Mental models and deduction

According to the mental-model theory of deductive reasoning, reasoners use the meanings of assertions together with general knowledge to construct mental models of the possibilities compatible with the premises. Each model represents what is true in a possibility. A conclusion is held to be valid if it holds in all the models of the premises. Recent evidence described here shows that the fewer models an inference calls for, the easier the inference is. Errors arise because reasoners fail to consider all possible models, and because models do not normally represent what is false, even though reasoners can construct counterexamples to refute invalid conclusions.     

Naive probability: a mental model theory of extensional reasoning

This article outlines a theory of naive probability. According to the theory, individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an extensional way: They construct mental models of what is true in the various possibilities. Each model represents an equiprobable alternative unless individuals have beliefs to the contrary, in which case some models will have higher probabilities than others. The probability of an event depends on the proportion of models in which it occurs. The theory predicts several phenomena of reasoning about absolute probabilities, including typical biases. It correctly predicts certain cognitive illusions in inferences about relative probabilities. It accommodates reasoning based on numerical premises, and it explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem. Finally, it dispels some common misconceptions of probabilistic reasoning.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

Disjunctive illusory inferences and how to eliminate them

The mental model theory of reasoning postulates that individuals construct mental models of the possibilities in which the premises of an inference hold and that these models represent what is true but not what is false. An unexpected consequence of this assumption is that certain premises should yield systematically invalid inferences. This prediction is unique among current theories of reasoning, because no alternative theory, whether based on formal rules of inference or on probabilistic considerations, predicts these illusory inferences. We report three studies of novel illusory inferences that depend on embedded disjunctions—for example, premises of this sort: A or else (B or else C). The theory distinguishes between those embedded disjunctions that should yield illusions and those that should not. In Experiment 1, we corroborated this distinction. In Experiment 2, we extended the illusory inferences to a more stringently controlled set of problems. In Experiment 3, we established a novel method for reducing illusions by calling for participants to make auxiliary inferences.     

Illusory inferences: a novel class of erroneous deductions.

The mental model theory postulates that reasoners build models of the situations described in premises, and that these models normally make explicit only what is true. The theory has an unexpected consequence: it predicts the occurrence of inferences that are compelling but invalid. They should arise from reasoners failing to take into account what is false. Three experiments corroborated the systematic occurrence of these illusory inferences, and eliminated a number of alternative explanations for them. Their results illuminate the controversy among various current theories of reasoning.     

Transitive and pseudo-transitive inferences

Given that A is longer than B, and that B is longer than C, even 5-year-old children can infer that A is longer than C. Theories of reasoning based on formal rules of inference invoke simple axioms ("meaning postulates") to capture such transitive inferences. An alternative theory proposes instead that reasoners construct mental models of the situation described by the premises in order to draw such inferences. An unexpected consequence of the model theory is that if adult reasoners construct simple models of typical situations, then they should infer transitive relations where, in certain cases, none exists. We report four studies corroborating the occurrence of these "pseudo-transitive" fallacies. Experiment 1 established that individuals' diagrams of certain non-transitive relations yield transitive conclusions. Experiment 2 showed that these premises also give rise to fallacious transitive inferences. Experiment 3 established that when the context suggested alternatives to the simple models, the participants made fewer errors. Experiment 4 showed that tense is an important aspect of meaning which affects whether individuals draw transitive conclusions. We discuss the implications of these results for various theories of reasoning.     

The psychological puzzle of Sudoku

Sudoku puzzles, which are popular worldwide, require individuals to infer the missing digits in a 9 × 9 array according to the general rule that every digit from 1 to 9 must occur once in each row, in each column, and in each of the 3-by-3 boxes in the array. We present a theory of how individuals solve these puzzles. It postulates that they rely solely on pure deductions, and that they spontaneously acquire various deductive tactics, which differ in their difficulty depending on their “relational complexity”, i.e., the number of constraints on which they depend. A major strategic shift is necessary to acquire tactics for more difficult puzzles: solvers have to keep track of possible digits in a cell. We report three experiments corroborating this theory. We also discuss their implications for theories of reasoning that downplay the role of deduction in everyday reasoning.     

Systems of modal logic for impossible worlds

The intuitive notion behind the usual semantics of most systems of modal logic is that of ‘possible worlds’. Loosely speaking, an expression is necessary if and only if it holds in all possible worlds; it is possible if and only if it holds in some possible world. Of course, contradictory expressions turn out to hold in no possible worlds, and logically true expressions turn out to hold in every possible world. A method is presented for transforming standard modal systems into systems of modal logic for impossible worlds. To each possible world there corresponds an impossible world such that an expression holds in the impossible world if and only if it does not hold in the possible world. One can then talk about such worlds quite consistently, and there seems to be no logical reason for excluding them from consideration.     

Negation: A theory of its meaning,representation, and use

This article presents a model-based theory of what negation means, how it is mentally represented, and how it is understood. The theory postulates that negation takes a single argument that refers to a set of possibilities and returns the complement of that set. Individuals therefore tend to assign a small scope to negation in order to minimize the number of models of possibilities that they have to consider. Individuals untrained in logic do not know the possibilities corresponding to the negation of compound assertions formed with if, or, and and, and have to infer the possibilities one by one. It follows that negations are easier to understand, and to formulate, when individuals already have in mind the possibilities to be negated. The paper shows that the evidence, including the results of recent studies, corroborates the theory.     

A critique of van Fraassen's voluntaristic epistemology

Van Fraassen's epistemology is forged from two commitments, one to a type of Bayesianism and the other to what he terms voluntarism. Van Fraassen holds that if one is going to follow a rule in belief-revision, it must be a Bayesian rule, but that one does not need to follow a rule in order to be rational. It is argued that van Fraassen's arguments for rejecting non-Bayesian rules is unsound, and that his voluntarism is subject to a fatal dilemma arising from the non-monotonic character of reasoning.     

Aligning Logical and Psychological Contributions to the Understanding of Human Reasoning

Modern logic is an abstract study of consequence rather than a mechanistic study of reasoning. This abstract view has much to offer psychological studies of the representations which implement human reasoning. This paper reviews recent results showing the abstract equivalence of all the main psychological competence models of syllogistic reasoning. Their apparently contrasting representations are computationally equivalent for the kind of data presented. The true empirical contrasts between theories lie in other details of implementation — in whether integrated representations are constructed and in the sequence of cases considered. These contrasts are independent of sentential, graphical or mental model representation.Logic is also a guide to representation requirements. Linear syllogism models are representable by sequences of tokens, whereas categorial syllogism models generally require attribute binding. However, one consequence of the present semantic analysis is that all categorial syllogism competence models converge on an algorithm which avoids representation of more than a single sequence.The paper goes on to consider what evidence could distinguish the representations implementing reasoning in working memory. The example of dual task paradigms is considered and some conceptual questions are raised about the identification of the modalities of working memory subsystems with linguistic and spatial representations. Carefully distinguishing memory medium from memory interpretation leads to the conclusion that it is the interpretation of the contents of memory which determines whether it holds spatial or linguistic information. It is argued that the critical dimension for understanding reasoning processes is the expressiveness of interpretation of representations.     

Constructivism about Practical Reasons1

Philosophers commonly wonder what a constructivist theory as applied to practical reasons might look like. For the methods or procedures of reasoning familiar from moral constructivism do not clearly apply generally, to all practical reasons. The paper argues that procedural specification is not necessary, so long as our aims are not first‐order but explanatory. We can seek to explain how there could be facts of the matter about reasons for action without saying what reasons we have. Explanatory constructivism must assume constructive “norms of practical reasoning” which yield particular truths without assuming them. But philosophers often mistakenly assume that only “formal” norms of reasoning could fulfill this role. The paper describes a further possibility: norms of reasoning can be “situation‐specific” and yet retain truth‐independent authority. Though we might doubt whether such norms can be independently defended, we should not doubt the possibility or coherence of constructivism about practical reasons.     

Reasoning from inconsistency to consistency

This article presents a theory of how individuals reason from inconsistency to consistency. The theory is based on 3 main principles. First, individuals try to construct a single mental model of a possibility that satisfies a current set of propositions, and if the task is impossible, they infer that the set is inconsistent. Second, when an inconsistency arises from an incontrovertible fact, they retract any singularly dubious proposition or any proposition that is inconsistent with the fact; otherwise, they retract whichever proposition mismatches the fact. A mismatch can arise from a proposition that has only mental models that conflict with the fact or fail to represent it. Third, individuals use their causal knowledge-in the form of models of possibilities-to create explanations of what led to the inconsistency. A computer program implements the theory, and experimental results support each of its principles.     

An antidote to illusory inferences

The mental model theory predicts that reasoners normally represent what is true, but not what is false. One consequence is that reasoners should make “illusory” inferences, which are compelling but invalid. Three experiments confirmed the existence of such illusions based on disjunctions of disjunctions. They also established a successful antidote to them: Reasoners are much less likely to succumb to illusions if the inferences concern disjunctions of physical objects (alternative newspaper advertisements) rather disjunctions of the truth values of assertions. The results shed light both on the cause of the illusions and on the current controversy among different theories of reasoning.     

Propositional reasoning by model.

This article describes a new theory of propositional reasoning, that is, deductions depending on if, or, and, and not. The theory proposes that reasoning is a semantic process based on mental models. It assumes that people are able to maintain models of only a limited number of alternative states of affairs, and they accordingly use models representing as much information as possible in an implicit way. They represent a disjunctive proposition, such as "There is a circle or there is a triangle," by imagining initially 2 alternative possibilities: one in which there is a circle and the other in which there is a triangle. This representation can, if necessary, be fleshed out to yield an explicit representation of an exclusive or an inclusive disjunction. The theory elucidates all the robust phenomena of propositional reasoning. It also makes several novel predictions, which were corroborated by the results of 4 experiments.     

Reasoning as simulation

The theory that human cognition proceeds through mental simulations, if true, would provide a parsimonious explanation of how the mechanisms of reasoning and problem solving integrate with and develop from mechanisms underlying forms of cognition that occur earlier in evolution and development. However, questions remain about whether simulation mechanisms are powerful enough to exhibit human-level reasoning and inference. In order to investigate this issue, we show that it is possible to characterize some of the most powerful modern artificial intelligence algorithms for logical and probabilistic inference as methods of simulating alternate states of the world. We show that a set of specific human perceptual mechanisms, even if not implemented using mechanisms described in artificial intelligence, can nevertheless perform the same operations as those algorithms. Although this result does not demonstrate that simulation theory is true, it does show that whatever mechanisms underlie perception have at least as much power to explain non-perceptual human reasoning and problem solving as some of the most powerful known algorithms.     

Explanatory Consolidation: From ‘Best’ to ‘Good Enough’

In science and everyday life, we often infer that something is true because it would explain some set of facts better than any other hypothesis we can think of. But what if we have reason to believe that there is a better way to explain these facts that we just haven’t thought of? Wouldn’t that undermine our warrant for believing the best available explanation? Many philosophers have assumed that we can solve such underconsideration problems by stipulating that a hypothesis should not only be ‘the best’ explanation available; rather, it should also be ‘good enough’. Unfortunately, however, the only current suggestion for what it might mean to say that an explanation is ‘good enough’ is, well, not good enough. This paper aims to provide a better account of what is required for an explanatory hypothesis to be considered ‘good enough’. In brief, the account holds that a ‘good enough’ hypothesis is one that has gone through a process that I call explanatory consolidation, in which accumulating evidence and failed attempts to formulate better alternatives gradually make it more plausible that the explanation we currently have is better than any other that could be formulated.