Johnson-Laird on Reasoning and Logic

Rules of Inference and Reasoning

Despite believing reasoning is much broader, Johnson-Laird (2006, p. 3) follows the trend in psychological research on reasoning that makes reasoning synonymous with inference and deduction with valid inferences. An inference can be analysed into input, premises, and output, conclusions, and a rule of inference governing the transition from premises to conclusions. Inferences are then valid if the transition from premises to conclusions occurs according to the rules of inference; deductive inferences, on the other hand, considered paradigmatic of rational thought, are those whose conclusions are necessarily true if the premises are true (Hintikka and Sandu 2007, sec. 1).

A preliminary characterisation of logic is the study of inferences (Hintikka and Sandu 2007, sec. 1); however, logic is also characterised as the study of logical truths. Logical truths differ from ordinary truth. Ordinarily, a sentence is true if a constellation of facts verifies the proposition it expresses; logical truths, on the other hand, are not contingent on any particular constellation of facts. They are true in all possible worlds not just the actual world like ordinary truths (Hintikka and Sandu 2007, sec. 5).

Regardless of whether rules of inference or logical truth correctly characterise it, deduction is a matter of form in classical logic. Particular deductive inferences are valid if they instantiate the form of an argument deemed valid. The form itself is independent of the content involved in particular inferences; it is a relationship existing solely between the logical constants—e.g. negation, ‘if’, ‘and’, ‘or’, as well as first-order quantifiers—involved and not the particulars related by them (Read 1995, pp. 35–6). Perhaps because it realised this ideal to a large extent, first-order logic is still the paragon of logic despite logicians accepting a plurality of logics today (Restall and Beall 2000, 2001; Russell 2019). Thus logic drives an ‘irremovable wedge between form and content’ according to Johnson-Laird (2006, p. 10).

Philosophers, logicians, and psychologists alike have equated the formal rules of logic with laws of thought (e.g., George 1997; Posy 1997). However, logically, infinitely many valid conclusions can be deduced from any set of premises; given premises ‘Ali was the greatest boxer’ and ‘Today is Wednesday’, for example, ‘Ali was the greatest boxer and today is Wednesday’; ‘Ali was the greatest boxer and today is Wednesday and Ali was the greatest boxer’; ‘Ali was the greatest boxer and today is Wednesday and Ali was the greatest boxer and today is Wednesday’; etc. (Johnson-Laird 2006, p. 11) can be deduced. However, when we reason, all but a few of the logically possible conclusions are actually considered (Johnson-Laird 2006, p. 4). Using the surface information available, the information, that is, that can be read off without logical means, deductive inferences raise depth information to the surface. In other words, individuals employ deduction in conjunction with surface information to reveal depth information implicit in premises. As the example above shows, logically possible conclusions, on the other hand, do not necessarily reveal depth information the reasoner is not yet aware of. According to Johnson-Laird (2006, p. 11), ‘parsimony’ distinguishes actual reasoning from logic.

On the other hand, we make deductive inferences like ‘Nothing is both round and square (at the same time)’ when we reason. The inference is clearly valid, but its validity cannot be a matter of form since the logical constants ‘Nothing is both … and …’ remain after eliminating the particular content, and it is easy to substitute non-logical terms into the form to generate a false proposition: ‘Nothing is both a terrestrial and an aquatic animal’, for example, is clearly false since amphibians are both (Read 1995, pp. 49–50). These deductive inferences are valid by virtue of content, the meaning of the proposition, that is, constituted by the meanings of the non-logical terms substituted into the form. Collectively, such inferences are known as analytic inferences, and they are distinguished from deductive inferences based solely on logical constants. Deductive logic in the strict sense is usually confined to the latter (Hintikka and Sandu 2007, pp. 14–5; Read 1995, pp. 52–3).

Although logic is the study of inferences, the preceding paragraphs indicate a mismatch between logical inferences and how we actually reason when inferring deductively. Moreover, the mismatch is due to content and not form alone playing a role in deduction. The full extent of the role content plays in reasoning dawned on Johnson-Laird after experimental anomalies forced him to reconsider explanations of reasoning in terms of rules of inference. The anomalies occurred in the so-called ‘selection’ task conceived by Peter Wason (1966). In it, test persons are required to determine the truth or falsity of the statement ‘If a card has an “A” on one side, then it has a “2” on its other side’ by selecting evidence from four cards laid out in front of them. A, B, 2, and 3 are on display on the cards laid out, and the test persons know that each card has a letter on one side and a number on the other. The outcome was that test persons tend to select the A-card and perhaps the 2-card but failed to recognise the significance of the 3-card. From a logical point of view, the omission is puzzling since the 3-card is the only selection besides the A-card that could falsify the statement. Initially, the anomaly did not shake the rule-based, mental-logic theory of reasoning; however, the error of omission persisted stubbornly, despite attempts to eliminate possible sources of confusion. Eventually, Wason suggested changing the content of the general claim, a heretical suggestion according to the view that laws of thought are formal rules of inference.

In order to test his hypothesis, Wason conducted the same experiment but this time the test persons were asked to determine whether ‘Every time I go to Manchester I travel by train’ is true or false. Despite the statement having the same logical form, the test persons recognised the relevance of the car-card, the equivalent to the 3-card. Johnson-Laird expresses the import of these findings as follows:

A change in content alone had a striking effect on reasoning, even though the two sorts of contents had the same logical form. These findings were embarrassing for the formal theory. On the one hand, the systematic error of omission with the letters and numbers was contrary to the laws of thought embodied in formal rules. On the other hand, a change in content alone should have no effect on performance, because by definition formal rules are blind to content (Johnson-Laird 2006, p. 15).

Johnson-Laird concluded, ‘[l]ogic is an essential tool for all sciences, but it is not a psychological theory of reasoning’ (Johnson-Laird 2006, p. 17). Having rejected theories of reasoning that see in the laws of thought nothing but the formal rules of logic, he conceived an alternative psychological theory of reasoning.

The Mental-Model Theory of Reasoning

Johnson-Laird (e.g., 2001, 2006, 2010; Johnson-Laird et al. 1992) proposed a mental-model theory of reasoning. A mental model is not a visual image, but like visual images it is iconic. In other words, the structure of the model corresponds to the structure of what it represents. Unlike an image, on the other hand, it is an abstraction, underlying images and representing content, even content that cannot be visualised (Johnson-Laird 2006, p. 418).

Reasoning is based on mental models. Each mental model represents a possibility in as iconic a manner as possible, and the model theory does not simply claim that we often think of possibilities when we reason and take them into consideration in our deliberations; more controversially, it maintains that consideration of possibilities is fundamental to the way we think (Johnson-Laird 2006, pp. 38–40). Experimental evidence corroborates this claim since the more possibilities reasoners have to take into account the more time they need to draw conclusions and the more mistakes they make (Johnson-Laird 2006, p. 47). Moreover, the mistakes made are largely erroneous conclusions based on some possibilities while overlooking others (Johnson-Laird 2006, p. 45).

Johnson-Laird (2006, p. 417) was led to the theory by our reliance on perception, the meaning of words and sentences, the significance of the propositions that they express, and our knowledge when reasoning. If reasoning were based solely on logical forms, the use, let alone reliance, on the content of arguments would be inexplicable. Consideration of content is, on the other hand, consistent with elaborating a set of possibilities compatible with a given state of affairs.

The principle of truth is a fundamental tenet of the mental-model theory (Johnson-Laird 2006, p. 112). Although reasoning is based on the consideration of possibilities, our capacity to deal with them is surprisingly limited. Computational power lies in the capacity to hold the results of intermediate computations in memory; however, our working memory is very limited. In fact, holding more than one possibility in mind at any time already causes substantial mental strain. For Johnson-Laird (2006, pp. 40–1), working memory is therefore a bottle neck in intelligence, and, in order to ease the flow, mental models only represent a simple proposition contained in the premises when it is true. However, mental notes are made as reminders that not all possibilities have been represented explicitly by this heuristic (Johnson-Laird 2006, p. 113).

A consequence of the principle of truth is that mental models can diverge, sometimes radically, from complete models. A complete model of a propositional connective elaborates all possibilities compatible with the truth of the complex proposition formed whereas the mental model only represents a subset of them explicitly, namely those possibilities in which the elementary propositions connected are true, while making a mental note that other possibilities are implicit. For conjunctions, such as ‘Theresa May is the PM of the UK (P) and the UK is in the European Union (U)’, the complete model P∧U coincides with the mental model P∧U. However, the mental model of the conditional ‘If MPs vote for the Brexit agreement (B), then Theresa May will resign as Prime Minister (R)’ only represents B∧R of the three possibilities B∧R,  ~B∧R, and ~B∧~R comprising the complete model (see Table 1) (Johnson-Laird 2006, p. 115). Despite mental notes being made that additional possibilities are to be considered, mental and complete models diverge in practice and their systematic divergence explains the occurrence of common errors in reasoning. In fact, some errors are so compelling from the point of view of mental models that Johnson-Laird has dubbed them ‘“illusory” inferences’ (Johnson-Laird 2006, p. 117).

Table 1: Mental models and complete models of conjunctions and conditionals. The columns of the table contrast the mental model and the complete model of two propositional connectives. The ellipsis stands for the possibilities omitted in the mental model due to the principle of truth but whose existence has been noted mentally (Johnson-Laird 2006, p. 115 Box 8.3).

According to the model theory, reasoners construct mental models of premises and draw conclusions based on them; however, premises can have several models. Johnson-Laird (2006, p. 44) calls a single model compatible with the premises ‘an example.’ For premises with several models, a single example is therefore enough to demonstrate that a conclusion is possible. However, inferences are valid if given true premises the conclusion cannot be false; to establish the validity of a conclusion, then, all models of the premises must be examples. Conversely, a single model that is not an example refutes the validity of an inference; however, to refute the possibility of a conclusion consistent with the premises no models may be examples. The model theory, therefore, reflects what Johnson-Laird holds to be the foundation of human rationality: ‘an inference is valid if its conclusion holds in all possibilities compatible with the premises, and it is invalid if there is a possibility compatible with the premises but not with the conclusion’ (Johnson-Laird 2006, p. 112). The latter is a counterexample, and the search for counterexamples is an integral part of the model theory.

We grasp the force of counterexamples in rational argumentation even without training, according to (Johnson-Laird 2006, p. 5); however, there do not appear to be any fixed procedures for finding counterexamples. Intellectual ability is one factor affecting their use since only reasoners with a modicum of competence search for them. But our ability to imagine counterexamples is also affected by experience and whether we draw the conclusion ourselves or are judging the inferences of others. Another factor is whether an invalid conclusion is consistent or not with the premises. For conclusions inconsistent with their premises, contradiction with the premises is usually sought; attempts to find counterexamples, on the other hand, are made for conclusions that are consistent with their premises (Johnson-Laird 2006, Chapter 16).

Mental models are not only at the heart of propositional reasoning; they also play an important role in reasoning on the innards of propositions. Since mental models are based on the meaning of the premises, they take context and knowledge into account and embody such information in iconic representations. (Johnson-Laird 2006, pp. 134–5) Implicit relations are therefore embodied in the iconic representations, and novel relational inferences emerge on the basis of this content. Although the method used in syllogistic reasoning is controversial (Johnson-Laird 2006, p. 149), mental models also facilitate reasoning with properties, even with non-standard quantifiers that are difficult to capture with formal rules (Johnson-Laird 2006, pp. 136–7).

In summary, the model theory explains how we reason in terms of mental models, and, by virtue of the principle of truth, it also accounts for the typical errors reasoners often make when reasoning. In opposition to psychological theories of reasoning based on formal rules of inference, the content of the inferences plays a seminal role in elaborating possibilities in the mental-model theory.

References

Carl J. Posy. (1997). Between Leibniz and Mill: Kant’s Logic and the Rhetoric of Psychologism. Philosophy & Rhetoric, (3), 243.

George, R. (1997). Psychologism in Logic: Bacon to Bolzano. Philosophy & Rhetoric, 30(3), 213–242.

Hintikka, J., & Sandu, G. (2007). What is Logic? In D. Jacquette (Ed.), Philosophy of Logic (pp. 13–40). Netherlands: North Holland.

Jaakko, H., & Sandu, G. (2007). What is Logic? In D. Jacquette (Ed.), Philosophy of Logic (pp. 13–40). Netherlands: North Holland.

Johnson-Laird, P. N. (2001). Mental models and deduction. Trends in cognitive sciences, 5(10), 434–442.

Johnson-Laird, P. N. (2006). How We Reason. USA: Oxford University Press.

Johnson-Laird, P. N. (2010). Mental models and human reasoning. Proceedings of the National Academy of Sciences of the United States of America, 107(43), 18243–18250. https://doi.org/10.1073/pnas.1012933107

Johnson-Laird, P. N., Byrne, R. M. J., & Schaeken, W. (1992). Propositional Reasoning by Model. Psychological Review, 99(3), 418–439.

Posy, C. J. (1997). Between Leibniz and Mill: Kant’s Logic and the Rhetoric of Psychologism. Philosophy & Rhetoric, (3), 243.

Read, S. (1995). Thinking About Logic: An Introduction to the Philosophy of Logic. New York: Oxford University Press.

Restall, G., & Beall, J. C. (2000). Logical pluralism. Australasian journal of philosophy, 78(4), 475–493. https://doi.org/10.1080/00048400012349751

Restall, G., & Beall, J. C. (2001). Defending Logical Pluralism. In Proceedings of the 1999 Conference of the Society of Exact Philosophy (pp. 1–22). Presented at the Logical Consequence: Rival Approaches, Stanmore: Hermes Science Publishers.

Russell, G. (2019). Logical Pluralism. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2019.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/sum2019/entries/logical-pluralism/. Accessed 31 January 2020

Wason, P. C. (1966). Reasoning. In B. M. Foss (Ed.), New Horizons in Psychology (pp. 135–151). Harmondsworth, Middx.: Penguin Books.