The Psychology of Uncertainty and Three-Valued Truth Tables.

Psychological research on people's understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant" (devoid of value) when A is false (whatever the value of C). This was called the "psychological defective table of the conditional." Here we show that far from being anomalous the "defective" table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants' truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people's interpretations of natural language connectives. We find that de Finetti's (1936/1995) three-valued system is the best approximation to participants' truth tables.

Preadolescents Solve Natural Syllogisms Proficiently.

"Natural syllogisms" are arguments formally identifiable with categorical syllogisms that have an implicit universal affirmative premise retrieved from semantic memory rather than explicitly stated. Previous studies with adult participants (Politzer, 2011) have shown that the rate of success is remarkably high. Because their resolution requires only the use of a simple strategy (known as ecthesis in classic logic) and an operational use of the concept of inclusion (the recognition that an element that belongs to a subset must belong to the set but not vice versa), it was hypothesized that these syllogisms would be within the grasp of non-adult participants, provided they have acquired the notion of deductive validity. Here, 11-year-old children were presented with natural syllogisms embedded in short dialogs. The first experiment showed that their performance was equivalent to adults' highest level of performance in standard experiments on syllogisms. The second experiment, while confirming children's proficiency in solving natural syllogisms, showed that they outperformed children who solved non-natural matched syllogisms in the same experimental setting. The results are also in agreement with the argumentation theory of reasoning.

The class inclusion question: a case study in applying pragmatics to the experimental study of cognition.

For more than 70 years, Piaget's class-inclusion task (given, e.g., five asters and three tulips, the child is asked whether "there are more asters or more flowers") has been the object of experimental investigation. Inclusion is of considerable importance for cognitive science as it is a key concept for logical operations and knowledge representation. It is shown that the question can be characterised by a kind of privative ambiguity which is at the source of the younger children's answer, "more asters". A relevance-theoretic explanation of children's interpretation of the question and of the subsequent responses is expounded. This account can explain the effect of all the factors that are known to influence performance (e.g., role of collections, counting, typicality, qualification, syntax, etc.), a review of which is presented. It is further tested experimentally. The development of performance is explained on the basis of the way children disambiguate the question. This study exemplifies the two ways in which pragmatic analysis is pertinent to the study of children's (as well as adults') reasoning and judgement, namely in explaining and predicting participants' comprehension of the statements and questions, and in taking into account attribution processes that occur in the experimental setting.

The interpretation of classically quantified sentences: a set-theoretic approach.

We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. We show that although these sentences are known to have a diagrammatic expression (in the form of the Gergonne circles) that constitutes a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. We hypothesized that the quantified sentences have an abstract underlying representation common to the formulas and their associated sets of diagrams (models). We derived 9 predictions (3 semantic, 2 pragmatic, and 4 mixed) regarding people's assessment of how well each of the 5 diagrams expresses the meaning of each of the quantified sentences. We report the results from 3 experiments using Gergonne's (1817) circles or an adaptation of Leibniz (1903/1988) lines as external representations and show them to support the predictions.

No problem for Aristotle's subject and predicate.

It is argued that, in the traditional subject-predicate sentence, two interpretations of the subject term coexist, one intensional and the other extensional, which explains the superficial difference between the traditional S-P relation and the predication of predicate logic. Data from psychological studies of syllogistic reasoning support the view that the contrast between predicate and argument is carried over to the traditional S-P sentence.

Deductive reasoning from uncertain conditionals.

This paper begins with a review of the literature on plausible reasoning with deductive arguments containing a conditional premise. There is concurring evidence that people presented with valid conditional arguments such as Modus Ponens and Modus Tollens generally do not endorse the conclusion, but rather find it uncertain, in case (1) the plausibility of the major conditional premise is debatable, (2) the major conditional premise is formulated in frequentist or probabilistic terms, or (3) an additional premise introduces uncertainty about the major conditional premise. This third situation gives rise to non-monotonic effects by a mechanism that can be characterized as follows: the reasoner is invited to doubt the major conditional premise by doubting the satisfaction of a tacit condition, which is necessary for the consequent to occur. Three experiments are presented. The first two aim to generalize the latter result using various types of conditionals and the last shows that performance in conditional reasoning is significantly affected by the representation of the task. This third point is discussed along with various other issues: we propose a pragmatic account of how the tacit conditions mentioned earlier are treated in plausible reasoning; the relationship of this account with the conditional probability view on conditional sentences is examined; an application of the same account to the Suppression Effect (Byrne, 1989) is proposed and compared with the counter-example availability explanation; and finally, some suggestions on how uncertainty could be implemented in a mental logic system are presented.