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Chapter 3.
Psychological Theories of Syllogistic Reasoning
Sangeet Khemlani
Summary Psychologists have studied syllogistic inferences for more than a century, because they can serve as a microcosm of human rationality. “Syllogisms” is a term that refers to a set of 64 reasoning arguments, each of which is comprised of two premises, such as: “All of the designers are women. Some of the women are not employees. What, if anything, follows?” People make systematic mistakes on such problems, and they appear to reason using different strategies. A meta-analysis showed that many existing theories fail to explain such patterns. To address the limitations of previous accounts, two recent theories synthesized both heuristic and deliberative processing. This chapter reviews both accounts and addresses their strengths. It concludes by arguing that if syllogistic reasoning serves as a sensible microcosm of rationality, the synthesized theories may
provide directions on how to resolve broader conflicts that vex psychologists of reasoning and human thinking.
1. Introduction
In 1908, the German scholar Gustav Störring published a 130-page manuscript detailing the results of the first known experiments on human reasoning. His main purpose in conducting them was to develop solutions to long-standing debates between logicians and philosophers, such as what people imagine when they reason. The studies worked like this: volunteers entered a dark room alone, sat down, and received a battery of deductive reasoning problems called “syllogisms,” one after another. Störring recorded his observations of their verbal responses, reaction times, eye movements, gestures, and even their breathing patterns (Störring, 1908). The research would likely be rejected were it to be submitted to any contemporary psychology journal. For one thing, Störring investigated only four participants. For another, he used an arbitrary experimental design, and he failed to present any quantitative analysis of their behaviors except for a single table that listed averaged reaction times. But what Störring learned from his research was remarkable (see Clark, 1922; Knauff, 2013; Politzer, 2004). He noticed, for instance, that his volunteers were biased by the structure of the different reasoning
and colleagues ran a neuroimaging experiment in which they gave participants syllogistic reasoning problems with and without meaningful contents to discover that certain brain regions—such as temporal and frontal regions—systematically respond to semantic information (Goel, Buchel, Frith, & Dolan, 2000). Perhaps syllogisms serve as an attractive microcosm of thinking behavior because of their simplicity and that there is only a finite number of them. Classical syllogisms, i.e., those investigated by Aristotle and Scholastic logicians, are reasoning arguments comprised of multiple premises, such as (1) All of the women are designers. Some of the employees are not women. What, if anything, follows? These syllogisms contain a quantified noun phrase, such as “all of the women,” and these quantifiers can be in one of four separate moods , i.e., expressions comprised of quantifiers and negations, as shown below: All of the a are b. (A ab ) None of the a is b. (E ab ) Some of the a are b. (I ab ) Some of the a are not b. (O ab )
The parentheses indicate the abbreviation conventions adopted by Scholastic logicians, i.e., the 12th-century university scholars who gained access to Aristotle’s works. Contemporary psychologists adopted those conventions, and we retain them here. Since syllogisms consist of two premises, the terms in the premises (e.g., “women,” “designers,” “employees”) can occur in four different arrangements. These different arrangements are known as figures : Figure 1 Figure 2 Figure 3 Figure 4 a – b b – a a – b b – a b – c c – b c – b b – c Other psychologists use different numbering systems for figures—and they sometimes include the conclusion as part of their numbering systems. Here we state the figures in terms of the premises only. In sum, syllogisms concern 64 separate reasoning problems (4 moods of the first premise × 4 moods of the second premise × 4 separate figures). Many experiments on syllogisms focus on only these 64 problems, i.e., they provide participants with the pairs of premises and then ask them to infer what follows from them. Typically, reasoners do not consider all the possible valid and invalid responses when they generate conclusions; they tend to describe just one or two. But across the problems overall, their conclusions can be classified into
question, psychologists have run many studies on which inferences people conclude from the 64 syllogisms. Khemlani and Johnson-Laird (2012) compiled six of them together in a meta-analysis, which shows that the most common response to ( 2 ) is “Some of the designers are undergraduates” (see Figure 1 , left panel, row I ab A bc ) and the most common response to (3) is “None of the designers are undergraduates” (see Figure 1, left panel, row E ab A bc )—the response is an error, since (3) does not rule out the possibility that some of the designers are undergraduates.
Figure 1: The percentages of responses to 64 syllogisms in the meta-analysis in Khemlani & Johnson-Laird (2012). Each of the 64 pairs of premises occurs in a row, and each of the possible responses occurs in a column. Abbreviations for premises are as follows: A ac = All of the A are C , I ac = Some of the A are C , E ac = None of the A is a C , O ac = Some of the A are not C , and NVC = No valid conclusion. The left panel denotes the 27 syllogisms with a valid definite conclusion and the right panel denotes the 37 syllogisms without a valid definite conclusion. The grey-scale in each cell indicates the proportion of corresponding conclusions (black = 100 % and white = 16 % or below). Hence, for the top-most valid syllogism, A ab A bc , nearly 100 % of participants in the meta-analysis responded that A ac follows. Oba Obc Oab Ocb Oba Ocb Oab Obc Oba Ebc Oab Ecb Oba Ecb Oab Ebc Oba Ibc Oab Icb Oba Icb Oab Ibc Oba Acb Oab Abc Eba Obc Eab Ocb Eba Ocb Eab Obc Eba Ebc Eab Ecb Eba Ecb Eab Ebc Iba Obc Iab Ocb Iba Ocb Iab Obc Iba Ibc Iab Icb Iba Icb Iab Ibc Iab Acb Iba Acb Aba Ocb Aab Obc Aab Icb Aab Ibc Aab Acb Oba Abc Oab Acb Eba Ibc Eab Icb Eba Icb Eab Ibc Eba Abc Eab Acb Eba Acb Eab Abc Iba Ebc Iab Ecb Iba Ecb Iab Ebc Iba Abc Iab Abc Aba Obc Aab Ocb Aba Ebc Aab Ecb Aba Ecb Aab Ebc Aba Ibc Aba Icb Aba Abc Aba Acb Aab Abc Aac Iac Eac Oa Aca Ica Eca Oca NV Aac Iac Eac Oa Aca Ica Eca Oca NV Aac Iac Eac Oa Aca Ica Eca Oca NV Ica Eca Oca NVC alysis Aac Iac Eac Oac Aca Ica Eca Oca NVC mReasoner (r = 0.82) Aac Iac Eac Oac Aca Ica Eca Oca NVC Johnson-Laird & Steedman (1978) Oba Obc Oab Ocb Oba Ocb Oab Obc Oba Ebc Oab Ecb Oba Ecb Oab Ebc Oba Ibc Oab Icb Oba Icb Oab Ibc Oba Acb Oab Abc Eba Obc Eab Ocb Eba Ocb Eab Obc Eba Ebc Eab Ecb Eba Ecb Eab Ebc Iba Obc Iab Ocb Iba Ocb Iab Obc Iba Ibc Iab Icb Iba Icb Iab Ibc Iab Acb Iba Acb Aba Ocb Aab Obc Aab Icb Aab Ibc Aab Acb Oba Abc Oab Acb Eba Ibc Eab Icb Eba Icb Eab Ibc Eba Abc Eab Acb Eba Acb Eab Abc Iba Ebc Iab Ecb Iba Ecb Iab Ebc Aac Iac Eac Oac Aca Ica Eca Oca NVC Meta-analysis Aac Iac Eac Oac Aca Ica Eca Oca NVC mReasoner (r = 0.82) Aac J S Oba Obc Oab Ocb Oba Ocb Oab Obc Oba Ebc Oab Ecb Oba Ecb Oab Ebc Oba Ibc Oab Icb Oba Icb Oab Ibc Oba Acb Oab Abc Eba Obc Eab Ocb Eba Ocb Eab Obc Eba Ebc Eab Ecb Eba Ecb Eab Ebc Iba Obc Iab Ocb Iba Ocb Iab Obc Iba Ibc Iab Icb Iba Icb Iab Ibc Iab Acb Iba Acb Aba Ocb Aab Obc Aab Icb Aab Ibc Aab Acb Oba Abc Oab Acb Eba Ibc Eab Icb Eba Icb Eab Ibc Eba Abc Eab Acb Eba Acb Eab Abc Iba Ebc Iab Ecb Iba Ecb Aac Iac Eac Oac Aca Ica Eca Oca NVC Meta-analysis Aac Iac Eac Oac Aca Ica Eca Oca NVC mReasoner (r = 0.82) Aac Iac Eac Oac Aca Ica Eca Oca NVC Johnson-Laird &
Valid syllogisms Invalid syllogisms^ Steedman (1978)
(see Chapter 2.3 by Johnson-Laird, in this volume). The result sparked a fascination with the extent to which human reasoning could be characterized as rational (see Chapter 1.2 by Evans and Chapter 3.1 by Steinberger, both in this volume), and theorists began to devise accounts of the phenomena underlying syllogistic reasoning. The use of computational and formal tools helped some researchers implement psychological theories of the syllogism and test them against human data. As a result, after decades of research, nearly a dozen theories of the phenomenon had been proposed, and there existed a dire need to sort out the different theoretical proposals. Khemlani and Johnson-Laird (2012) surveyed existing psychological accounts of syllogistic reasoning to discover broad trends between them. The survey suggested that theories tended to fall into one of three groups: one group of theories explained syllogistic reasoning by appealing to sets of heuristics in how quantified statements were processed (e.g., Begg & Denny, 1969; Chater & Oaksford, 1999; Revlis, 1975; Wetherick & Gilhooly, 1995). For example, the so-called “matching” strategy (Wetherick & Gilhooly, 1995) posited that for syllogisms such as (4) Some of the designers are women. Some of the women are employees. What, if anything, follows?
people should conclude—erroneously—that “some of the designers are employees.” The reason is because the conclusion matches the mood of the most “conservative” premise, i.e., the premise that presupposes the existence of the fewest entities. And indeed, reasoners draw the predicted conclusion 61 % of the time (see Figure 1, right panel, row I ab I bc ). But about a third of the time they also accurately infer that “No valid conclusion” follows, and accounts based on heuristics have difficulty explaining the deliberative processes by which reasoners correct their mistakes (see also Ragni, Dames, Brand, & Riesterer, 2019). In order to account for deliberative reasoning, another group of psychological theories proposed that reasoners mentally simulate the situation described in the premises when they reason about syllogisms (Bucciarelli & Johnson-Laird, 1999; Guyote & Sternberg, 1981; Johnson-Laird & Steedman, 1978; Polk & Newell, 1995). The theories posited that mental simulations help explain both errors and correct responses: reasoners construct, and can make inferences from, initial simulations, but difficult syllogisms demand reasoners to consider alternative simulations (Johnson-Laird, 1983). A third group of theories assumed that syllogistic reasoning depends on mental proofs and rules of inference akin to those in formal logic (see, e.g., Braine & Rumain, 1983; Geurts, 2003; Politzer, 2007; Rips, 1994)—but such theories have systematic difficulty explaining how reasoners draw the conclusion that nothing follows from a set of premises, and so we presently address only the first two groups of theories.
distinct, inter-reliant processes (Johnson-Laird & Wason, 1970; Wason & Evans, 1974). As Evans (200 8 , p. 263 ) notes, the dichotomy between heuristics and deliberation is closely related to dual processes because heuristics are thought to be a fast, shallow form of processing and deliberation is thought to be a slower, deeper form of processing. In practice, heuristics and deliberative thinking often occur sequentially, i.e., a heuristic response is proposed and a deliberative process validates or falsifies it. More general accounts of dual processing are not committed to sequential processing—they permit that fast processes and slow processes can operate in parallel and interact with one another. The introduction noted that previous theories of syllogistic reasoning tended to account for one type of process over the other. It may be that previous theories were easier to formulate because it is difficult to anticipate the interactive effects of two interdependent processes. Yet, if it is indeed the case that human thinking depends on two inextricable processes, those theories were doomed to fail. Two recent theories of syllogistic reasoning are unique in that they seek to model interactive processing. Both theories are built around the integrative idea that fast, heuristic processing is the result of a biased sampling procedure that can be formalized using probabilistic constraints, and that slower, deliberative processing suggests that reasoning depends on representations referred to as “mental models.” The theory that people construct mental models when they reason originates from Johnson-Laird (1983; see also Chapter 2.3 by Johnson-
Laird, in this volume), who computationally developed earlier proposals that people build “small-scale models” of reality to anticipate events (Craik, 1943). Johnson-Laird’s “model theory” posits that each mental model represents a distinct possibility or situation in the world. In other words, when reasoners draw inferences from syllogisms, they mentally simulate the situation referred to by the premises. The model theory predicts that problems which require reasoners to consider multiple mental models should be more difficult relative to those that require fewer models. As a result, models help explain reasoning difficulty in many domains (see, e.g., Johnson-Laird & Khemlani, 201 3 ; Khemlani & Johnson-Laird, 2017). But, as Khemlani and Johnson-Laird (2012) show, previous implementations of the model theory tend to make overly liberal predictions of the kinds of syllogistic inferences people are likely to draw. Hence, the two latest theories of syllogistic reasoning add additional constraints that explain why reasoners are reticent to draw overly liberal conclusions from model- based representations. We describe each theory in turn.
2.1 The Probability Sampling Model
Masasi Hattori developed a recent account of syllogistic reasoning called the “probability sampling model” (PSM; Hattori, 2016). The account holds that reasoners interpret a set of syllogistic premises by constructing a prototypical representation of them (referred to as a “probability prototype model”). The
designer woman ¬ designer woman designer woman ¬ designer woman Each row of the diagram depicts the results of a random draw from the prototype model, and so the first row depicts a designer who is also a woman. The “¬” denotes the symbol for negation, and so the second row depicts a woman who is not a designer. The additional rows depict additional random draws from the prototype model. In the PSM, the establishment of a sample mental model is the central representation on which a unitary reasoning process operates. The algorithm works by applying a series of tests, one after another, to the sample mental model in order to generate a conclusion. Figure 2 provides a schematic of how the full theory works. To test the theory’s predictions, Hattori implemented the theory computationally and then ran simulations that compared the theory’s predictions against eight separate datasets on syllogistic reasoning. He also compared the PSM’s predictions against two other theories, i.e., Chater and Oaksford’s probability heuristics model (Chater & Oaksford, 1999) and a parameterized version of mental model theory (Hattori’s implementation of Johnson-Laird &
Bara, 1984). His analyses show that the PSM matches the performance of both theories (Hattori, 2016, p. 308).
One major strength of the PSM is that it can explain how contents affect the kinds of inferences reasoners draw. For instance, consider the following problem: ( 5 ) Some of the Frenchmen are wine-drinkers. Some of the wine-drinkers are Italians. What, if anything, follows? A robust result from studies of syllogistic reasoning is that reasoners should be less apt to infer “Some of the Frenchmen are Italians,” since they likely consider it either implausible or rare that a person is both French and Italian at the same time. Hattori’s PSM can account for the effect as follows: the meanings of the premises bias the way people construct the probability prototype model such that implausible areas of the representation are assigned low probabilities. Hattori implemented the process, and he described two studies that corroborate the predictions of his implementation (Hattori, 2016). However, Hattori’s theory is not without its limitations. One limitation is that the theory cannot explain how reasoners might draw inferences from more than two premises: in these cases, the two-dimensional probability sampling model may be unable to represent all of the possible different individuals. Another limitation is that under the PSM, deliberation occurs by applying a set of
logical tests, one after another, to a sample mental model. The process is akin to the way heuristic processing operates in previous theories (e.g., Chater & Oaksford, 1999). No evidence at present suggests that reasoners apply such tests in a fixed, systematic order, for each and every syllogism; indeed, the evidence suggests that reasoners tend to develop strategies over the course of a study on syllogisms, which would seem to conflict with the PSM. Moreover, running such tests in a fixed order is bound to produce a conclusion for any sample mental model—and so a clear consequence of the tests is that the PSM cannot account for why reasoners often spontaneously respond that “No valid conclusion” follows (Ragni, Dames, Brand, & Riesterer, 2019). Other accounts based on sampling and constructing mental models suffer from similar issues (see, e.g., Tessler & Goodman, 2014). In general, it is too taxing for people to constantly and repeatedly apply the same set of tests to the representations they construct, and so the algorithm, though tractable, is limited and not cognitively plausible. In essence, the PSM does not explain the psychological processes that underlie how reasoners deliberate on syllogistic inferences. The primary theoretical contribution of Hattori’s (2016) probabilistic sampling model is that it integrates probabilistic machinery and sampling procedures with the construction of mental models. Another recent computational theory—mReasoner—likewise integrates probabilistic sampling with procedures with the construction of mental models.
