MMT has been applied to reasoning in a range of deductive (Bucciarelli & Johnson-Laird, 1999; Johnson-Laird & Byrne, 2002; Johnson-Laird, Legrenzi, Girotto, Legrenzi, & Caverni, 1999; Johnson-Laird & Savary, 1999) and inductive (Johnson-Laird, 1994) tasks, including ones about spatial relationships (Ragni & Knauff, 2013) and about causality (Khemlani, Barbey, & Johnson-Laird, 2014). It has sometimes been criticised because the MMT specifications for these tasks can appear independent, task specific accounts (Baratgin et al., 2015). But the general assumptions of the theory may be summarised as follows.
People reason by making and manipulating isomorphic representations of aspects of the reasoning domain, called models. In the theory's earlier account of deductive reasoning with sentence particles like if, and, or, and not (Byrne & Johnson-Laird, 2009; Johnson-Laird & Byrne, 2002), each model refers to a logical possibility for a statement, corresponding to the truth table cases that make the statement true in classical logic. For example, a conjunction is true in one possibility, that in which both p and q are true, so it is represented with one model: p q
The inclusive disjunction is true in three possibilities, so it is represented with three models: p
q p q
Note that the above mental models do not explicitly represent negations. But they can be
92 are false at each possibility in which the statement as a whole is true. For the disjunction this yields:
p not-q not-p q
p q
The logical possibilities in which the statement as a whole is false (in the case of the disjunction the not-p, not-q possibility) are generally not included in any models. The assumption that people tend to represent what is true in a model, rather than what is false, and that they represent the models that render a statement true but not those that render a statement false, is called the principle of truth in MMT. Logically, the fully explicit models are equivalent to the disjunctive normal form for the statement. For example, this form for the disjunction p or q is (p & not-q) or (not-p & q) or (p & q).
Conditionals in MMT
The initial mental model for conditionals is represented as a conjunction, followed by an
implicit model represented as an ellipsis:
p q ...
The implicit model symbolises that there are further possibilities in which the statement could be true, that are not yet fleshed out. The fully explicit model for the conditional is equivalent to the cases of the truth table in which the material conditional is true:
p q not-p q not-p not-q
Consequently these models are the same as the models for not-p or q.
The above models represent what the theory calls the basic meanings of conjunctions, disjunctions and conditionals. These are the meanings people are thought to give to the statements when their contents are "abstract", or arbitrary. For example, a basic conditional about figures drawn on a black board could be "If there is a circle then there is a square". The theory posits that these meanings can change through semantic and pragmatic modulation, i.e. through features of the content or context (Byrne & Johnson-Laird, 2010; Johnson-Laird & Byrne, 2002). For example, "If it rained then it poured" would be represented only by two models:
it rained it poured it did not-rain it did not pour
93 with the model "it did not rain, it poured" ruled out by the content, which indicates that pouring is a form of raining. Modulation is assumed to rule out models and also annotate models with extra information, e.g. about temporal or causal order, but it is assumed not to lead to the addition of models (Byrne & Johnson-Laird, 2010).
It is argued that although the meanings of basic conditionals, conjunctions and disjunctions are truth functional, and have the same truth conditions as in classical logic, the meanings of these statement types after semantic and pragmatic modulation is not truth functional, so that overall, conditionals and other statement types are not truth functional in the theory (Johnson-Laird & Byrne, 2002). There are two difficulties with this assumption. The first is that MMT categorically endorsed the paradoxes of the material conditional as logically valid inferences (Johnson-Laird & Byrne, 1991, 2002), but the validity of these inferences (plus the inference from the truth of p and falsity of q to the falsity of if p then q) entails truth functionality (Evans, Over, & Handley, 2005). The second is that when and how semantic and pragmatic modulation affects the meaning of conditionals, and other statement types, is not clearly specified. Byrne and Johnson-Laird (2010, p. 66; see also Johnson-Laird & Byrne, 2002, p. 674) are explicit on this point: "One consequence of modulation is that conditionals have an indefinite number of meanings – ten sets of possibilities and a variety of relations between the antecedent if-clause and the consequent then-clause." The result is that when modulation is taken into account, any experimental finding can be explained, and none can be predicted. There has consequently been little research testing the theory's account of non-basic conditionals other than by proponents of the theory (for a recent example see Orenes & Johnson-Laird, 2012).
Proponents of MMT (Barrouillet, Gauffroy, & Lecas, 2008; Girotto & Johnson-Laird, 2010) have argued that one of the interpretations of conditionals posited by the theory corresponds to the de Finetti table, and so explains people's responses in truth table tasks. They held that the de Finetti response pattern for conditionals, if p then q, may be based on the initial mental model in which only the p & q possibility is represented explicitly, and the other two possibilities remain implicit (denoted by the ellipsis). Being implicit, these two possibilities are considered irrelevant, while at the same time the p & not-q possibility is somehow acknowledged as the only case that makes the conditional false.
However, this position seems inconsistent with other parts of MMT. If the implicit possibilities for the conditional are treated as irrelevant, then implicit possibilities should also be treated as irrelevant in conjunctions and disjunctions, which would change their meanings (for example, the not-p, q model should then be judged irrelevant, and not false, for conjunctions). And if the p, not-q model for the conditional is fleshed out alongside the p, q model, then this would violate the principle of truth. There is no mechanism within MMT for distinguishing between false and irrelevant cases (Oberauer & Oaksford, 2008), and it is
94 generally difficult to find a representation for a statement that is not a full proposition using the propositional tools for representing mental models.
Reasoning with conditional syllogisms in MMT
The earlier version of MMT (Johnson-Laird & Byrne, 2002) assumed that when people draw inferences with conditionals in conditional syllogisms, they first build mental models for the two premises, and then integrate these models. The integration is done by adding the models of one premise to the models of the other, and removing any models that are inconsistent. Next people take a few (typically one or two) of the logical possibilities in which the combined models are true, and assess whether the conclusion is true in these possibilities. If it is, the process stops, and they take the inference to be deductively valid. If it is not, the process also stops, and they take the inference to be invalid. But if the conclusion does not feature in the premises, and people have enough time and resources, then they look at further models of the premises, possibly fleshing them out into fully explicit models, to see whether they can find a counterexample to their initial conclusion. A counterexample is a logical possibility in which the premises are true but the conclusion false. Markovits & Barrouillet (2002) proposed that counterexamples are based not just on abstract logical possibilities, but are retrieved from long term memory, with ease of retrieval varying with the specific content of the conditional (c.f. Cummins, 1995).
The above procedure implies that, if people flesh out all the models of the premises, and modulation is not an issue, then they will respond exactly in the way predicted by the rules of classical logic. But the theory posits that people make errors because they fail to take into account relevant logical possibilities. For instance, they might not flesh out mental models into fully explicit models, or not flesh out the negations within a model. This is assumed to occur mainly because of working memory limitations, and it implies that the more models are required to fully flesh out the premises of an inference, the more difficult the inference will be.
Work in developmental psychology (e.g. Barrouillet & Gauffroy, 2015; Barrouillet & Lecas, 1999; Markovits & Barrouillet, 2002) suggested a developmental trend in the interpretation of conditionals, going from a conjunctive interpretation in young children, over a biconditional interpretation, to a conditional interpretation in adults – with the conditional interpretation being in accordance with the material conditional under binary paradigm instructions, and in accordance with the probability conditional under probabilistic instructions. MMT proposed that this trend comes about because, when people flesh out the initial conjunctive model of the conditional, they first add the not-p, not-q model that yields a biconditional interpretation, and only afterwards add the not-p, q model that results in the conditional interpretation. There is no rationale within the theory for this specific ordering
95 (e.g. why is the not-p, not-q model added before and not after the not-p, q model?), but the interpretation of the developmental trend, in the successive addition of models, corresponds to a situation in which increasing numbers of models are considered, in line with the predictions of the theory.
The above ordering can also be used to explain the pattern of responses in the conditional inference task. A conjunctive interpretation of the conditional in which the ellipsis is forgotten would lead to the acceptance of MP and AC. The addition of the not-p, not-q model would lead to the acceptance of all four inferences, and the subsequent addition of the not-p, q model would result in participants accepting MP and MT and rejecting AC and DA. This account of conditional syllogisms has found some support in the literature, but mostly when combined with an additional assumption about conditionals that is not part of MMT, called directionality (Barrouillet, Grosset, & Lecas, 2000; Evans et al., 2005; Oberauer, 2006). This is the assumption that people tend to process a conditional in direction from antecedent to consequent, and that the outcome of processing the conditional can differ if it is instead carried out from consequent to antecedent. Directionality is hard to explain within MMT when reasoning with basic conditionals, but it is a feature that follows from the Ramsey test (see also Verschueren et al., 2005).
Mental models and probabilities
MMT was extended early on to cover situations in which the premises are not certain but only probable to a higher or lower degree. It proposed that by default models are equiprobable, so that the probability of a statement corresponds to the proportion of models that make the statement true. But it was also held that people are able to assign distinct probabilities to models. In simple cases, they could do this by creating repetitions of models of the same kind, and letting the proportion of models of each kind stand for their probability. In more complex cases, tags with numerical probabilities could be added to the models (Geiger & Oberauer, 2010; Girotto & Johnson-Laird, 2004; Johnson-Laird et al., 1999). Conditional probabilities can be represented by using the subset principle, in which a person focuses on the subset of models in which the denominator is true, and assesses the proportion of models in this subset in which the numerator is also true (Johnson-Laird et al., 1999). This procedure seems functionally equivalent to a Ramsey test. Later on Khemlani et al. (2015) also described a way of representing subjective probabilities, using a scale set a priori to have eight subdivisions, on which positive and negative evidence for a statement could be averaged (see also Juslin, Nilsson, & Winman, 2009, on the averaging hypothesis). But MMT never proposed a rational procedure for transferring premise probabilities to conclusion probabilities for relations more complex than that of a set to a subset.
96 New MMT
Johnson-Laird and colleagues (Johnson-Laird et al., 2015; see also Johnson-Laird & Ragni, 2017, July; Khemlani, Hinterecker, & Johnson-Laird, 2017) have recently proposed a radical revision of MMT, in which the paradoxes of the material conditional, but also other classically valid inferences, notably or-introduction, p, therefore p or q, are declared invalid in the theory. The revision is based on a change in the meanings of and, or, and if. They are still represented by the same models, but whereas before a statement was true when one of the possibilities used to represent it was actually the case, now a statement is held to be equivalent to the conjunction of those possibilities. For example, as pointed out above, the disjunction p or q was at first fully represented, in effect, as its disjunctive normal form, (p & not-q) or (not-p &
q) or (p & q). But it is now taken to be equivalent to the conjunction of possibilities: possibly
(p & not-q) & possibly (not-p & q) & possibly (p & q). One consequence of this change is that the paradox of the material conditional not-p, therefore if p then q, is argued to be invalid, because the model used to represent not-p does not guarantee that all the models used to represent the material conditional are possible. Or-introduction is declared invalid for the same reason: the model for p does not establish that the three models for the disjunction are possible.
The revision arguably has the advantage that the highly counterintuitive paradoxes are no longer considered valid. But it apparently entails inconsistent and counterintuitive consequences that make it not worth pursuing (for overviews see Baratgin et al., 2015; Cruz, Over, & Oaksford, 2017; Oaksford, Over, & Cruz, 2018). For example, if or-introduction is invalid, then so is and-elimination, p & q, therefore p, because the two inferences can be derived from one another by De Morgan's rules. But and-elimination remains valid in the revised MMT, as the models of the premise do establish that the model for the conclusion is possible. The claim that or-introduction is invalid may be based on experiments with binary paradigm instructions, showing that it is then endorsed less often than other valid inferences (e.g. Orenes & Johnson-Laird, 2012). But under probabilistic instructions, asking people directly about their degrees of belief, or-introduction is accepted to the same degree as and- elimination, and to a much higher degree than the paradoxes of the material conditional (Cruz et al., 2017). Further, if p, therefore p or q is invalid, then the premise p should be consistent with the negation of the conclusion, not-(p or q), which is equivalent by De Morgan's rules to
not-p & not-q. But MMT continues to consider two statements consistent when they have at
least one model in common, and p does not share any model with not-p & not-q (Baratgin et al., 2015).
It is also not clear what Johnson-Laird et al. (2015) mean when they state that or-
introduction is invalid "in MMT". An inference is valid or invalid within a specific logical
97 the theory would have to specify either a known logical system on which it bases itself when making assertions about validity, or specify its own provably consistent system, or substitute for "validity" a non-logical term to characterise inferences, e.g. "endorsed often by most people". In the current formulation of the theory, it is unclear whether the proposals made about validity are at the computational or the algorithmic level (c.f. Oaksford et al., 2018).
MMT is referred to at points in the interpretation of the results of the thesis, but there are perhaps two main reasons why the theory has no strong bearing on the work of the thesis as a whole. The first is that the mental model account of reasoning with probabilities has not yet been specified for relations more complex than that between a set and a subset. This means that it is not clear, for example, how the probabilities of two-premise inferences, like the conditional syllogisms, could be transferred in a principled way to the conclusion. The second is that the current revision of MMT raises many questions that make it difficult to derive predictions from it. However, the revision is still under development, and removing the apparent inconsistencies may be a part of that process, with the implications for reasoning from uncertain premises becoming clearer over time.