Temporal and spatial relations in sentential reasoning

The mental model theory postulates that the meanings of assertions, and knowledge about their context can modulate the logical meaning of sentential connectives, such as "if" and "or". One known effect of modulation is to block the representation of possibilities to which a proposition refers. But, modulation should also add relational information, such as temporal order, to models of possibilities. Three experiments tested this prediction. Experiment 1 showed that individuals spontaneously matched the tense of their conclusions (in Portuguese) to embody implied, but unexpressed, temporal relations in conditional premises. Experiment 2 demonstrated the same phenomenon in inferences from disjunctions. Experiment 3 showed that the number of such implicit relations in inferences from conditionals affects both accuracy and the speed of reasoning. These results support the modulation hypothesis.     

充分条件句语义关系知觉对条件推理的影响

作者通过实验考察了大学生被试对五种充分条件句语义关系知觉对其条件推理的影响。两个实验的结果表明,被试对条件句中前后件之间语义关系的知觉理解影响和调节其相应的条件推理的成绩。实验1新发现,被试对充分条件句后件对前件必要性的知觉理解存在难易差异,对许可句和定义句的后件必要性容易知觉,对偶然句、义务句和因果句的后件必要性难以知觉。实验2新发现,被试对充分条件句后件对前件必要性的知觉理解的难易差异导致被试在否定后件式推理成绩上的差异,对许可句和定义句的否定后件式推理成绩显著高于对偶然句、义务句和因果句的否定后件式推理成绩,对充分条件句否定后件式推理成绩随对后件必要性知觉增加而增加。     

The negated conditional: a litmus test for the suppositional conditional?

Under the suppositional account of conditionals, when people think about a conditional assertion, "if p then q," they engage in a mental simulation in which they imagine p holds and evaluate the probability that q holds under this supposition. One implication of this account is that belief in a conditional equates to conditional probability [P(q/p)]. In this paper, the authors examine a further implication of this analysis with respect to the wide-scope negation of conditional assertions, "it is not the case that if p then q." Under the suppositional account, nothing categorically follows from the negation of a conditional, other than a second conditional, "if p then not-q." In contrast, according to the mental model theory, a negated conditional is consistent only with the determinate state of affairs, p and not-q. In 4 experiments, the authors compare the contrasting predictions that arise from each of these accounts. The findings are consistent with the suppositional theory but are incongruent with the mental model theory of conditionals.     

Unless reasoning

We report the results of two experiments investigating conditional inferences from conditional unless assertions, such as Juan is not in León unless Nuria is in Madrid. Experiments 1 and 2 check Fillenbaum's hypothesis about the semantic similarity of unless with if not and only if assertions; both also examine inferential endorsements (Experiment 1) and endorsements and latencies (Experiment 2) of the four logically equivalent conditional formulations: if A then B, if not-B then not-A, A only if B and notA unless B. The results of these experiments show the similarity of unless and only if confirming that the representation of both conditionals from the outset probably include two possibilities directionally oriented from B to A; results also confirm the especial difficulty of unless assertions. The implications of the results are discussed in the context of recent psychological and linguistic theories of the meaning of unless.     

To do or not to do? A cognitive consistency model for drawing conclusions from conditional instructions and advice

This paper investigates a cognitive consistency model of the directionality of conditional instructions and advice that use probability expressions to express uncertainty about the antecedent p. The proposed model combines world knowledge (conveyed by causal direction) with linguistic information (conveyed by polarity and negation), and predicts whether the complex sentence antecedent has a positive or negative directionality, which in turn predicts whether a positive or negative conclusion q will be drawn. The first experiment uses Do q if p conditionals to show that given a consequent q participants complete antecedents p with a probability expression in line with expected sentence directionality. The second experiment uses If p then do q conditionals to show similar effects in a reverse direction. A third experiment uses If p then do q conditionals to show that participants draw conclusions predicted by the cognitive consistency model but not by a decision-theoretic approach to reasoning.     

Two meanings of “if”? Individual differences in the interpretation of conditionals

This work investigates the nature of two distinct response patterns in a probabilistic truth table evaluation task, in which people estimate the probability of a conditional on the basis of frequencies of the truth table cases. The conditional-probability pattern reflects an interpretation of conditionals as expressing a conditional probability. The conjunctive pattern suggests that some people treat conditionals as conjunctions, in line with a prediction of the mental-model theory. Experiments 1 and 2 rule out two alternative explanations of the conjunctive pattern. It does not arise from people believing that at least one case matching the conjunction of antecedent and consequent must exist for a conditional to be true, and it does not arise from people adding the converse to the given conditional. Experiment 3 establishes that people's response patterns in the probabilistic truth table task are very consistent across different conditionals, and that the two response patterns generalize to conditionals with negated antecedents and consequents. Individual differences in rating the probability of a conditional were loosely correlated with corresponding response patterns in a classical truth table evaluation task, but there was little association with people's evaluation of deductive inferences from conditionals as premises. A theoretical framework is proposed that integrates elements from the conditional-probability view with the theory of mental models.     

Two meanings of "if"? Individual differences in the interpretation of conditionals

This work investigates the nature of two distinct response patterns in a probabilistic truth table evaluation task, in which people estimate the probability of a conditional on the basis of frequencies of the truth table cases. The conditional-probability pattern reflects an interpretation of conditionals as expressing a conditional probability. The conjunctive pattern suggests that some people treat conditionals as conjunctions, in line with a prediction of the mental-model theory. Experiments 1 and 2 rule out two alternative explanations of the conjunctive pattern. It does not arise from people believing that at least one case matching the conjunction of antecedent and consequent must exist for a conditional to be true, and it does not arise from people adding the converse to the given conditional. Experiment 3 establishes that people's response patterns in the probabilistic truth table task are very consistent across different conditionals, and that the two response patterns generalize to conditionals with negated antecedents and consequents. Individual differences in rating the probability of a conditional were loosely correlated with corresponding response patterns in a classical truth table evaluation task, but there was little association with people's evaluation of deductive inferences from conditionals as premises. A theoretical framework is proposed that integrates elements from the conditional-probability view with the theory of mental models.     

Mental models in deductive reasoning

We report research investigating the role of mental models in deduction. The first study deals with conjunctive inferences (from one conjunction and two conditional premises) and disjunctive inferences (from one disjunction and the same two conditionals). The second study examines reasoning from multiple conditionals such as: If e then b; If a then b; If b then c; What follows between a and c? The third study addresses reasoning from different sorts of conditional assertions, including conditionals based on if then, only if, and unless. The paper also presents research on figural effects in syllogistic reasoning, on the effects of structure and believability in reasoning from double conditionals, and on reasoning from factual, counterfactual, and semifactual conditionals. The findings of these studies support the model theory, pose some difficulties for rule theories, and show the influence on reasoning of the linguistic structure and the semantic content of problems.     

Rephrasing between Disjunctives and Conditionals: Mental Models and the Effects of Thematic Content

Two experiments are reported thatinvestigate whether the logical equivalence of conditionals and disjunctives is paralleled by a psychological equivalence. In these experiments, subjects rephrased from one form into the other. Experiment 1 demonstrated strong effects of familiarity and causality of rule content. Similar findings were found in Experiment 2 with a different conditional rule syntax. An account of the experiments is given in terms of mental models theory: In this account, task performance can be seen to depend upon the extent to which the model sets used by subjects to generate rephrasings are complete, task content being the most important factor affecting model set completion. A 'Minimal Completion' strategy is proposed to operate in the absence of thematic content. The experiments also falsify the long-held assumption that conditionals with negative antecedents are always interpreted as their disjunctive equivalents. This raises doubts about the mental models explanation for matching bias in conditional reasoning.     

Concessive and semifactual interpretations during reasoning with multiple conditionals

The present research evaluates how people integrate factual ‘if then’ and semifactual ‘even if’ conditional premises in an inference task. The theory of mental models establishes that semifactual statements are represented by two mental models with different epistemic status: ‘A & B’ is conjectured and ‘not-A & B’ is presupposed. However, following the principle of cognitive economy in tasks with a high working memory load such as reasoning with multiple conditionals, people could simplify the deduction process in two ways, by discarding: (a) the presupposed case and/or (b) the epistemic status information. In Experiment 1 and Experiment 2, we evaluated each of these hypotheses. In Experiment 1, participants make inferences from two conditionals: two factual conditionals or one factual and one semifactual, with different representations. In Experiment 2, participants make inferences with a factual conditional followed by two different semifactual conditionals that share the same representations but differ in their epistemic status. Accuracy and latency data suggest that people think of both the conjectured and the presupposed situations, but do not codify the epistemic status of either when the task does not require it. The results are discussed through theoretical predictions about how people make inferences from different connected conditionals.     

Perceptual simulations can be as expressive as first-order logic

Theories asserting that human reasoning is based on perceptual simulations often suppose these simulations are of concrete individual objects and the specific relations that hold among them. However, much human knowledge involves assertions about which relations do not hold, generalities over large numbers of objects and conditional facts. Can simulation theories explain how the mind represents these forms of knowledge, or are they inferior in their expressive power to knowledge representation schemes based on logical formalisms designed specifically to deal with negative, conditional and quantificational knowledge? In this paper, we show how assertions about mental simulations can in fact straightforwardly express all the concepts that comprise first-order logic, including negation, conditionals and both universal and existential quantification. We also speculate on how to extend this approach to deal with probabilistic and more expressive logics.     

Consequential conditionals: invited and suppressed inferences from valued outcomes

Consequential conditionals are defined as "if P then Q" statements, where P is an action, and Q a predicted outcome of this action, which is either desirable or undesirable to the agent. Experiment 1 shows that desirable (viz. undesirable) outcomes invite an inference to the truth (viz. falsity) of their antecedent. Experiment 2 shows that the more extreme the outcome is, the stronger the invited inference is. Experiment 3 shows that modus ponens from premises "If A then C, A" can be suppressed with the introduction of a consequential conditional, "If C then Q," where Q is an undesirable outcome. Experiment 4 shows that the more undesirable Q is, the larger the suppression is. The authors discuss how these results can enrich current approaches of conditional inference on the basis of mental models, complementary necessary conditions, and conditional probabilities.     

Heuristic and analytic processes in mental models for conditionals: An integrative developmental theory

This article presents a developmental dual-process theory of the understanding of conditionals that integrates Evans’ heuristic–analytic theory within the revised mental model theory of conditional proposed by Barrouillet, Gauffroy, and Lecas (2008). According to this theory, the interpretation of a conditional sentence is driven by unconscious and implicit heuristic processes that provide individuals with an initial representation that captures its meaning by representing the cases that make it true. This initial model can be enriched with additional models (a process named fleshing out within the mental model theory) through the intervention of conscious and demanding analytic processes. Being optional, these processes construct representations of cases that are only compatible with the conditional, leaving its truth-value indeterminate when they occur. Because heuristic processes are relatively immune to developmental changes, while analytic processes strongly develop with age, the initial model remains stable through development whereas the number of additional models that can be constructed increases steadily. Thus, the dual-process mental model theory predicts in which cases conditionals will be deemed true, indeterminate, or false and how these cases evolve with age. These predictions were verified in children, adolescents and adults who were asked to evaluate the truth value and the probability of several types of conditionals. The results reveal a variety of developmental trajectories in the way different conditionals are interpreted, which can all be accounted for by our revised mental model theory.     

The negations of conjunctions,conditionals, and disjunctions

How do reasoners understand and formulate denials of compound assertions, such as conjunctions and disjunctions? A theory based on mental models postulates that individuals enumerate models of the various possibilities consistent with the assertions. It therefore predicts a novel interaction: in affirmations, conjunctions, A and B, which refer to one possibility, should be easier to understand than disjunctions, A or B, which refer to more than one possibility; in denials, conjunctions, not(A and B), which refer to more than one possibility, should be harder to understand than disjunctions, not(A or B), which do not. Conditionals are ambiguous and they should be of intermediate difficulty. Experiment 1 corroborated this trend with a task in which the participants selected which possibilities were consistent with assertions, such as: Bob denied that he wore a yellow shirt and he wore blue pants on Tuesday. Experiment 2 likewise showed that participants' own formulations of verbal denials yielded the same trend in which denials of conjunctions were harder than denials of conditionals, which in turn were harder than denials of disjunctions.     

Conjunctive bias in memory representations of logical connectivesa

The paper presents the conjunctive bias in memory-a novel phenomenon that helps to clarify representations of logical connectives. The conjunctive bias is a tendency toward more accurate recall and recognition of conjunctive forms than of forms based on other logical connectives and a tendency to recall and recognize other logical forms as if they were conjunctions. Three experiments, in which participants' memory representations associated with different logical connectives were examined, were conducted to test the conjunctive bias hypothesis. In Experiment 1, participants learned picture-proposition pairs involving either conjunctions or disjunctions and then had to recall each proposition when cued with its picture. In Experiments 2 and 3, recognition memory for conjunctions, disjunctions, and conditionals was examined with an old/new recognition procedure. The findings of these experiments provide evidence for the conjunctive bias. Furthermore, the results of Experiment 3 suggest that corjunctive bias is not simply a pragmatically caused preference for conjunctions. The discussion focuses on the implications of these findings for current theories of deductive reasoning.     

On the Mental Representation of Conditional Sentences

Four experiments are reported which attempt to externalize subjects' mental representation of conditional sentences, using novel research methods. In Experiment 1, subjects were shown arrays of coloured shapes and asked to rate the degree to which they appeared to be true of conditional statements such as 'If the figure is green then it is a triangle'. The arrays contained different distributions of the four logically possible cases in which the antecedent or consequent is true or false: TT, TF, FT, and FF. For example, a blue triangle would be FT for the conditional quoted above. In Experiments 2 to 4, subjects were able to construct their own arrays to make conditionals either true or false with any distribution of the four cases they wished to choose. The presence and absence of negative components was varied, as was the form of the conditional, being either 'if then' as above or 'only if': 'The figure is green only if it is a triangle'. The first finding was that subjects represent conditionals in fuzzy way: conditionals that include some counter-example TF cases (Experiment 1) may be rated as true, and such cases are often included when subjects construct an array to make the rule true (Experiments 2 to 4). Other findings included a strong tendency to include psychologically irrelevant FT and FF cases in constructed arrays, presumably to show that conditional statements only apply some of the time. A tendency to construct cases in line with the 'matching bias' reported on analogous tasks in the literature was found, but only in Experiment 4, where the number of symbols available to construct each case was controlled. The findings are discussed in relation to the major contemporary theories of conditional reasoning based upon inference rules and mental models, neither of which can account for all the results.     

Counterfactual antecedent falsity and the epistemic sensitivity of counterfactuals

Why do utterances of counterfactual conditionals typically, but not universally, convey the message that their antecedents are false? I demonstrate that two common theoretical commitments–commitment to the existence of scalar implicature and of informative presupposition—can be supplemented with an independently motivated theory of the presuppositions of competing conditional alternatives to jointly predict this information when and only when it appears. The view works best if indicative and counterfactual conditionals have a closely related semantics, so I conclude by undermining two familiar arguments for a nonunified semantics of indicative and counterfactual conditionals.     

A Syntactically Motivated Theory of Conditionals

In this paper I shall propose and discuss a new theory of indicative conditionals in English, concentrating on the connectives if, only if, even if , and unless. By ".a theory of these connectives I mean a systematic assignment of logical forms to sentences containing them, which (a) accounts for such sentences' felt implications and other intuitive semantical properties, (b) explains the ways in which their truth-values depend upon context, and (c) accords with and accounts for noteworthy aspects of their surface-syntactic behavior. I shall also comment on the difference between indicative and subjunctive conditionals and shall suggest that indicatives and subjunctives are semantically identical though pragmatically distinct.     

Conditionals and directionality: On the meaning of if vs. only if

The aim of this study was to test the predictions of the current theories of reasoning about the comprehension of conditional statements. We used two types of conditional statement that are logically equivalent: if p then q and p only if q . The model theory of reasoning considers that these conditional forms differ in their initial meaning, because the negative contingency is considered only in the p only if q form. Mental-rule theories maintain that the interpretation of p only if q depends on a rephrasing of the statement as: if not q then not p . Alternatively, a directional bias may explain the differences between if p then q and p only if q . We report three experiments that demonstrate the existence of a directional bias in the comprehension of the conditionals. The results were not predicted by either the mental-rules theories or the model theory; they could, however, be easily assimilated by the model theory.     

Reasoning about conditional sentences: development of understanding of cues to quantification

The previous literature has reported that when children are asked to judge the truth or falsity of universally quantified conditional sentences of the form If a thing is P then it is Q they typically give responses, e.g., responding "true" whenever there is a case of P and Q even if there are also cases of P and not-Q. Three experiments are reported that address possible sources of this error. Experiment 1 shows that the error survives on sentences that refer to particular things as well as to things of a particular kind, and further shows that articulating the necessity of the consequent (... then it has to be Q) eliminates the error for adults and reduces it for fifth graders, although it does not affect second grade performance. Experiment 2 shows that for second and fifth graders the error survives to problems that are not universally quantified and for second graders to problems that are not conditionals although are otherwise structurally similar. Experiment 3 compares various verbal formulations of such universally quantified conditionals: Second and fifth graders do not make the error when the quantification is expressed with the surface structure that makes its universality most explicit (all things ...); the error tendency is greatest when the indefinite article is used (if a thing ...); and formulations using any fall in between. We argue that such erroneous evaluations of universally quantified conditionals have more to do with the quantificational aspect than the conditional aspect of the problems; children interpret the indefinite article as existential, although they resist the error when the cue to universal quantification is completely clear. The error appears to result more from the surface-structure form of the stimuli than from an inability of children to appreciate the logic of universally quantified conditionals.