The processes of inference

System 0 processes: parsing premises to compose intensional representations

Models are built by consulting the representation of the meaning of each premise, i.e. an intensional representation, which is composed out of the meanings of words and the grammatical relations among them. Accordingly, the first process in mReasoner is a shift-and-reduce parse that makes use of a context-free grammar and a lexicon (Hopcroft and Ullman 1979). It uses the meanings of words in the lexicon to compose an intensional representation that depends on the grammatical relations among the words. The lexical entries consist of a word (such as “all”), its part of speech (“determiner”), and a specification of its semantics. The grammatical rules specify how a string of syntactic constituents can be reduced to a higher order grammatical constituent, culminating in a well-formed sentence, and each grammatical rule is paired with an appropriate semantic rule. The parser applies the matching semantic rule as it uses a grammatical rule to reduce a string of words to a single constituent, such as a “noun phrase”. The uses of a standard parser, a context-free grammar, and a rule-by-rule compositional semantics, are neither novel nor empirical claims of the theory. The point instead is to illustrate how the meanings of assertions containing determiners, both orthodox ones, such as “all” and “some”, and unorthodox ones, e.g. “most” and “few”, can be captured using the values of parameters. Orthodox determiners are those that can be represented in the first-order predicate calculus, which is the standard version of logic in which variables range over individuals. The unorthodox determiners cannot be captured in this calculus, but call for one in which variables range over predicates, i.e. sets of individuals. The model theory accordingly postulates that quantified assertions express relations between sets (for a summary of this view, see Cohen and Nagel 1934, pp. 124–125; Khemlani and Johnson-Laird in press). Other psychological theories likewise treat quantifiers as relations between sets; some of them make use of diagrammatic representations to handle relations (Ceraso and Provitera 1971; Erickson 1974; Ford 1995), and others rely on formal rules of inference (Stenning and Yule 1997; Guyote and Sternberg 1981; Geurts 2003; Politzer, van der Henst, Luche, and Noveck 2006). What distinguishes the model theory is that it relies on models of individual entities and properties to represent sets.

The program captures the meanings of quantifiers in the values of six parameters. They constrain various operations, including building models, which we describe in the next section of the article. As an illustration of the parameters, consider the assertion:

The first parameter in the intension is the cardinality of the entities in a model representing the set in the initial noun phrase, e.g. the number of tokens for the set of artists in the example above. This value is set by default to 4 in the lexical entry of the quantifier, and so the initial model contains four such tokens. The system generates the same predictions for syllogistic reasoning regardless of whether the default value is 3, 4, or more, but future studies may allow researchers to determine the parameter empirically. The value is mutable, and it can be changed at a later stage of processing (see the account below of searching for counterexamples). This parameter also includes the boundary conditions on the cardinality, e.g. it must be greater than or equal to 1 in the case of “some”. Unlike Aristotle, modern logic treats universally quantified assertions, such as “all artists are bohemians”, as making no claims about the existence of artists. Likewise, in daily life, an assertion such as “all trespassers are prosecuted”, can be true even if there are none. At present, mReasoner finesses this problem, but in principle it can be handled in the parameters.

The second parameter in the intension is the cardinality of the set referred to by the quantified phrase as a whole, such as “some artists”, which for this determiner has a default setting of 2, which is less than the value of the first parameter. However, in the case of “all artists”, the second parameter is the same as the cardinality of “artists,” i.e. the default value of the first parameter. Of course, the second parameter changes if a change is made to the first parameter. The third parameter states the constraints on the relation between the two cardinalities, e.g. “some artists” is represented with fewer tokens than the set of artists as a whole, but the number must be greater than zero in order to capture the existential force of the determiner. The fourth parameter states the polarity of the determiner, that is, whether it is affirmative or negative. The fifth parameter states whether the determiner is universal (e.g. “all,” “no”) or existential (e.g. “some,” “most”), which affects the strategies used to search for counterexamples. Also, the sixth parameter states the relation between the sets referred to in the subject and in the predicate of the assertion. In the case of monadic assertions, the relation is usually set-inclusion or its negation, e.g. a subset of artists is included, or not included, in the set of bohemians. However, other relations, such as set membership do also occur, e.g. “artists are of varying abilities”, which means that the set of artists is a member of the set of those sets of individuals who vary in ability from one to the other.

The set of parameters may seem complicated, but readers should bear in mind that the parameters can capture the meaning of other sorts of determiner such as: “most”, “many”, “at least three”, “more than half”, which include determiners that cannot be expressed in the first-order predicate calculus. The present set of parameters is illustrated in Table 3 for the assertions that occur in Aristotelian syllogisms: “All As are Bs”, “Some As are Bs”, “No As are Bs”, “Some As are not Bs”, and three representative examples of assertions that occur outside syllogisms, “Most As are Bs”, “Neither A is a B”, and “Exactly five of the As are Bs”. The set of parameters is incomplete, because an extension of the theory to deal with quantified relations, such as, “All philosophers have read some books”, calls for further parameters to represent the respective scopes of the quantifiers. The likely interpretation of the preceding example contrasts in scope with an assertion in the passive voice: “Some books have been read by all philosophers” (see, e.g. Johnson-Laird and Byrne 1991). Unlike the first assertion, the second implies that philosophers have read the same books.

In summary, intensions are collections of parameters that, together with the semantic content of the open-class words in a sentence, such as “artists” and “bohemians”, capture the meaning of the sentence. They provide the data needed to build and to modify models. The order of the parameters has no bearing on how the intension is used in these processes. The compositionality of intensions follows the tradition of formal semantics, as does the assignment of truth values to assertions if their intensions can be mapped into independent models of the world (Partee 1996). From this perspective, a mental model captures what is common to a set of possibilities (Barwise 1993). But, the relation between sentences and their intensions is also compatible with cognitive and constructionist approaches to grammar (see e.g. Goldberg 2003; Langacker 2008). The system can be extended to deal with subtle distinctions in meaning among determiners, such as, “all”, “each”, “every”, and “any” (Langacker 2008, p. 292). Likewise, it can be extended to deal with numerical, proportional, and scalar quantifiers. However, our focus in the next section is on the monadic assertions that occur in Aristotelian syllogisms. These inferences are from two premises, in which each has one of the four initial forms in Table 3.

System 1 processes: the construction and interpretation of initial models

System 1 uses the intension of the first premise in a syllogism to build an initial model, and it updates this model given the subsequent premise. As an illustration, consider the assertion:

The second parameter specifies that the number of artists who are also bohemians is two by default. Because the fourth parameter states that the polarity of the determiner is positive, two of the artists are updated as bohemians. (Had the parameter been negative, the artists would have been updated with the property of not being bohemians.) The model is accordingly updated to:

The fifth parameter states that the assertion is not universal, and so it is possible that there are artists that are not bohemians, and bohemians who are not artists. The model is updated with an extra individual bohemian who is not an artist:

System 1 also accommodates a second premise, such as:

Once system 1 has an initial model of this sort, it can draw a conclusion establishing a new set-theoretic relation, that is, a relation that is not asserted in the premises. Researchers often place heuristics at the forefront of theories of reasoning (see also Ford 1995; Stenning and Yule 1997; Chater and Oaksford 1999; Politzer et al. 2006), but until now proponents of the model theory have downplayed their use. In an effort to bridge the two approaches, mReasoner embodies heuristics in its system 1 processes. However, the system abides by the constraint that any conclusion that heuristics generate must hold in the initial model. We have described the specific heuristics in detail elsewhere (Khemlani et al. under review a), and so here we only outline their general principles. For the inference above, heuristics need to deliver both the quantifier in the conclusion (its mood) and the order of the terms that occur in it: “artists” and “cadgers” (its figure). Previous heuristics, such as the atmosphere effect (Revlis 1975), have been based on superficial aspects of sentences, such as the determiners that occur in them, or on the informativeness of premises with a view to yielding probabilistic conclusions (Chater and Oaksford 1999). The heuristics in system 1 depend on a very different idea: individuals use their knowledge of the meaning of premises, i.e. their intensions, to guide them to initial conclusions. If a negative premise occurs in a syllogism, any valid conclusion is bound to be negative too. If a premise containing the determiner “some” occurs in a syllogism, any valid conclusion is bound to include it too. So, given that individuals are sensitive to the nature of potentially valid conclusions, they should have acquired the knowledge that as soon as a premise contains a negation, or an existential determiner, the conclusion must be in a negative and existential mood. The premises in our example above are:

The premise in the dominant mood also determines the order of the terms in the conclusion (its figure). Given the preceding premises, the two end terms, which are those that occur in only one premise, are “artists” and “cadgers”. The first premise is the dominant one, and so it determines the figure of the conclusion. System 1 uses the grammatical role of the end term in the dominant premise to play the same role the conclusion, and so it yields the initial conclusion:

This conclusion holds in the initial model of the premises, and so it is the output from system 1. Analogous principles apply to other sorts of premises. They account for the well-known figural effect that occurs in syllogistic reasoning, e.g. the tendency to infer the conclusion above rather than its converse, “Some cadgers are artists”. Indeed, conclusions in the predicted figure occurred 82% of the time in six different experiments (see the meta-analysis in Khemlani and Johnson-Laird in press). A similar heuristic determining the figure of conclusions is due to Chater and Oaksford (1999), i.e. the “attachment” heuristic according to which if the least informative premise has an end-term as its subject, it is also the subject of the conclusion; otherwise, the end-term in the other premise is the subject of the conclusion. Once again, however, the theories diverge, because of Chater and Oaksford's assumption that individuals avoid inferring conclusions of the form, Some _ are not _.

The heuristics in mReasoner rely on the intensions of the premises and the initial mental model. They operate without storing any information in working memory, and so they are rapid. However, the heuristics are fallible. Consider the following inference:

System 2 processes: the search for counterexamples

In the preceding section, we focused on how mReasoner embodies system 1 and uses heuristics to draw conclusions that are true in the initial model of the premises. However, these conclusions are often not true in other models of the premises. For that reason, the program embodies this second system, which makes a recursive search for alternative models that might falsify a heuristic conclusion. When it finds a counterexample, it also formulates a new conclusion if one is possible, or else declares that no definite conclusion follows about the relation between the end terms. It searches for counterexamples using three operations: adding, breaking, and moving properties in a model. These operations were embodied in an earlier program that dealt solely with syllogisms, and subsequent research in which participants manipulated external models of premises showed that they used these three operations too (Bucciarelli and Johnson-Laird 1999). We describe and illustrate each of the operations in Table 4.

Reasoners may adopt additional strategies for searching for counterexamples. Indeed, when they reason from premises that concern spatial relations (e.g. in front of, to the left of, above) individuals often make minimal, systematic changes to their initial models by “chunking” multiple entities within a model and operating upon those entities as though they were a single unit (Jahn, Knauff, and Johnson-Laird 2007). At present, system 2 implements only the search operations for models based on monadic assertions, but in principle, it can support other sorts of operation. The difficulty is to find empirical methods that reveal the nature of the operations underlying a search for counterexamples.

When system 2 succeeds in finding a counterexample to a conclusion, it attempts to formulate a weaker conclusion in the same figure by adjusting the parameters in the intension of the conclusion (Table 3). For instance, if it finds a counterexample to the conclusion, “All artists are cadgers”, it reduces the value of the parameter specifying the default number of artists who are cadgers from 4 to 3. The result is an intension for the assertion, “Some of the artists are cadgers”. The weaker conclusion is then checked against both the initial model and the counterexample. If it does not hold in both these models, system 2 weakens it still further. Also, if it is ultimately weakened until it expresses no information, the program responds that no valid conclusion exists. In this way, mReasoner accounts for how individuals infer that no valid conclusion follows from some premises, which many theories of monadic reasoning cannot do. In contrast, if the search fails to find a counterexample, the system asserts that the conclusion is valid, i.e. it holds in all models of the premises.

mReasoner predicts that certain valid inferences should be more difficult than others, and it even predicts that certain valid inferences are beyond the ability of logically naïve individuals. Consider, for instance, the following problem:

We have assessed the theory as it applies to syllogistic reasoning. In our meta-analysis (Khemlani and Johnson-Laird in press), we examined seven extant theories: the atmosphere hypothesis (Begg and Denny 1969), an analogous hypothesis in which reasoners are supposed to draw conclusions matching the mood of a premise (Wetherick and Gilhooly 1990), the hypothesis that reasoners make illicit conversions of premises (Revlis 1975), the probability heuristics model (Chater and Oaksford 1999), a theory based on rules of inference (Rips 1994), a program implementing an earlier mental model theory (Johnson-Laird and Byrne 1991), and another model-based program in which verbal formulations are central and no search for alternative models occurs (Polk and Newell 1995). Other theories of syllogisms exist, but their proponents did not consider them complete enough to be entered into the meta-analysis, and the predictions of still another theory have never been published (Khemlani and Johnson-Laird in press). In the meta-analysis, we compared the predictions of each of the seven theories to the conclusions that the participants had drawn in six studies of syllogistic reasoning. The accuracy of a theory depends on the extent to which its predicted responses occur in the data, and the extent to which the responses that it does not predict do not occur in the data. Figure 1 combines these two measures into a single measure of “prediction accuracy”. We have recently examined the performance of mReasoner, and, as Figure 1 shows, it outperforms in accuracy all seven of the theories in our previous meta-analysis.

Figure 1.

The percentages of accurate predictions of seven different theories of syllogistic reasoning for six experiments (in a meta-analysis in Khemlani and Johnson-Laird in press) to which we have added the predictions of mReasoner. The theories in the original meta-analysis are the matching hypothesis, the probability heuristics model: probability heuristics model PSYCOP, the atmosphere hypothesis, the original mental models theory, the illicit conversion model, and verbal models theory (see the text for references to these theories).

Other inferential tasks in mReasoner

We have described how mReasoner makes inferences from monadic premises. However, the general procedure of a rapid heuristic inference followed by an attempted falsification is general. It applies to many sorts of valid inference in many sorts of reasoning. Of course, there are limits on the kinds of heuristics the system can implement. At present, the system does not use heuristics for models of three or more premises. The constraint reflects the intuition that individuals use heuristics only for limited sets of premises, and tend to be lost about what to conclude from a large set of premises.

Psychological experiments on reasoning typically call for the participants to draw a valid conclusion from premises, or else to evaluate the validity of a given conclusion. Sometimes the conclusion is about what is necessarily the case, and sometimes it is about what is possibly the case, though this latter task occurs less often, and some theories of reasoning offer no account of how reasoners carry it out, e.g. Rips's (1994) formal rule theory. It is straightforward in mReasoner: a conclusion about what is possible is valid if there is a model of the premises in which it holds. In daily life, many other sorts of reasoning occur. Individuals may need to infer a likely conclusion, to create an explanation that resolves an inconsistency, and even to detect the inconsistency in the first place. mReasoner is already able to carry out many of these tasks. They include: