eroica symphony? Or discover general relativity? Or even replicate a child’s effortless ability to perceive the environment, tie her shoelaces, and discern the emotions of others? Intuitive, creative, or skillful human activity may seem to resist formalization by a computer program (Dreyfus 1972, 1992). More generally, one might worry that crucial aspects of human cognition elude computational modeling, especially classical computational modeling.

Ironically, Fodor promulgates a forceful version of this critique. Even in his earliest statements of CCTM, Fodor (1975: 197–205) expresses considerable skepticism that CCTM can handle all important cognitive phenomena. The pessimism becomes more pronounced in his later writings (1983, 2000), which focus especially on abductive reasoning as a mental phenomenon that potentially eludes computational modeling. His core argument may be summarized as follows:

Some critics deny (1), arguing that suitable Turing-style computations can be sensitive to “nonlocal” properties (Schneider 2011; Wilson 2005). Some challenge (2), arguing that typical abductive inferences are sensitive only to “local” properties (Carruthers 2003; Ludwig and Schneider 2008; Sperber 2002). Some concede step (3) but dispute step (4), insisting that we have promising non-Turing-style models of the relevant mental processes (Pinker 2005). Partly spurred by such criticisms, Fodor elaborates his argument in considerable detail. To defend (2), he critiques theories that model abduction by deploying “local” heuristic algorithms (2005: 41–46; 2008: 115–126) or by positing a profusion of domain-specific cognitive modules (2005: 56–100). To defend (4), he critiques various theories that handle abduction through non-Turing-style models (2000: 46–53; 2008), such as connectionist networks.

The scope and limits of computational modeling remain controversial. We may expect this topic to remain an active focus of inquiry, pursued jointly with AI.

Mental activity unfolds in time. Moreover, the mind accomplishes sophisticated tasks (e.g., perceptual estimation) very quickly. Many critics worry that computationalism, especially classical computationalism, does not adequately accommodate temporal aspects of cognition. A Turing-style model makes no explicit mention of the time scale over which computation occurs. One could physically implement the same abstract Turing machine with a silicon-based device, or a slower vacuum-tube device, or an even slower pulley-and-lever device. Critics recommend that we reject CCTM in favor of some alternative framework that more directly incorporates temporal considerations. van Gelder and Port (1995) use this argument to promote a non-computational dynamical systems framework for modeling mental activity. Eliasmith (2003, 2013: 12–13) uses it to support his Neural Engineering Framework.

Computationalists respond that we can supplement an abstract computational model with temporal considerations (Piccinini 2010; Weiskopf 2004). For example, a Turing machine model presupposes discrete “stages of computation”, without describing how the stages relate to physical time. But we can supplement our model by describing how long each stage lasts, thereby converting our non-temporal Turing machine model into a theory that yields detailed temporal predictions. Many advocates of CTM employ supplementation along these lines to study temporal properties of cognition (Newell 1990). Similar supplementation figures prominently in computer science, whose practitioners are quite concerned to build machines with appropriate temporal properties. Computationalists conclude that a suitably supplemented version of CTM can adequately capture how cognition unfolds in time.

A second temporal objection highlights the contrast between discrete and continuous temporal evolution (van Gelder and Port 1995). Computation by a Turing machine unfolds in discrete stages, while mental activity unfolds in a continuous time. Thus, there is a fundamental mismatch between the temporal properties of Turing-style computation and those of actual mental activity. We need a psychological theory that describes continuous temporal evolution.

Computationalists respond that this objection assumes what is to be shown: that cognitive activity does not fall into explanatory significant discrete stages (Weiskopf 2004). Assuming that physical time is continuous, it follows that mental activity unfolds in continuous time. It does not follow that cognitive models must have continuous temporal structure. A personal computer operates in continuous time, and its physical state evolves continuously. A complete physical theory will reflect all those physical changes. But our computational model does not reflect every physical change to the computer. Our computational model has discrete temporal structure. Why assume that a good cognitive-level model of the mind must reflect every physical change to the brain? Even if there is a continuum of evolving physical states, why assume a continuum of evolving cognitive states? The mere fact of continuous temporal evolution does not militate against computational models with discrete temporal structure.

Embodied cognition is a research program that draws inspiration from the continental philosopher Maurice Merleau-Ponty, the perceptual psychologist J.J. Gibson, and other assorted influences. It is a fairly heterogeneous movement, but the basic strategy is to emphasize links between cognition, bodily action, and the surrounding environment. See Varela, Thompson, and Rosch (1991) for an influential early statement. In many cases, proponents deploy tools of dynamical systems theory. Proponents typically present their approach as a radical alternative to computationalism (Chemero 2009; Kelso 1995; Thelen and Smith 1994). CTM, they complain, treats mental activity as static symbol manipulation detached from the embedding environment. It neglects myriad complex ways that the environment causally or constitutively shapes mental activity. We should replace CTM with a new picture that emphasizes continuous links between mind, body, and environment. Agent-environment dynamics, not internal mental computation, holds the key to understanding cognition. Often, a broadly eliminativist attitude towards intentionality propels this critique.

Computationalists respond that CTM allows due recognition of cognition’s embodiment. Computational models can take into account how mind, body, and environment continuously interact. After all, computational models can incorporate sensory inputs and motor outputs. There is no obvious reason why an emphasis upon agent-environment dynamics precludes a dual emphasis upon internal mental computation (Clark 2014: 140–165; Rupert 2009). Computationalists maintain that CTM can incorporate any legitimate insights offered by the embodied cognition movement. They also insist that CTM remains our best overall framework for explaining numerous core psychological phenomena.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Graves, A., G. Wayne, and I. Danihelko, 2014, “ Neural Turing Machines ”, manuscript at arXiv.org.
• Horst, Steven, “The Computational Theory of Mind”, Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), URL = < https://plato.stanford.edu/archives/sum2015/entries/computational-mind/ >. [This is the previous entry on the Computational Theory of Mind in the Stanford Encyclopedia of Philosophy — see the version history .]
• Marcin Milkowski, “ The Computational Theory of Mind ,” in the Internet Encyclopedia of Philosophy .
• Pozzi, I., S. Bohté, and P. Roelfsema, 2019, “ A Biologically Plausible Learning Rule for Deep Learning in the Brain ”, manuscript at arXiv.org.
• Bibliography on philosophy of artificial intelligence , in Philpapers.org.

analogy and analogical reasoning | anomalous monism | causation: the metaphysics of | Chinese room argument | Church-Turing Thesis | cognitive science | computability and complexity | computation: in physical systems | computer science, philosophy of | computing: modern history of | connectionism | culture: and cognitive science | externalism about the mind | folk psychology: as mental simulation | frame problem | functionalism | Gödel, Kurt | Gödel, Kurt: incompleteness theorems | Hilbert, David: program in the foundations of mathematics | language of thought hypothesis | mental causation | mental content: causal theories of | mental content: narrow | mental content: teleological theories of | mental imagery | mental representation | mental representation: in medieval philosophy | mind/brain identity theory | models in science | multiple realizability | other minds | reasoning: automated | reasoning: defeasible | reduction, scientific | simulations in science | Turing, Alan | Turing machines | Turing test | zombies

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The psychology of thinking : reasoning, decision-making & problem-solving

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• SECTION I: THE ORGANIZATION OF HUMAN THOUGHT
• Chapter 1. The Psychology of Thinking
• Chapter 2. The Psychology of Similarity
• Chapter 3. Knowledge and Memory
• Chapter 4. Concepts and Categories
• Chapter 5. Language and Thought SECTION II: THINKING AND REASONING
• Chapter 6. Inference and Induction
• Chapter 7. Deductive Reasoning
• Chapter 8. Context, Motivation and Mood SECTION III: THINKING IN ACTION - DECISION MAKING, PROBLEM SOLVING, AND EXPERTISE
• Chapter 9. Decision Making
• Chapter 10. Problem Solving
• Chapter 11. Expertise and Expert Thinking.
• (source: Nielsen Book Data)

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The Psychology of Thinking Reasoning, Decision-Making and Problem-Solving

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The Psychology of Thinking is an engaging, interesting and easy-to-follow guide into the essential concepts behind our reasoning, decision-making and problem-solving. Clearly structured into 3 sections, this book will;

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Complex cognition: the science of human reasoning, problem-solving, and decision-making

• Published: 23 March 2010
• Volume 11 , pages 99–102, ( 2010 )

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Climate change, globalization, policy of peace, and financial market crises—often we are faced with very complex problems. In order to tackle these complex problems, the responsible people should first come to mutual terms. An additional challenge is that typically the involved parties have different (often conflicting) interests and relate the problems to different emotions and wishes. These factors certainly do not ease the quest for a solution to these complex problems.

It is needless to say that the big problems of our time are not easy to solve. Less clear, however, is identifying the causes that led to these problems. Interest conflicts between social groups, the economic and social system or greed—one can think of many responsible factors for the large-scale problems we are currently confronted with.

The present “Special Corner: complex cognition” deals with questions in this regard that have often received little consideration. Under the headline “complex cognition”, we summarize mental activities such as thinking, reasoning, problem - solving, and decision - making that typically rely on the combination and interaction of more elementary processes such as perception, learning, memory, emotion, etc. (cf. Sternberg and Ben-Zeev 2001 ). However, even though complex cognition relies on these elementary functions, the scope of complex cognition research goes beyond the isolated analysis of such elementary mental processes. Two aspects are essential for “complex cognition”: The first aspect refers to the interaction of different mental activities such as perception, memory, learning, reasoning, emotion, etc. The second aspect takes the complexity of the situation into account an agent is confronted with. Based on these two aspects, the term “complex cognition” can be defined in the following way:

Complex psychological processes: We talk about “complex cognition”, when thinking, problem-solving, or decision-making falls back on other cognitive processes such as “perception”, “working memory”, “long-term memory”, “executive processes”, or when the cognitive processes are in close connection with other processes such as “emotion” and “motivation”. The complexity also results from an interaction from a multitude of processes that occur simultaneously or at different points in time and can be realized in different cognitive and/or neuronal structures.

Complex conditions: We also talk about “complex cognition” when the conditions are complex in which a person finds himself and in which conclusions need to be drawn, a problem needs to be solved, or decisions need to be made. The complexity of the conditions or constraints can have different causes. The situation structure itself can be difficult to “see”, or the action alternatives are difficult “to put into effect”. The conditions can themselves comprise of many different variables. These variables can exhibit a high level of interdependence and cross-connection, and it can, as time passes by, come to a change of the original conditions (e.g. Dörner and Wearing 1995 ; Osman 2010 ). It can also be the case that the problem is embedded in a larger social context and can be solved only under certain specifications (norms, data, legislations, culture, etc.) or that the problem can only be solved in interaction with other agents, be it other persons or technical systems.

When one summarizes these two aspects, this yields the following view of what should be understood as “complex cognition”.

As “complex cognition” we define all mental processes that are used by individuals for deriving new information out of given information, with the intention to solve problems, make decision, and plan actions. The crucial characteristic of “complex cognition” is that it takes place under complex conditions in which a multitude of cognitive processes interact with one another or with other noncognitive processes.

The “Special Corner: complex cognition” deals with complex cognition from many different perspectives. The typical questions of all contributions are: Does the design of the human mind enable the necessary thinking skills to solve the truly complex problems we are faced with? Where lay the boundaries of our thinking skills? How do people derive at conclusions? What makes a problem a complex problem? How can we improve our skills to effectively solve problems and make sound judgements?

It is for sure too much to expect that the Special Corner answers these questions. If it were that easy, we would not be still searching for an answer. It is, however, our intention with the current collection of articles to bring to focus such questions to a larger extent than has been done so far.

An important starting point is the fact that people’s skills to solve the most complex of all problems and to ponder about the most complex issues is often immense—humankind would not otherwise be there were she is now. Yet, on the other hand, it has become more clear in the past few years that often people drift away from what one would identify as “rational” (Kahneman 2003 ). People hardly ever adhere to that what the norms of logic, the probability calculus, or the mathematical decision theory state. For example, most people (and organizations) typically accept more losses for a potential high gain than would be the case if they were to take into account the rules of the probability theory. Similarly, they draw conclusions from received information in a way that is not according to the rules of logic. When people, for example, accept the rule “If it rains, then the street is wet”, they most often conclude that when the street is wet, it must have rained. That, however, is incorrect from a logical perspective: perhaps a cleaning car just drove by. In psychology, two main views are traditionally put forward to explain how such deviations from the normative guidelines occur. One scientific stream is interested in how deviations from the normative models can be explained (Evans 2005 ; Johnson-Laird 2008 ; Knauff 2007 ; Reason 1990 ). According to this line of research, deviations are caused by the limitations of the human cognitive system. The other psychological stream puts forward as the main criticism that the deviations can actually be regarded as mistakes (Gigerenzer 2008 ). The deviations accordingly have a high value, because they are adjusted to the information structure of the environment (Gigerenzer et al. 1999 ). They have probably developed during evolution, because they could ensure survival as for example the specifications of formal logic (Hertwig and Herzog 2009 ). We, the editors of the special corner, are very pleased that we can offer an impression of this debate with the contributions from Marewski, Gaissmaier, and Gigerenzer and the commentaries to this contribution from Evans and Over. Added to this is a reply from Marewski, Gaissmaier, and Gigerenzer to the commentary from Evans and Over.

Another topic in the area of complex cognition can be best illustrated by means of the climate protection. To be successful in this area, the responsible actors have to consider a multitude of ecological, biological, geological, political, and economical factors, the basic conditions are constantly at change, and the intervention methods are not clear. Because the necessary information is not readily available for the person dealing with the problem, the person is forced to obtain the relevant information from other sources. Furthermore, intervention in the complex variable structure of the climate can trigger processes whose impact was likely not intended. Finally, the system will not “wait” for intervention of the actors but will change itself over time. The special corner is also concerned with thinking and problem-solving in such complex situations. The article by Funke gives an overview of the current state of research on this topic from the viewpoint of the author, in which several research areas are covered that have internationally not received much acknowledgement (but see, for example, Osman 2010 ).

Although most contributions to the special corner come from the area of psychology, the contribution by Ragni and Löffler illustrates that computer science can provide a valuable addition to the understanding of complex cognition. Computer science plays an important role in complex cognition. In general, computer science, which is used to investigate computational processes central to all research approaches, can be placed in a “computational theory of cognition” framework. This is true especially for the development of computational theories of complex cognitive processes. In many of our modern knowledge domains, the application of simulations and modelling has become a major part of the methods inventory. Simulations help forecast the weather and climate change, help govern traffic flow and help comprehend physical processes. Although modelling in these areas is a vastly established method, it has been very little applied in the area of human thinking (but see e.g. Anderson 1990 ; Gray 2007 ). However, exactly in the area of complex cognition, the method of cognitive modelling offers empirical research an additional methodological access to the description and explanation of complex cognitive processes. While the validity of psychological theories can be tested with the use of empirical research, cognitive models, with their internal coherence, make possible to test consistency and completeness (e.g. Schmid 2008 ). They will also lead to new hypotheses that will in turn be possible to test experimentally. The contribution of Ragni and Löffler demonstrates with the help of an interesting example, finding the optimal route, the usefulness of simulation and modelling in psychology.

A further problem in the area of complex cognition is that many problems are solvable only under certain social conditions (norms, values, laws, culture) or only in interaction with other actors (cf. Beller 2008 ). The article on deontic reasoning by Beller is concerned with this topic. Deontic reasoning is thinking about whether actions are forbidden or allowed, obligatory or not obligatory. Beller proposes that social norms, imposing constraints on individual actions, constitute the fundamental concept for deontic thinking and that people reason from such norms flexibly according to deontic core principles. The review paper shows how knowing what in a certain situation is allowed or forbidden can influence how people derive at conclusions.

The article of Waldmann, Meder, von Sydow, and Hagmayer is concerned with the important topic of causal reasoning. More specifically, the authors explore the interaction between category and causal induction in causal model learning. The paper is a good example of how experimental work in psychology can combine different research traditions that typically work quite isolated. The paper goes beyond a divide and conquers approach and shows that causal knowledge plays an important role in learning, categorization, perception, decision-making, problem-solving, and text comprehension. In each of these fields, separate theories have been developed to investigate the role of causal knowledge. The first author of the paper is internationally well known for his work on the role of causality in other cognitive functions, in particular in categorization and learning (e.g. Lagnado et al. 2007 ; Waldmann et al. 1995 ). In a number of experimental studies, Waldmann and his colleagues have shown that people when learning about causal relations do not simply form associations between causes and effects but make use of abstract prior assumptions about the underlying causal structure and functional form (Waldmann 2007 ).

We, the guest editors, are very pleased that we have the opportunity with this Special corner to make accessible the topic “complex cognition” to the interdisciplinary readership of Cognitive Processing . We predict a bright future for this topic. The research topic possesses high research relevance in the area of basic research for a multitude of disciplines, for example psychology, computer science, and neuroscience. In addition, this area forms a good foundation for an interdisciplinary cooperation.

A further important reason for the positive development of the area is that the relevance of the area goes beyond fundamental research. In that way, the results of the area can for example also contribute to better understanding of the possibilities and borders of human thinking, problem-solving, and decisions in politics, corporations, and economy. In the long term, it might even lead to practical directions on how to avoid “mistakes” and help us better understand the global challenges of our time—Climate change, globalization, financial market crises, etc.

We thank all the authors for their insightful and inspiring contributions, a multitude of reviewers for their help, the editor-in-chief Marta Olivetti Belardinelli that she gave us the opportunity to address this topic, and the editorial manager, Thomas Hünefeldt, for his support for accomplishing the Special Corner. We wish the readers of the Special Corner lots of fun with reading the contributions!

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Knauff, M., Wolf, A.G. Complex cognition: the science of human reasoning, problem-solving, and decision-making. Cogn Process 11 , 99–102 (2010). https://doi.org/10.1007/s10339-010-0362-z

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21 Problem Solving

Miriam Bassok, Department of Psychology, University of Washington, Seattle, WA

Laura R. Novick, Department of Psychology and Human Development, Vanderbilt University, Nashville, TN

• Published: 21 November 2012
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This chapter follows the historical development of research on problem solving. It begins with a description of two research traditions that addressed different aspects of the problem-solving process: ( 1 ) research on problem representation (the Gestalt legacy) that examined how people understand the problem at hand, and ( 2 ) research on search in a problem space (the legacy of Newell and Simon) that examined how people generate the problem's solution. It then describes some developments in the field that fueled the integration of these two lines of research: work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. Next, it presents examples of recent work on problem solving in science and mathematics that highlight the impact of visual perception and background knowledge on how people represent problems and search for problem solutions. The final section considers possible directions for future research.

People are confronted with problems on a daily basis, be it trying to extract a broken light bulb from a socket, finding a detour when the regular route is blocked, fixing dinner for unexpected guests, dealing with a medical emergency, or deciding what house to buy. Obviously, the problems people encounter differ in many ways, and their solutions require different types of knowledge and skills. Yet we have a sense that all the situations we classify as problems share a common core. Karl Duncker defined this core as follows: “A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action [i.e., by the performance of obvious operations], then there has to be recourse to thinking” (Duncker, 1945 , p. 1). Consider the broken light bulb. The obvious operation—holding the glass part of the bulb with one's fingers while unscrewing the base from the socket—is prevented by the fact that the glass is broken. Thus, there must be “recourse to thinking” about possible ways to solve the problem. For example, one might try mounting half a potato on the broken bulb (we do not know the source of this creative solution, which is described on many “how to” Web sites).

The above definition and examples make it clear that what constitutes a problem for one person may not be a problem for another person, or for that same person at another point in time. For example, the second time one has to remove a broken light bulb from a socket, the solution likely can be retrieved from memory; there is no problem. Similarly, tying shoes may be considered a problem for 5-year-olds but not for readers of this chapter. And, of course, people may change their goal and either no longer have a problem (e.g., take the guests to a restaurant instead of fixing dinner) or attempt to solve a different problem (e.g., decide what restaurant to go to). Given the highly subjective nature of what constitutes a problem, researchers who study problem solving have often presented people with novel problems that they should be capable of solving and attempted to find regularities in the resulting problem-solving behavior. Despite the variety of possible problem situations, researchers have identified important regularities in the thinking processes by which people (a) represent , or understand, problem situations and (b) search for possible ways to get to their goal.

A problem representation is a model constructed by the solver that summarizes his or her understanding of the problem components—the initial state (e.g., a broken light bulb in a socket), the goal state (the light bulb extracted), and the set of possible operators one may apply to get from the initial state to the goal state (e.g., use pliers). According to Reitman ( 1965 ), problem components differ in the extent to which they are well defined . Some components leave little room for interpretation (e.g., the initial state in the broken light bulb example is relatively well defined), whereas other components may be ill defined and have to be defined by the solver (e.g., the possible actions one may take to extract the broken bulb). The solver's representation of the problem guides the search for a possible solution (e.g., possible attempts at extracting the light bulb). This search may, in turn, change the representation of the problem (e.g., finding that the goal cannot be achieved using pliers) and lead to a new search. Such a recursive process of representation and search continues until the problem is solved or until the solver decides to abort the goal.

Duncker ( 1945 , pp. 28–37) documented the interplay between representation and search based on his careful analysis of one person's solution to the “Radiation Problem” (later to be used extensively in research analogy, see Holyoak, Chapter 13 ). This problem requires using some rays to destroy a patient's stomach tumor without harming the patient. At sufficiently high intensity, the rays will destroy the tumor. However, at that intensity, they will also destroy the healthy tissue surrounding the tumor. At lower intensity, the rays will not harm the healthy tissue, but they also will not destroy the tumor. Duncker's analysis revealed that the solver's solution attempts were guided by three distinct problem representations. He depicted these solution attempts as an inverted search tree in which the three main branches correspond to the three general problem representations (Duncker, 1945 , p. 32). We reproduce this diagram in Figure 21.1 . The desired solution appears on the rightmost branch of the tree, within the general problem representation in which the solver aims to “lower the intensity of the rays on their way through healthy tissue.” The actual solution is to project multiple low-intensity rays at the tumor from several points around the patient “by use of lens.” The low-intensity rays will converge on the tumor, where their individual intensities will sum to a level sufficient to destroy the tumor.

A search-tree representation of one subject's solution to the radiation problem, reproduced from Duncker ( 1945 , p. 32).

Although there are inherent interactions between representation and search, some researchers focus their efforts on understanding the factors that affect how solvers represent problems, whereas others look for regularities in how they search for a solution within a particular representation. Based on their main focus of interest, researchers devise or select problems with solutions that mainly require either constructing a particular representation or finding the appropriate sequence of steps leading from the initial state to the goal state. In most cases, researchers who are interested in problem representation select problems in which one or more of the components are ill defined, whereas those who are interested in search select problems in which the components are well defined. The following examples illustrate, respectively, these two problem types.

The Bird-and-Trains problem (Posner, 1973 , pp. 150–151) is a mathematical word problem that tends to elicit two distinct problem representations (see Fig. 21.2a and b ):

Two train stations are 50 miles apart. At 2 p.m. one Saturday afternoon two trains start toward each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead to the front of the second train. When the bird reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet? Fig. 21.2 Open in new tab Download slide Alternative representations of Posner's ( 1973 ) trains-and-bird problem. Adapted from Novick and Hmelo ( 1994 ).

Some solvers focus on the back-and-forth path of the bird (Fig. 21.2a ). This representation yields a problem that would be difficult for most people to solve (e.g., a series of differential equations). Other solvers focus on the paths of the trains (Fig. 21.2b ), a representation that yields a relatively easy distance-rate-time problem.

The Tower of Hanoi problem falls on the other end of the representation-search continuum. It leaves little room for differences in problem representations, and the primary work is to discover a solution path (or the best solution path) from the initial state to the goal state .

There are three pegs mounted on a base. On the leftmost peg, there are three disks of differing sizes. The disks are arranged in order of size with the largest disk on the bottom and the smallest disk on the top. The disks may be moved one at a time, but only the top disk on a peg may be moved, and at no time may a larger disk be placed on a smaller disk. The goal is to move the three-disk tower from the leftmost peg to the rightmost peg.

Figure 21.3 shows all the possible legal arrangements of disks on pegs. The arrows indicate transitions between states that result from moving a single disk, with the thicker gray arrows indicating the shortest path that connects the initial state to the goal state.

The division of labor between research on representation versus search has distinct historical antecedents and research traditions. In the next two sections, we review the main findings from these two historical traditions. Then, we describe some developments in the field that fueled the integration of these lines of research—work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. In the fifth section, we present some examples of recent work on problem solving in science and mathematics. This work highlights the role of visual perception and background knowledge in the way people represent problems and search for problem solutions. In the final section, we consider possible directions for future research.

Our review is by no means an exhaustive one. It follows the historical development of the field and highlights findings that pertain to a wide variety of problems. Research pertaining to specific types of problems (e.g., medical problems), specific processes that are involved in problem solving (e.g., analogical inferences), and developmental changes in problem solving due to learning and maturation may be found elsewhere in this volume (e.g., Holyoak, Chapter 13 ; Smith & Ward, Chapter 23 ; van Steenburgh et al., Chapter 24 ; Simonton, Chapter 25 ; Opfer & Siegler, Chapter 30 ; Hegarty & Stull, Chapter 31 ; Dunbar & Klahr, Chapter 35 ; Patel et al., Chapter 37 ; Lowenstein, Chapter 38 ; Koedinger & Roll, Chapter 40 ).

All possible problem states for the three-disk Tower of Hanoi problem. The thicker gray arrows show the optimum solution path connecting the initial state (State #1) to the goal state (State #27).

Problem Representation: The Gestalt Legacy

Research on problem representation has its origins in Gestalt psychology, an influential approach in European psychology during the first half of the 20th century. (Behaviorism was the dominant perspective in American psychology at this time.) Karl Duncker published a book on the topic in his native German in 1935, which was translated into English and published 10 years later as the monograph On Problem-Solving (Duncker, 1945 ). Max Wertheimer also published a book on the topic in 1945, titled Productive Thinking . An enlarged edition published posthumously includes previously unpublished material (Wertheimer, 1959 ). Interestingly, 1945 seems to have been a watershed year for problem solving, as mathematician George Polya's book, How to Solve It , also appeared then (a second edition was published 12 years later; Polya, 1957 ).

The Gestalt psychologists extended the organizational principles of visual perception to the domain of problem solving. They showed that various visual aspects of the problem, as well the solver's prior knowledge, affect how people understand problems and, therefore, generate problem solutions. The principles of visual perception (e.g., proximity, closure, grouping, good continuation) are directly relevant to problem solving when the physical layout of the problem, or a diagram that accompanies the problem description, elicits inferences that solvers include in their problem representations. Such effects are nicely illustrated by Maier's ( 1930 ) nine-dot problem: Nine dots are arrayed in a 3x3 grid, and the task is to connect all the dots by drawing four straight lines without lifting one's pencil from the paper. People have difficulty solving this problem because their initial representations generally include a constraint, inferred from the configuration of the dots, that the lines should not go outside the boundary of the imaginary square formed by the outer dots. With this constraint, the problem cannot be solved (but see Adams, 1979 ). Without this constraint, the problem may be solved as shown in Figure 21.4 (though the problem is still difficult for many people; see Weisberg & Alba, 1981 ).

The nine-dot problem is a classic insight problem (see van Steenburgh et al., Chapter 24 ). According to the Gestalt view (e.g., Duncker, 1945 ; Kohler, 1925 ; Maier, 1931 ; see Ohlsson, 1984 , for a review), the solution to an insight problem appears suddenly, accompanied by an “aha!” sensation, immediately following the sudden “restructuring” of one's understanding of the problem (i.e., a change in the problem representation): “The decisive points in thought-processes, the moments of sudden comprehension, of the ‘Aha!,’ of the new, are always at the same time moments in which such a sudden restructuring of the thought-material takes place” (Duncker, 1945 , p. 29). For the nine-dot problem, one view of the required restructuring is that the solver relaxes the constraint implied by the perceptual form of the problem and realizes that the lines may, in fact, extend past the boundary of the imaginary square. Later in the chapter, we present more recent accounts of insight.

The entities that appear in a problem also tend to evoke various inferences that people incorporate into their problem representations. A classic demonstration of this is the phenomenon of functional fixedness , introduced by Duncker ( 1945 ): If an object is habitually used for a certain purpose (e.g., a box serves as a container), it is difficult to see

A solution to the nine-dot problem.

that object as having properties that would enable it to be used for a dissimilar purpose. Duncker's basic experimental paradigm involved two conditions that varied in terms of whether the object that was crucial for solution was initially used for a function other than that required for solution.

Consider the candles problem—the best known of the five “practical problems” Duncker ( 1945 ) investigated. Three candles are to be mounted at eye height on a door. On the table, for use in completing this task, are some tacks and three boxes. The solution is to tack the three boxes to the door to serve as platforms for the candles. In the control condition, the three boxes were presented to subjects empty. In the functional-fixedness condition, they were filled with candles, tacks, and matches. Thus, in the latter condition, the boxes initially served the function of container, whereas the solution requires that they serve the function of platform. The results showed that 100% of the subjects who received empty boxes solved the candles problem, compared with only 43% of subjects who received filled boxes. Every one of the five problems in this study showed a difference favoring the control condition over the functional-fixedness condition, with average solution rates across the five problems of 97% and 58%, respectively.

The function of the objects in a problem can be also “fixed” by their most recent use. For example, Birch and Rabinowitz ( 1951 ) had subjects perform two consecutive tasks. In the first task, people had to use either a switch or a relay to form an electric circuit. After completing this task, both groups of subjects were asked to solve Maier's ( 1931 ) two-ropes problem. The solution to this problem requires tying an object to one of the ropes and making the rope swing as a pendulum. Subjects could create the pendulum using either the object from the electric-circuit task or the other object. Birch and Rabinowitz found that subjects avoided using the same object for two unrelated functions. That is, those who used the switch in the first task made the pendulum using the relay, and vice versa. The explanations subjects subsequently gave for their object choices revealed that they were unaware of the functional-fixedness constraint they imposed on themselves.

In addition to investigating people's solutions to such practical problems as irradiating a tumor, mounting candles on the wall, or tying ropes, the Gestalt psychologists examined how people understand and solve mathematical problems that require domain-specific knowledge. For example, Wertheimer ( 1959 ) observed individual differences in students' learning and subsequent application of the formula for finding the area of a parallelogram (see Fig. 21.5a ). Some students understood the logic underlying the learned formula (i.e., the fact that a parallelogram can be transformed into a rectangle by cutting off a triangle from one side and pasting it onto the other side) and exhibited “productive thinking”—using the same logic to find the area of the quadrilateral in Figure 21.5b and the irregularly shaped geometric figure in Figure 21.5c . Other students memorized the formula and exhibited “reproductive thinking”—reproducing the learned solution only to novel parallelograms that were highly similar to the original one.

The psychological study of human problem solving faded into the background after the demise of the Gestalt tradition (during World War II), and problem solving was investigated only sporadically until Allen Newell and Herbert Simon's ( 1972 ) landmark book Human Problem Solving sparked a flurry of research on this topic. Newell and Simon adopted and refined Duncker's ( 1945 ) methodology of collecting and analyzing the think-aloud protocols that accompany problem solutions and extended Duncker's conceptualization of a problem solution as a search tree. However, their initial work did not aim to extend the Gestalt findings

Finding the area of ( a ) a parallelogram, ( b ) a quadrilateral, and ( c ) an irregularly shaped geometric figure. The solid lines indicate the geometric figures whose areas are desired. The dashed lines show how to convert the given figures into rectangles (i.e., they show solutions with understanding).

pertaining to problem representation. Instead, as we explain in the next section, their objective was to identify the general-purpose strategies people use in searching for a problem solution.

Search in a Problem Space: The Legacy of Newell and Simon

Newell and Simon ( 1972 ) wrote a magnum opus detailing their theory of problem solving and the supporting research they conducted with various collaborators. This theory was grounded in the information-processing approach to cognitive psychology and guided by an analogy between human and artificial intelligence (i.e., both people and computers being “Physical Symbol Systems,” Newell & Simon, 1976 ; see Doumas & Hummel, Chapter 5 ). They conceptualized problem solving as a process of search through a problem space for a path that connects the initial state to the goal state—a metaphor that alludes to the visual or spatial nature of problem solving (Simon, 1990 ). The term problem space refers to the solver's representation of the task as presented (Simon, 1978 ). It consists of ( 1 ) a set of knowledge states (the initial state, the goal state, and all possible intermediate states), ( 2 ) a set of operators that allow movement from one knowledge state to another, ( 3 ) a set of constraints, and ( 4 ) local information about the path one is taking through the space (e.g., the current knowledge state and how one got there).

We illustrate the components of a problem space for the three-disk Tower of Hanoi problem, as depicted in Figure 21.3 . The initial state appears at the top (State #1) and the goal state at the bottom right (State #27). The remaining knowledge states in the figure are possible intermediate states. The current knowledge state is the one at which the solver is located at any given point in the solution process. For example, the current state for a solver who has made three moves along the optimum solution path would be State #9. The solver presumably would know that he or she arrived at this state from State #5. This knowledge allows the solver to recognize a move that involves backtracking. The three operators in this problem are moving each of the three disks from one peg to another. These operators are subject to the constraint that a larger disk may not be placed on a smaller disk.

Newell and Simon ( 1972 ), as well as other contemporaneous researchers (e.g., Atwood & Polson, 1976 ; Greeno, 1974 ; Thomas, 1974 ), examined how people traverse the spaces of various well-defined problems (e.g., the Tower of Hanoi, Hobbits and Orcs). They discovered that solvers' search is guided by a number of shortcut strategies, or heuristics , which are likely to get the solver to the goal state without an extensive amount of search. Heuristics are often contrasted with algorithms —methods that are guaranteed to yield the correct solution. For example, one could try every possible move in the three-disk Tower of Hanoi problem and, eventually, find the correct solution. Although such an exhaustive search is a valid algorithm for this problem, for many problems its application is very time consuming and impractical (e.g., consider the game of chess).

In their attempts to identify people's search heuristics, Newell and Simon ( 1972 ) relied on two primary methodologies: think-aloud protocols and computer simulations. Their use of think-aloud protocols brought a high degree of scientific rigor to the methodology used by Duncker ( 1945 ; see Ericsson & Simon, 1980 ). Solvers were required to say out loud everything they were thinking as they solved the problem, that is, everything that went through their verbal working memory. Subjects' verbalizations—their think-aloud protocols—were tape-recorded and then transcribed verbatim for analysis. This method is extremely time consuming (e.g., a transcript of one person's solution to the cryptarithmetic problem DONALD + GERALD = ROBERT, with D = 5, generated a 17-page transcript), but it provides a detailed record of the solver's ongoing solution process.

An important caveat to keep in mind while interpreting a subject's verbalizations is that “a protocol is relatively reliable only for what it positively contains, but not for that which it omits” (Duncker, 1945 , p. 11). Ericsson and Simon ( 1980 ) provided an in-depth discussion of the conditions under which this method is valid (but see Russo, Johnson, & Stephens, 1989 , for an alternative perspective). To test their interpretation of a subject's problem solution, inferred from the subject's verbal protocol, Newell and Simon ( 1972 ) created a computer simulation program and examined whether it solved the problem the same way the subject did. To the extent that the computer simulation provided a close approximation of the solver's step-by-step solution process, it lent credence to the researcher's interpretation of the verbal protocol.

Newell and Simon's ( 1972 ) most famous simulation was the General Problem Solver or GPS (Ernst & Newell, 1969 ). GPS successfully modeled human solutions to problems as different as the Tower of Hanoi and the construction of logic proofs using a single general-purpose heuristic: means-ends analysis . This heuristic captures people's tendency to devise a solution plan by setting subgoals that could help them achieve their final goal. It consists of the following steps: ( 1 ) Identify a difference between the current state and the goal (or subgoal ) state; ( 2 ) Find an operator that will remove (or reduce) the difference; (3a) If the operator can be directly applied, do so, or (3b) If the operator cannot be directly applied, set a subgoal to remove the obstacle that is preventing execution of the desired operator; ( 4 ) Repeat steps 1–3 until the problem is solved. Next, we illustrate the implementation of this heuristic for the Tower of Hanoi problem, using the problem space in Figure 21.3 .

As can be seen in Figure 21.3 , a key difference between the initial state and the goal state is that the large disk is on the wrong peg (step 1). To remove this difference (step 2), one needs to apply the operator “move-large-disk.” However, this operator cannot be applied because of the presence of the medium and small disks on top of the large disk. Therefore, the solver may set a subgoal to move that two-disk tower to the middle peg (step 3b), leaving the rightmost peg free for the large disk. A key difference between the initial state and this new subgoal state is that the medium disk is on the wrong peg. Because application of the move-medium-disk operator is blocked, the solver sets another subgoal to move the small disk to the right peg. This subgoal can be satisfied immediately by applying the move-small-disk operator (step 3a), generating State #3. The solver then returns to the previous subgoal—moving the tower consisting of the small and medium disks to the middle peg. The differences between the current state (#3) and the subgoal state (#9) can be removed by first applying the move-medium-disk operator (yielding State #5) and then the move-small-disk operator (yielding State #9). Finally, the move-large-disk operator is no longer blocked. Hence, the solver moves the large disk to the right peg, yielding State #11.

Notice that the subgoals are stacked up in the order in which they are generated, so that they pop up in the order of last in first out. Given the first subgoal in our example, repeated application of the means-ends analysis heuristic will yield the shortest-path solution, indicated by the large gray arrows. In general, subgoals provide direction to the search and allow solvers to plan several moves ahead. By assessing progress toward a required subgoal rather than the final goal, solvers may be able to make moves that otherwise seem unwise. To take a concrete example, consider the transition from State #1 to State #3 in Figure 21.3 . Comparing the initial state to the goal state, this move seems unwise because it places the small disk on the bottom of the right peg, whereas it ultimately needs to be at the top of the tower on that peg. But comparing the initial state to the solver-generated subgoal state of having the medium disk on the middle peg, this is exactly where the small disk needs to go.

Means-ends analysis and various other heuristics (e.g., the hill-climbing heuristic that exploits the similarity, or distance, between the state generated by the next operator and the goal state; working backward from the goal state to the initial state) are flexible strategies that people often use to successfully solve a large variety of problems. However, the generality of these heuristics comes at a cost: They are relatively weak and fallible (e.g., in the means-ends solution to the problem of fixing a hole in a bucket, “Dear Liza” leads “Dear Henry” in a loop that ends back at the initial state; the lyrics of this famous song can be readily found on the Web). Hence, although people use general-purpose heuristics when they encounter novel problems, they replace them as soon as they acquire experience with and sufficient knowledge about the particular problem space (e.g., Anzai & Simon, 1979 ).

Despite the fruitfulness of this research agenda, it soon became evident that a fundamental weakness was that it minimized the importance of people's background knowledge. Of course, Newell and Simon ( 1972 ) were aware that problem solutions require relevant knowledge (e.g., the rules of logical proofs, or rules for stacking disks). Hence, in programming GPS, they supplemented every problem they modeled with the necessary background knowledge. This practice highlighted the generality and flexibility of means-ends analysis but failed to capture how people's background knowledge affects their solutions. As we discussed in the previous section, domain knowledge is likely to affect how people represent problems and, therefore, how they generate problem solutions. Moreover, as people gain experience solving problems in a particular knowledge domain (e.g., math, physics), they change their representations of these problems (e.g., Chi, Feltovich, & Glaser, 1981 ; Haverty, Koedinger, Klahr, & Alibali, 2000 ; Schoenfeld & Herrmann, 1982 ) and learn domain-specific heuristics (e.g., Polya, 1957 ; Schoenfeld, 1979 ) that trump the general-purpose strategies.

It is perhaps inevitable that the two traditions in problem-solving research—one emphasizing representation and the other emphasizing search strategies—would eventually come together. In the next section we review developments that led to this integration.

The Two Legacies Converge

Because Newell and Simon ( 1972 ) aimed to discover the strategies people use in searching for a solution, they investigated problems that minimized the impact of factors that tend to evoke differences in problem representations, of the sort documented by the Gestalt psychologists. In subsequent work, however, Simon and his collaborators showed that such factors are highly relevant to people's solutions of well-defined problems, and Simon ( 1986 ) incorporated these findings into the theoretical framework that views problem solving as search in a problem space.

In this section, we first describe illustrative examples of this work. We then describe research on insight solutions that incorporates ideas from the two legacies described in the previous sections.

Relevance of the Gestalt Ideas to the Solution of Search Problems

In this subsection we describe two lines of research by Simon and his colleagues, and by other researchers, that document the importance of perception and of background knowledge to the way people search for a problem solution. The first line of research used variants of relatively well-defined riddle problems that had the same structure (i.e., “problem isomorphs”) and, therefore, supposedly the same problem space. It documented that people's search depended on various perceptual and conceptual inferences they tended to draw from a specific instantiation of the problem's structure. The second line of research documented that people's search strategies crucially depend on their domain knowledge and on their prior experience with related problems.

Problem Isomorphs

Hayes and Simon ( 1977 ) used two variants of the Tower of Hanoi problem that, instead of disks and pegs, involved monsters and globes that differed in size (small, medium, and large). In both variants, the initial state had the small monster holding the large globe, the medium-sized monster holding the small globe, and the large monster holding the medium-sized globe. Moreover, in both variants the goal was for each monster to hold a globe proportionate to its own size. The only difference between the problems concerned the description of the operators. In one variant (“transfer”), subjects were told that the monsters could transfer the globes from one to another as long as they followed a set of rules, adapted from the rules in the original Tower of Hanoi problem (e.g., only one globe may be transferred at a time). In the other variant (“change”), subjects were told that the monsters could shrink and expand themselves according to a set of rules, which corresponded to the rules in the transfer version of the problem (e.g., only one monster may change its size at a time). Despite the isomorphism of the two variants, subjects conducted their search in two qualitatively different problem spaces, which led to solution times for the change variant being almost twice as long as those for the transfer variant. This difference arose because subjects could more readily envision and track an object that was changing its location with every move than one that was changing its size.

Recent work by Patsenko and Altmann ( 2010 ) found that, even in the standard Tower of Hanoi problem, people's solutions involve object-bound routines that depend on perception and selective attention. The subjects in their study solved various Tower of Hanoi problems on a computer. During the solution of a particular “critical” problem, the computer screen changed at various points without subjects' awareness (e.g., a disk was added, such that a subject who started with a five-disc tower ended with a six-disc tower). Patsenko and Altmann found that subjects' moves were guided by the configurations of the objects on the screen rather than by solution plans they had stored in memory (e.g., the next subgoal).

The Gestalt psychologists highlighted the role of perceptual factors in the formation of problem representations (e.g., Maier's, 1930 , nine-dot problem) but were generally silent about the corresponding implications for how the problem was solved (although they did note effects on solution accuracy). An important contribution of the work on people's solutions of the Tower of Hanoi problem and its variants was to show the relevance of perceptual factors to the application of various operators during search for a problem solution—that is, to the how of problem solving. In the next section, we describe recent work that documents the involvement of perceptual factors in how people understand and use equations and diagrams in the context of solving math and science problems.

Kotovsky, Hayes, and Simon ( 1985 ) further investigated factors that affect people's representation and search in isomorphs of the Tower of Hanoi problem. In one of their isomorphs, three disks were stacked on top of each other to form an inverted pyramid, with the smallest disc on the bottom and the largest on top. Subjects' solutions of the inverted pyramid version were similar to their solutions of the standard version that has the largest disc on the bottom and the smallest on top. However, the two versions were solved very differently when subjects were told that the discs represent acrobats. Subjects readily solved the version in which they had to place a small acrobat on the shoulders of a large one, but they refrained from letting a large acrobat stand on the shoulders of a small one. In other words, object-based inferences that draw on people's semantic knowledge affected the solution of search problems, much as they affect the solution of the ill-defined problems investigated by the Gestalt psychologists (e.g., Duncker's, 1945 , candles problem). In the next section, we describe more recent work that shows similar effects in people's solutions to mathematical word problems.

The work on differences in the representation and solution of problem isomorphs is highly relevant to research on analogical problem solving (or analogical transfer), which examines when and how people realize that two problems that differ in their cover stories have a similar structure (or a similar problem space) and, therefore, can be solved in a similar way. This research shows that minor differences between example problems, such as the use of X-rays versus ultrasound waves to fuse a broken filament of a light bulb, can elicit different problem representations that significantly affect the likelihood of subsequent transfer to novel problem analogs (Holyoak & Koh, 1987 ). Analogical transfer has played a central role in research on human problem solving, in part because it can shed light on people's understanding of a given problem and its solution and in part because it is believed to provide a window onto understanding and investigating creativity (see Smith & Ward, Chapter 23 ). We briefly mention some findings from the analogy literature in the next subsection on expertise, but we do not discuss analogical transfer in detail because this topic is covered elsewhere in this volume (Holyoak, Chapter 13 ).

Expertise and Its Development

In another line of research, Simon and his colleagues examined how people solve ecologically valid problems from various rule-governed and knowledge-rich domains. They found that people's level of expertise in such domains, be it in chess (Chase & Simon, 1973 ; Gobet & Simon, 1996 ), mathematics (Hinsley, Hayes, & Simon, 1977 ; Paige & Simon, 1966 ), or physics (Larkin, McDermott, Simon, & Simon, 1980 ; Simon & Simon, 1978 ), plays a crucial role in how they represent problems and search for solutions. This work, and the work of numerous other researchers, led to the discovery (and rediscovery, see Duncker, 1945 ) of important differences between experts and novices, and between “good” and “poor” students.

One difference between experts and novices pertains to pattern recognition. Experts' attention is quickly captured by familiar configurations within a problem situation (e.g., a familiar configuration of pieces in a chess game). In contrast, novices' attention is focused on isolated components of the problem (e.g., individual chess pieces). This difference, which has been found in numerous domains, indicates that experts have stored in memory many meaningful groups (chunks) of information: for example, chess (Chase & Simon, 1973 ), circuit diagrams (Egan & Schwartz, 1979 ), computer programs (McKeithen, Reitman, Rueter, & Hirtle, 1981 ), medicine (Coughlin & Patel, 1987 ; Myles-Worsley, Johnston, & Simons, 1988 ), basketball and field hockey (Allard & Starkes, 1991 ), and figure skating (Deakin & Allard, 1991 ).

The perceptual configurations that domain experts readily recognize are associated with stored solution plans and/or compiled procedures (Anderson, 1982 ). As a result, experts' solutions are much faster than, and often qualitatively different from, the piecemeal solutions that novice solvers tend to construct (e.g., Larkin et al., 1980 ). In effect, experts often see the solutions that novices have yet to compute (e.g., Chase & Simon, 1973 ; Novick & Sherman, 2003 , 2008 ). These findings have led to the design of various successful instructional interventions (e.g., Catrambone, 1998 ; Kellman et al., 2008 ). For example, Catrambone ( 1998 ) perceptually isolated the subgoals of a statistics problem. This perceptual chunking of meaningful components of the problem prompted novice students to self-explain the meaning of the chunks, leading to a conceptual understanding of the learned solution. In the next section, we describe some recent work that shows the beneficial effects of perceptual pattern recognition on the solution of familiar mathematics problems, as well as the potentially detrimental effects of familiar perceptual chunks to understanding and reasoning with diagrams depicting evolutionary relationships among taxa.

Another difference between experts and novices pertains to their understanding of the solution-relevant problem structure. Experts' knowledge is highly organized around domain principles, and their problem representations tend to reflect this principled understanding. In particular, they can extract the solution-relevant structure of the problems they encounter (e.g., meaningful causal relations among the objects in the problem; see Cheng & Buehner, Chapter 12 ). In contrast, novices' representations tend to be bound to surface features of the problems that may be irrelevant to solution (e.g., the particular objects in a problem). For example, Chi, Feltovich, and Glaser ( 1981 ) examined how students with different levels of physics expertise group mechanics word problems. They found that advanced graduate students grouped the problems based on the physics principles relevant to the problems' solutions (e.g., conservation of energy, Newton's second law). In contrast, undergraduates who had successfully completed an introductory course in mechanics grouped the problems based on the specific objects involved (e.g., pulley problems, inclined plane problems). Other researchers have found similar results in the domains of biology, chemistry, computer programming, and math (Adelson, 1981 ; Kindfield, 1993 / 1994 ; Kozma & Russell, 1997 ; McKeithen et al., 1981 ; Silver, 1979 , 1981 ; Weiser & Shertz, 1983 ).

The level of domain expertise and the corresponding representational differences are, of course, a matter of degree. With increasing expertise, there is a gradual change in people's focus of attention from aspects that are not relevant to solution to those that are (e.g., Deakin & Allard, 1991 ; Hardiman, Dufresne, & Mestre, 1989 ; McKeithen et al., 1981 ; Myles-Worsley et al., 1988 ; Schoenfeld & Herrmann, 1982 ; Silver, 1981 ). Interestingly, Chi, Bassok, Lewis, Reimann, and Glaser ( 1989 ) found similar differences in focus on structural versus surface features among a group of novices who studied worked-out examples of mechanics problems. These differences, which echo Wertheimer's ( 1959 ) observations of individual differences in students' learning about the area of parallelograms, suggest that individual differences in people's interests and natural abilities may affect whether, or how quickly, they acquire domain expertise.

An important benefit of experts' ability to focus their attention on solution-relevant aspects of problems is that they are more likely than novices to recognize analogous problems that involve different objects and cover stories (e.g., Chi et al., 1989 ; Novick, 1988 ; Novick & Holyoak, 1991 ; Wertheimer, 1959 ) or that come from other knowledge domains (e.g., Bassok & Holyoak, 1989 ; Dunbar, 2001 ; Goldstone & Sakamoto, 2003 ). For example, Bassok and Holyoak ( 1989 ) found that, after learning to solve arithmetic-progression problems in algebra, subjects spontaneously applied these algebraic solutions to analogous physics problems that dealt with constantly accelerated motion. Note, however, that experts and good students do not simply ignore the surface features of problems. Rather, as was the case in the problem isomorphs we described earlier (Kotovsky et al., 1985 ), they tend to use such features to infer what the problem's structure could be (e.g., Alibali, Bassok, Solomon, Syc, & Goldin-Meadow, 1999 ; Blessing & Ross, 1996 ). For example, Hinsley et al. ( 1977 ) found that, after reading no more than the first few words of an algebra word problem, expert solvers classified the problem into a likely problem category (e.g., a work problem, a distance problem) and could predict what questions they might be asked and the equations they likely would need to use.

Surface-based problem categorization has a heuristic value (Medin & Ross, 1989 ): It does not ensure a correct categorization (Blessing & Ross, 1996 ), but it does allow solvers to retrieve potentially appropriate solutions from memory and to use them, possibly with some adaptation, to solve a variety of novel problems. Indeed, although experts exploit surface-structure correlations to save cognitive effort, they have the capability to realize that a particular surface cue is misleading (Hegarty, Mayer, & Green, 1992 ; Lewis & Mayer, 1987 ; Martin & Bassok, 2005 ; Novick 1988 , 1995 ; Novick & Holyoak, 1991 ). It is not surprising, therefore, that experts may revert to novice-like heuristic methods when solving problems under pressure (e.g., Beilock, 2008 ) or in subdomains in which they have general but not specific expertise (e.g., Patel, Groen, & Arocha, 1990 ).

Relevance of Search to Insight Solutions

We introduced the notion of insight in our discussion of the nine-dot problem in the section on the Gestalt tradition. The Gestalt view (e.g., Duncker, 1945 ; Maier, 1931 ; see Ohlsson, 1984 , for a review) was that insight problem solving is characterized by an initial work period during which no progress toward solution is made (i.e., an impasse), a sudden restructuring of one's problem representation to a more suitable form, followed immediately by the sudden appearance of the solution. Thus, solving problems by insight was believed to be all about representation, with essentially no role for a step-by-step solution process (i.e., search). Subsequent and contemporary researchers have generally concurred with the Gestalt view that getting the right representation is crucial. However, research has shown that insight solutions do not necessarily arise suddenly or full blown after restructuring (e.g., Weisberg & Alba, 1981 ); and even when they do, the underlying solution process (in this case outside of awareness) may reflect incremental progress toward the goal (Bowden & Jung-Beeman, 2003 ; Durso, Rea, & Dayton, 1994 ; Novick & Sherman, 2003 ).

“Demystifying insight,” to borrow a phrase from Bowden, Jung-Beeman, Fleck, and Kounios ( 2005 ), requires explaining ( 1 ) why solvers initially reach an impasse in solving a problem for which they have the necessary knowledge to generate the solution, ( 2 ) how the restructuring occurred, and ( 3 ) how it led to the solution. A detailed discussion of these topics appears elsewhere in this volume (van Steenburgh et al., Chapter 24 ). Here, we describe briefly three recent theories that have attempted to account for various aspects of these phenomena: Knoblich, Ohlsson, Haider, and Rhenius's ( 1999 ) representational change theory, MacGregor, Ormerod, and Chronicle's ( 2001 ) progress monitoring theory, and Bowden et al.'s ( 2005 ) neurological model. We then propose the need for an integrated approach to demystifying insight that considers both representation and search.

According to Knoblich et al.'s ( 1999 ) representational change theory, problems that are solved with insight are highly likely to evoke initial representations in which solvers place inappropriate constraints on their solution attempts, leading to an impasse. An impasse can be resolved by revising one's representation of the problem. Knoblich and his colleagues tested this theory using Roman numeral matchstick arithmetic problems in which solvers must move one stick to a new location to change a false numerical statement (e.g., I = II + II ) into a statement that is true. According to representational change theory, re-representation may occur through either constraint relaxation or chunk decomposition. (The solution to the example problem is to change II + to III – , which requires both methods of re-representation, yielding I = III – II ). Good support for this theory has been found based on measures of solution rate, solution time, and eye fixation (Knoblich et al., 1999 ; Knoblich, Ohlsson, & Raney, 2001 ; Öllinger, Jones, & Knoblich, 2008 ).

Progress monitoring theory (MacGregor et al., 2001 ) was proposed to account for subjects' difficulty in solving the nine-dot problem, which has traditionally been classified as an insight problem. According to this theory, solvers use the hill-climbing search heuristic to solve this problem, just as they do for traditional search problems (e.g., Hobbits and Orcs). In particular, solvers are hypothesized to monitor their progress toward solution using a criterion generated from the problem's current state. If solvers reach criterion failure, they seek alternative solutions by trying to relax one or more problem constraints. MacGregor et al. found support for this theory using several variants of the nine-dot problem (also see Ormerod, MacGregor, & Chronicle, 2002 ). Jones ( 2003 ) suggested that progress monitoring theory provides an account of the solution process up to the point an impasse is reached and representational change is sought, at which point representational change theory picks up and explains how insight may be achieved. Hence, it appears that a complete account of insight may require an integration of concepts from the Gestalt (representation) and Newell and Simon's (search) legacies.

Bowden et al.'s ( 2005 ) neurological model emphasizes the overlap between problem solving and language comprehension, and it hinges on differential processing in the right and left hemispheres. They proposed that an impasse is reached because initial processing of the problem produces strong activation of information irrelevant to solution in the left hemisphere. At the same time, weak semantic activation of alternative semantic interpretations, critical for solution, occurs in the right hemisphere. Insight arises when the weakly activated concepts reinforce each other, eventually rising above the threshold required for conscious awareness. Several studies of problem solving using compound remote associates problems, involving both behavioral and neuroimaging data, have found support for this model (Bowden & Jung-Beeman, 1998 , 2003 ; Jung-Beeman & Bowden, 2000 ; Jung-Beeman et al., 2004 ; also see Moss, Kotovsky, & Cagan, 2011 ).

Note that these three views of insight have received support using three quite distinct types of problems (Roman numeral matchstick arithmetic problems, the nine-dot problem, and compound remote associates problems, respectively). It remains to be established, therefore, whether these accounts can be generalized across problems. Kershaw and Ohlsson ( 2004 ) argued that insight problems are difficult because the key behavior required for solution may be hindered by perceptual factors (the Gestalt view), background knowledge (so expertise may be important; e.g., see Novick & Sherman, 2003 , 2008 ), and/or process factors (e.g., those affecting search). From this perspective, solving visual problems (e.g., the nine-dot problem) with insight may call upon more general visual processes, whereas solving verbal problems (e.g., anagrams, compound remote associates) with insight may call upon general verbal/semantic processes.

The work we reviewed in this section shows the relevance of problem representation (the Gestalt legacy) to the way people search the problem space (the legacy of Newell and Simon), and the relevance of search to the solution of insight problems that require a representational change. In addition to this inevitable integration of the two legacies, the work we described here underscores the fact that problem solving crucially depends on perceptual factors and on the solvers' background knowledge. In the next section, we describe some recent work that shows the involvement of these factors in the solution of problems in math and science.

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

Although the use of puzzle problems continues in research on problem solving, especially in investigations of insight, many contemporary researchers tackle problem solving in knowledge-rich domains, often in academic disciplines (e.g., mathematics, biology, physics, chemistry, meteorology). In this section, we provide a sampling of this research that highlights the importance of visual perception and background knowledge for successful problem solving.

The Role of Visual Perception

We stated at the outset that a problem representation (e.g., the problem space) is a model of the problem constructed by solvers to summarize their understanding of the problem's essential nature. This informal definition refers to the internal representations people construct and hold in working memory. Of course, people may also construct various external representations (Markman, 1999 ) and even manipulate those representations to aid in solution (see Hegarty & Stull, Chapter 31 ). For example, solvers often use paper and pencil to write notes or draw diagrams, especially when solving problems from formal domains (e.g., Cox, 1999 ; Kindfield, 1993 / 1994 ; S. Schwartz, 1971 ). In problems that provide solvers with external representation, such as the Tower of Hanoi problem, people's planning and memory of the current state is guided by the actual configurations of disks on pegs (Garber & Goldin-Meadow, 2002 ) or by the displays they see on a computer screen (Chen & Holyoak, 2010 ; Patsenko & Altmann, 2010 ).

In STEM (science, technology, engineering, and mathematics) disciplines, it is common for problems to be accompanied by diagrams or other external representations (e.g., equations) to be used in determining the solution. Larkin and Simon ( 1987 ) examined whether isomorphic sentential and diagrammatic representations are interchangeable in terms of facilitating solution. They argued that although the two formats may be equivalent in the sense that all of the information in each format can be inferred from the other format (informational equivalence), the ease or speed of making inferences from the two formats might differ (lack of computational equivalence). Based on their analysis of several problems in physics and math, Larkin and Simon further argued for the general superiority of diagrammatic representations (but see Mayer & Gallini, 1990 , for constraints on this general conclusion).

Novick and Hurley ( 2001 , p. 221) succinctly summarized the reasons for the general superiority of diagrams (especially abstract or schematic diagrams) over verbal representations: They “(a) simplify complex situations by discarding unnecessary details (e.g., Lynch, 1990 ; Winn, 1989 ), (b) make abstract concepts more concrete by mapping them onto spatial layouts with familiar interpretational conventions (e.g., Winn, 1989 ), and (c) substitute easier perceptual inferences for more computationally intensive search processes and sentential deductive inferences (Barwise & Etchemendy, 1991 ; Larkin & Simon, 1987 ).” Despite these benefits of diagrammatic representations, there is an important caveat, noted by Larkin and Simon ( 1987 , p. 99) at the very end of their paper: “Although every diagram supports some easy perceptual inferences, nothing ensures that these inferences must be useful in the problem-solving process.” We will see evidence of this in several of the studies reviewed in this section.

Next we describe recent work on perceptual factors that are involved in people's use of two types of external representations that are provided as part of the problem in two STEM disciplines: equations in algebra and diagrams in evolutionary biology. Although we focus here on effects of perceptual factors per se, it is important to note that such factors only influence performance when subjects have background knowledge that supports differential interpretation of the alternative diagrammatic depictions presented (Hegarty, Canham, & Fabricant, 2010 ).

In the previous section, we described the work of Patsenko and Altmann ( 2010 ) that shows direct involvement of visual attention and perception in the sequential application of move operators during the solution of the Tower of Hanoi problem. A related body of work documents similar effects in tasks that require the interpretation and use of mathematical equations (Goldstone, Landy, & Son, 2010 ; Landy & Goldstone, 2007a , b). For example, Landy and Goldstone ( 2007b ) varied the spatial proximity of arguments to the addition (+) and multiplication (*) operators in algebraic equations, such that the spatial layout of the equation was either consistent or inconsistent with the order-of-operations rule that multiplication precedes addition. In consistent equations , the space was narrower around multiplication than around addition (e.g., g*m + r*w = m*g + w*r ), whereas in inconsistent equations this relative spacing was reversed (e.g., s * n+e * c = n * s+c * e ). Subjects' judgments of the validity of such equations (i.e., whether the expressions on the two sides of the equal sign are equivalent) were significantly faster and more accurate for consistent than inconsistent equations.

In discussing these findings and related work with other external representations, Goldstone et al. ( 2010 ) proposed that experience with solving domain-specific problems leads people to “rig up” their perceptual system such that it allows them to look at the problem in a way that is consistent with the correct rules. Similar logic guides the Perceptual Learning Modules developed by Kellman and his collaborators to help students interpret and use algebraic equations and graphs (Kellman et al., 2008 ; Kellman, Massey, & Son, 2009 ). These authors argued and showed that, consistent with the previously reviewed work on expertise, perceptual training with particular external representations supports the development of perceptual fluency. This fluency, in turn, supports students' subsequent use of these external representations for problem solving.

This research suggests that extensive experience with particular equations or graphs may lead to perceptual fluency that could replace the more mindful application of domain-specific rules. Fisher, Borchert, and Bassok ( 2011 ) reported results from algebraic-modeling tasks that are consistent with this hypothesis. For example, college students were asked to represent verbal statements with algebraic equations, a task that typically elicits systematic errors (e.g., Clement, Lochhead, & Monk, 1981 ). Fisher et al. found that such errors were very common when subjects were asked to construct “standard form” equations ( y = ax ), which support fluent left-to-right translation of words to equations, but were relatively rare when subjects were asked to construct nonstandard division-format equations (x = y/a) that do not afford such translation fluency.

In part because of the left-to-right order in which people process equations, which mirrors the linear order in which they process text, equations have traditionally been viewed as sentential representations. However, Landy and Goldstone ( 2007a ) have proposed that equations also share some properties with diagrammatic displays and that, in fact, in some ways they are processed like diagrams. That is, spatial information is used to represent and to support inferences about syntactic structure. This hypothesis received support from Landy and Goldstone's ( 2007b ) results, described earlier, in which subjects' judgments of the validity of equations were affected by the Gestalt principle of grouping: Subjects did better when the grouping was consistent rather than inconsistent with the underlying structure of the problem (order of operations). Moreover, Landy and Goldstone ( 2007a ) found that when subjects wrote their own equations they grouped numbers and operators (+, *, =) in a way that reflected the hierarchical structure imposed by the order-of-operations rule.

In a recent line of research, Novick and Catley ( 2007 ; Novick, Catley, & Funk, 2010 ; Novick, Shade, & Catley, 2011 ) have examined effects of the spatial layout of diagrams depicting the evolutionary history of a set of taxa on people's ability to reason about patterns of relationship among those taxa. We consider here their work that investigates the role of another Gestalt perceptual principle—good continuation—in guiding students' reasoning. According to this principle, a continuous line is perceived as a single entity (Kellman, 2000 ). Consider the diagrams shown in Figure 21.6 . Each is a cladogram, a diagram that depicts nested sets of taxa that are related in terms of levels of most recent common ancestry. For example, chimpanzees and starfish are more closely related to each other than either is to spiders. The supporting evidence for their close relationship is their most recent common ancestor, which evolved the novel character of having radial cleavage. Spiders do not share this ancestor and thus do not have this character.

Cladograms are typically drawn in two isomorphic formats, which Novick and Catley ( 2007 ) referred to as trees and ladders. Although these formats are informationally equivalent (Larkin & Simon, 1987 ), Novick and Catley's ( 2007 ) research shows that they are not computationally equivalent (Larkin & Simon, 1987 ). Imagine that you are given evolutionary relationships in the ladder format, such as in Figure 21.6a (but without the four characters—hydrostatic skeleton, bilateral symmetry, radial cleavage, and trocophore larvae—and associated short lines indicating their locations on the cladogram), and your task is to translate that diagram to the tree format. A correct translation is shown in Figure 21.6b . Novick and Catley ( 2007 ) found that college students were much more likely to get such problems correct when the presented cladogram was in the nested circles (e.g., Figure 21.6d ) rather than the ladder format. Because the Gestalt principle of good continuation makes the long slanted line at the base of the ladder appear to represent a single hierarchical level, a common translation error for the ladder to tree problems was to draw a diagram such as that shown in Figure 21.6c .

The difficulty that good continuation presents for interpreting relationships depicted in the ladder format extends to answering reasoning questions as well. Novick and Catley (unpublished data) asked comparable questions about relationships depicted in the ladder and tree formats. For example, using the cladograms depicted in Figures 21.6a and 21.6b , consider the following questions: (a) Which taxon—jellyfish or earthworm—is the closest evolutionary relation to starfish, and what evidence supports your answer? (b) Do the bracketed taxa comprise a clade (a set of taxa consisting of the most recent common ancestor and all of its descendants), and what evidence supports your answer? For both such questions, students had higher accuracy and evidence quality composite scores when the relationships were depicted in the tree than the ladder format.

Four cladograms depicting evolutionary relationships among six animal taxa. Cladogram ( a ) is in the ladder format, cladograms ( b ) and ( c ) are in the tree format, and cladogram ( d ) is in the nested circles format. Cladograms ( a ), ( b ), and ( d ) are isomorphic.

If the difficulty in extracting the hierarchical structure of the ladder format is due to good continuation (which leads problem solvers to interpret continuous lines that depict multiple hierarchical levels as depicting only a single level), then a manipulation that breaks good continuation at the points where a new hierarchical level occurs should improve understanding. Novick et al. ( 2010 ) tested this hypothesis using a translation task by manipulating whether characters that are the markers for the most recent common ancestor of each nested set of taxa were included on the ladders. Figure 21.6a shows a ladder with such characters. As predicted, translation accuracy increased dramatically simply by adding these characters to the ladders, despite the additional information subjects had to account for in their translations.

The Role of Background Knowledge

As we mentioned earlier, the specific entities in the problems people encounter evoke inferences that affect how people represent these problems (e.g., the candle problem; Duncker, 1945 ) and how they apply the operators in searching for the solution (e.g., the disks vs. acrobats versions of the Tower of Hanoi problem; Kotovsky et al., 1985 ). Such object-based inferences draw on people's knowledge about the properties of the objects (e.g., a box is a container, an acrobat is a person who can be hurt). Here, we describe the work of Bassok and her colleagues, who found that similar inferences affect how people select mathematical procedures to solve problems in various formal domains. This work shows that the objects in the texts of mathematical word problems affect how people represent the problem situation (i.e., the situation model they construct; Kintsch & Greeno, 1985 ) and, in turn, lead them to select mathematical models that have a corresponding structure. To illustrate, a word problem that describes constant change in the rate at which ice is melting off a glacier evokes a model of continuous change, whereas a word problem that describes constant change in the rate at which ice is delivered to a restaurant evokes a model of discrete change. These distinct situation models lead subjects to select corresponding visual representations (e.g., Bassok & Olseth, 1995 ) and solutions methods, such as calculating the average change over time versus adding the consecutive changes (e.g., Alibali et al., 1999 ).

In a similar manner, people draw on their general knowledge to infer how the objects in a given problem are related to each other and construct mathematical solutions that correspond to these inferred object relations. For example, a word problem that involves doctors from two hospitals elicits a situation model in which the two sets of doctors play symmetric roles (e.g., work with each other), whereas a mathematically isomorphic problem that involves mechanics and cars elicits a situation model in which the sets play asymmetric roles (e.g., mechanics fix cars). The mathematical solutions people construct to such problems reflect this difference in symmetry (Bassok, Wu, & Olseth, 1995 ). In general, people tend to add objects that belong to the same taxonomic category (e.g., doctors + doctors) but divide functionally related objects (e.g., cars ÷ mechanics). People establish this correspondence by a process of analogical alignment between semantic and arithmetic relations, which Bassok and her colleagues refer to as “semantic alignment” (Bassok, Chase, & Martin, 1998 ; Doumas, Bassok, Guthormsen, & Hummel, 2006 ; Fisher, Bassok, & Osterhout, 2010 ).

Semantic alignment occurs very early in the solution process and can prime arithmetic facts that are potentially relevant to the problem solution (Bassok, Pedigo, & Oskarsson, 2008 ). Although such alignments can lead to erroneous solutions, they have a high heuristic value because, in most textbook problems, object relations indeed correspond to analogous mathematical relations (Bassok et al., 1998 ). Interestingly, unlike in the case of reliance on specific surface-structure correlations (e.g., the keyword “more” typically appears in word problems that require addition; Lewis & Mayer, 1987 ), people are more likely to exploit semantic alignment when they have more, rather than less modeling experience. For example, Martin and Bassok ( 2005 ) found very strong semantic-alignment effects when subjects solved simple division word problems, but not when they constructed algebraic equations to represent the relational statements that appeared in the problems. Of course, these subjects had significantly more experience with solving numerical word problems than with constructing algebraic models of relational statements. In a subsequent study, Fisher and Bassok ( 2009 ) found semantic-alignment effects for subjects who constructed correct algebraic models, but not for those who committed modeling errors.

Conclusions and Future Directions

In this chapter, we examined two broad components of the problem-solving process: representation (the Gestalt legacy) and search (the legacy of Newell and Simon). Although many researchers choose to focus their investigation on one or the other of these components, both Duncker ( 1945 ) and Simon ( 1986 ) underscored the necessity to investigate their interaction, as the representation one constructs for a problem determines (or at least constrains) how one goes about trying to generate a solution, and searching the problem space may lead to a change in problem representation. Indeed, Duncker's ( 1945 ) initial account of one subject's solution to the radiation problem was followed up by extensive and experimentally sophisticated work by Simon and his colleagues and by other researchers, documenting the involvement of visual perception and background knowledge in how people represent problems and search for problem solutions.

The relevance of perception and background knowledge to problem solving illustrates the fact that, when people attempt to find or devise ways to reach their goals, they draw on a variety of cognitive resources and engage in a host of cognitive activities. According to Duncker ( 1945 ), such goal-directed activities may include (a) placing objects into categories and making inferences based on category membership, (b) making inductive inferences from multiple instances, (c) reasoning by analogy, (d) identifying the causes of events, (e) deducing logical implications of given information, (f) making legal judgments, and (g) diagnosing medical conditions from historical and laboratory data. As this list suggests, many of the chapters in the present volume describe research that is highly relevant to the understanding of problem-solving behavior. We believe that important advancements in problem-solving research would emerge by integrating it with research in other areas of thinking and reasoning, and that research in these other areas could be similarly advanced by incorporating the insights gained from research on what has more traditionally been identified as problem solving.

As we have described in this chapter, many of the important findings in the field have been established by a careful investigation of various riddle problems. Although there are good methodological reasons for using such problems, many researchers choose to investigate problem solving using ecologically valid educational materials. This choice, which is increasingly common in contemporary research, provides researchers with the opportunity to apply their basic understanding of problem solving to benefit the design of instruction and, at the same time, allows them to gain a better understanding of the processes by which domain knowledge and educational conventions affect the solution process. We believe that the trend of conducting educationally relevant research is likely to continue, and we expect a significant expansion of research on people's understanding and use of dynamic and technologically rich external representations (e.g., Kellman et al., 2008 ; Mayer, Griffith, Jurkowitz, & Rothman, 2008 ; Richland & McDonough, 2010 ; Son & Goldstone, 2009 ). Such investigations are likely to yield both practical and theoretical payoffs.

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7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

• Concepts and inferences (7.1)
• Procedural knowledge (7.1)
• Metacognition (7.1)
• Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
• Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
• Fixation:  functional fixedness, mental set  (7.3)
• Algorithms, heuristics, and the role of confirmation bias (7.3)
• Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

• Identify which type of knowledge a piece of information is (7.1)
• Recognize examples of deductive and inductive reasoning (7.2)
• Recognize judgments that have probably been influenced by the availability heuristic (7.2)
• Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

• Use the principles of critical thinking to evaluate information (7.1)
• Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
• Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

• Take a few minutes to write down everything that you know about dogs.
• Do you believe that:
• Psychic ability exists?
• Hypnosis is an altered state of consciousness?
• Magnet therapy is effective for relieving pain?
• Aerobic exercise is an effective treatment for depression?
• UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

• Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
• Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
• Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

• Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

• What percentage of kidnappings are committed by strangers?
• Which area of the house is riskiest: kitchen, bathroom, or stairs?
• What is the most common cancer in the US?
• What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

• Statement #1: The forecaster on Channel 2 said it is going to rain today.
• Statement #2: The forecaster on Channel 5 said it is going to rain today.
• Statement #3: It is very cloudy and humid.
• Statement #4: You just heard thunder.
• Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

• A particular class will be interesting/useful/difficult
• You will be able to finish writing a paper by next week if you go out tonight
• Your laptop’s battery will last through the next trip to the library
• You will not miss anything important if you skip class tomorrow
• Your instructor will not notice if you skip class tomorrow
• You will be able to find a book that you will need for a paper
• There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

• What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

• Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

• Please take a few minutes to list a number of problems that you are facing right now.
• Now write about a problem that you recently solved.
• What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

• Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
• Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
• Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
• Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
• Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
• Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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