Illusory inferences in conditional expressions.

A robber points a gun at a cashier and says: "Only one of these two options is true: If you conceal the combination to the safe, then I kill you; otherwise, if you don´t conceal the combination to the safe, then I kill you." Hearing this statement, most people conclude that, in either case, "I kill you." This is an illusory response, in fact; the valid conclusion states "I don´t kill you." The research reported here studied the roles that different expressions of conditionals ("if-then," "only if," and "if and only if") play in the illusory response. Three experiments show that participants inferred the conclusion "I kill you" from the conditional "if-then" and "I may or may not kill you" from the conditional "only if," while selecting both options with similar frequency for the biconditional "if and only if." These results shed light on the main theories of deductive reasoning.

Inferences from the negation of counterfactual and semifactual conditionals.

Our goal was to study how people understand the negation of counterfactuals (such as "Antonio denied/said that it is false that if Messi had played, then Barcelona would have won") and semifactuals (such as "Antonio denied that even if Messi had played, Barcelona would have won"). Previous studies have shown that participants negated basic conditionals using small-scope interpretations by endorsing a new conditional with the negated consequent, but also by making large-scope interpretations, endorsing a conjunction with the negated consequent. Three experiments showed that when participants were asked whether the negation of a counterfactual (Experiments 1 and 2) or semifactual (Experiment 3) conditional was followed by a new conditional, they made a small-scope interpretation, endorsing the same conditional with the negated consequent (e.g., "if/even if Messi had played, Barcelona would not have won"). However, they also accepted the conditional with the negated antecedent for semifactuals (e.g., "even if Messi had not played, Barcelona would have won"). When participants were asked whether the negation of a counterfactual or semifactual conditional is followed by a conjunction, they endorsed the conjunction with both the negated antecedent and the consequent (e.g., "Messi did not play and Barcelona did not win"), but again they accepted the conjunction with the negated antecedent only for semifactuals (e.g., "Messi did not play and Barcelona did win"). These results have implications for the main theories of reasoning.

Possibilities and the parallel meanings of factual and counterfactual conditionals.

The mental model theory postulates that the meanings of conditionals are based on possibilities. Indicative conditionals-such as "If he is injured tomorrow, then he will take some leave"-have a factual interpretation that can be paraphrased as It is possible, and remains so, that he is injured tomorrow, and in that case certain that he takes some leave. Subjunctive conditionals, such as, "If he were injured tomorrow, then he would take some leave," have a prefactual interpretation that has the same paraphrase. But when context makes clear that his injury will not occur, the subjunctive has a counterfactual paraphrase, with the first clause: It was once possible, but does not remain so, that he will be injured tomorrow. Three experiments corroborated these predictions for participants' selections of paraphrases in their native Spanish, for epistemic and deontic conditionals, for those referring to past and to future events, and for those with then clauses referring to what may or must happen. These results are contrary to normal modal logics. They are also contrary to theories based on probabilities, which are inapplicable to deontic conditionals, such as, "If you have a ticket, then you must enter the show."

The compatibility heuristic in non-categorical hypothetical reasoning: inferences between conditionals and disjunctions.

A new theory explains how people make hypothetical inferences from a premise consistent with several alternatives to a conclusion consistent with several alternatives. The key proposal is that people rely on a heuristic that identifies compatible possibilities. It is tested in 7 experiments that examine inferences between conditionals and disjunctions. Participants accepted inferences between conditionals and inclusive disjunctions when a compatible possibility was immediately available, in their binary judgments that a conclusion followed or not (Experiment 1a) and ternary judgments that included it was not possible to know (Experiment 1b). The compatibility effect was amplified when compatible possibilities were more readily available, e.g., for 'A only if B' conditionals (Experiment 2). It was eliminated when compatible possibilities were not available, e.g., for 'if and only if A B' bi-conditionals and exclusive disjunctions (Experiment 3). The compatibility heuristic occurs even for inferences based on implicit negation e.g., 'A or B, therefore if C D' (Experiment 4), and between universals 'All A's are B's' and disjunctions (Experiment 5a) and universals and conditionals (Experiment 5b). The implications of the results for alternative theories of the cognitive processes underlying hypothetical deductions are discussed.

Similarities and differences in understanding negative and affirmative counterfactuals and causal assertions: Evidence from eye-tracking.

Two eye-tracking experiments compared affirmative and negative counterfactuals, "if she had (not) arrived early, she would (not) have bought roses" and affirmative and negative causal assertions, "Because she arrived (did not arrive) early, she bought (did not buy) roses." When participants heard a counterfactual, they looked on screen at words corresponding to its conjecture ("roses"), and its presupposed facts ("no roses"), whereas for a causal assertion, they looked only at words corresponding to the facts. For counterfactuals, they looked at the conjecture first, and later the presupposed facts, and at the latter more than the former. The effect was more pronounced for negative counterfactuals than affirmative ones because the negative counterfactual's presupposed facts identify a specific item ("she bought roses"), whereas the affirmative counterfactual's presupposed facts do not ("she did not buy roses"). Hence, when participants were given a binary context, "she did not know whether to buy roses or carnations," they looked primarily at the presupposed facts for both sorts of counterfactuals. We discuss the implications for theories of negation, the dual meaning of counterfactuals, and their relation to causal assertions.

How people keep track of what is real and what is imagined: The epistemic status of counterfactual alternatives to reality.

When people understand a counterfactual such as "if it had been a good year, there would have been roses," they simulate the imagined alternative to reality, for example, "there were roses," and the actual reality, as known or presupposed, for example, "there were no roses." Seven experiments examined how people keep track of the epistemic status of these possibilities, by priming participants to anticipate a story would continue about one or the other. When participants anticipated the story would continue about how the current reality related to the past presupposed reality, they read a target description about reality more rapidly than one about the imagined alternative, indicating they had prioritized access to their mental representation of reality; but when they anticipated the story would continue about how the current reality related to the imagined alternative to reality, they read a target description about the imagined alternative and one about reality equally rapidly, indicating they had maintained access to both (Experiment 1), unlike for stories with no counterfactuals (Experiments 2 and 3). The tendency is not invariant: it appears immune to remote experience (Experiments 4 and 5), but it is influenced by immediate experience (Experiments 6 and 7). The results have implications for theories of reality monitoring, reasoning, and imagination. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

The Suppression of Inferences From Counterfactual Conditionals.

We examine two competing effects of beliefs on conditional inferences. The suppression effect occurs for conditionals, for example, "if she watered the plants they bloomed," when beliefs about additional background conditions, for example, "if the sun shone they bloomed" decrease the frequency of inferences such as modus tollens (from "the plants did not bloom" to "therefore she did not water them"). In contrast, the counterfactual elevation effect occurs for counterfactual conditionals, for example, "if she had watered the plants they would have bloomed," when beliefs about the known or presupposed facts, "she did not water the plants and they did not bloom" increase the frequency of inferences such as modus tollens. We report six experiments that show that beliefs about additional conditions take precedence over beliefs about presupposed facts for counterfactuals. The modus tollens inference is suppressed for counterfactuals that contain additional conditions (Experiments 1a and 1b). The denial of the antecedent inference (from "she did not water the plants" to "therefore they did not bloom") is suppressed for counterfactuals that contain alternatives (Experiments 2a and 2b). We report a new "switched-suppression" effect for conditionals with negated components, for example, "if she had not watered the plants they would not have bloomed": modus tollens is suppressed by alternatives and denial of the antecedent by additional conditions, rather than vice versa (Experiments 3a and 3b). We discuss the implications of the results for alternative theories of conditional reasoning.

The Comprehension of Counterfactual Conditionals: Evidence From Eye-Tracking in the Visual World Paradigm.

Three experiments tracked participants' eye-movements to examine the time course of comprehension of the dual meaning of counterfactuals, such as "if there had been oranges then there would have been pears." Participants listened to conditionals while looking at images in the visual world paradigm, including an image of oranges and pears that corresponds to the counterfactual's conjecture, and one of no oranges and no pears that corresponds to its presumed facts, to establish at what point in time they consider each one. The results revealed striking individual differences: some participants looked at the negative image and the affirmative one, and some only at the affirmative image. The first experiment showed that participants who looked at the negative image increased their fixation on it within half a second. The second experiment showed they do so even without explicit instructions, and the third showed they do so even for printed words.

Initial models in conditionals: evidence from priming.

We examined the comprehension of different types of conditionals. We measured the reading time of sentences primed by different types of conditionals (Experiments 1 and 2). We found that the participants read not-p and not-q faster when it was primed by the conditional form p if q and they were slower to read p and q when it was primed by the conditional form p only if q. This effect disappeared in the second experiment, where the order of the elements was reversed (q and p and not-q and not-p). These results suggest that the conditional form p if q elicits an initial representation "from p to q" with two possibilities, while the conditional form p only if q elicits a reverse representation with only one possibility. The third experiment showed that there were effects of the order only for the conditional if p then q, which confirms the reverse representation hypothesis. We discuss the implications of these results for different theories of conditional comprehension.

Thinking About the Opposite of What Is Said: Counterfactual Conditionals and Symbolic or Alternate Simulations of Negation.

When people understand a counterfactual such as "if the flowers had been roses, the trees would have been orange trees," they think about the conjecture, "there were roses and orange trees," and they also think about its opposite, the presupposed facts. We test whether people think about the opposite by representing alternates, for example, "poppies and apple trees," or whether models can contain symbols, for example, "no roses and no orange trees." We report the discovery of an inference-to-alternates effect-a tendency to make an affirmative inference that refers to an alternate even from a negative minor premise, for example, "there were no orange trees, therefore there were poppies." Nine experiments show the inference-to-alternates effect occurs in a binary context, but not a multiple context, and for direct and indirect reference; it can be induced and reduced by prior experience with similar inferences, and it also occurs for indicative conditionals. The results have implications for theories of counterfactual conditionals, and of negation.

Directional effect in double conditionals with a construction task: The semantic hypothesis.

The goal of this paper is to test the main predictions of the semantic hypothesis about the directional effect in double conditionals (such as, 'A only if B/only if C, B') with a construction task. The semantic hypothesis claims that directional effect can be explained by the inherent directionality of the relation between the relatum and the target object of the premises. According to this hypothesis, a directional effect should occur if only one of the end-terms of the premises takes the role of relatum: a) if the end-term that plays the role of relatum is in the first premise, a forward directional effect is predicted (from A to C); and b) if the end-term that plays the role of relatum is in the second premise, a backward directional effect is predicted (from C to A). On the other hand, it claims that there should be no directional effect when both end-terms take the role of relatum or when neither of the end-terms plays the role of relatum. Three experiments confirmed the main predictions of the semantic hypothesis in a construction task.

Priming effect in indicative and subjunctive exceptive conditionals.

We report the results of three experiments that examine the mental representations underlying the comprehension stages of negative exceptive conditionals using subjunctive mood ('B a menos que A', 'B a no ser que A'; 'B excepto que A'='B unless A') and indicative mood ('B excepto si A' and 'B salvo si A'='B except if A'). The mental representations during the comprehension stage were analyzed using a priming methodology. All experiments showed that participants read the true possibility 'not-B & A' faster when it was primed by exceptive conditionals requiring the subjunctive mood than when it was primed by exceptive conditionals requiring the indicative mood; other possibilities ('B & A', 'B & not-A', 'not-B & not-A') were primed equally by both connectives. The experiments showed that (a) when people understand negative exceptive conditionals using the subjunctive mood, such as 'B a menos que A'/'B a no ser que A'/'B excepto que A', they access the true possibilities 'not-B & A' and 'B & not-A', and (b) when they understand negative exceptive conditionals using the indicative mood, such as 'B excepto si A'/'B salvo si A', they access 'B & not-A', but not 'not-B & A'. We discuss the implications of this for current theories of reasoning.

Early and late processes in syllogistic reasoning: evidence from eye-movements.

An eye-movement monitoring experiment was carried out to examine the effects of the difficulty of the problem (simple versus complex problems) and the type of figure (figure 1 or figure 4) on the time course of processing categorical syllogisms. The results showed that the course of influence for these two factors is different. We found early processing effects for the figure but not for the difficulty of the syllogism and later processing effects for both the figure and the difficulty. These results lend support to the Model Theory (Johnson-Laird, P. N., Byrne, R. M. J. (1991). Deduction. Hillsdale, New Jersey: LEA.) as opposed to other theories of reasoning (Chater, N., Oaksford, M. (1999). The probability heuristics model of syllogistic reasoning. Cognitive Psychology, 38, 191-258; Rips, L. J. (1994). The psychology of proof. Cambridge, MA: MIT Press; Rips, L. J. (1994). The psychology of proof. Cambridge, MA: MIT Press).

Counterfactual and semifactual conditionals prime alternative possibilities.

The authors examined in 3 experiments the comprehension of counterfactuals, such as "If it had rained, the plants would have bloomed," and semifactuals, such as "Even if it had rained, the plants would have bloomed," compared with indicative conditionals, "If it rained, the plants bloomed." The first experiment showed that people read the negative conjunction, "not p and not q" faster when it was primed by a counterfactual than when it was primed by an indicative conditional. They read the affirmative conjunction, "p and q" equally quickly when it was primed by either conditional. The 2nd experiment showed that people read the negated-antecedent conjunction, "not p and q" faster when it was primed by a semifactual conditional. The 3rd experiment corroborated these results in a direct comparison of counterfactuals and semifactuals. The authors discuss the implications of the results for the mental representations of different conditionals.

Directionality Effect in Double Conditionals.

Directionality effect in deductive reasoning is a very well-known phenomenon that shows that the percentage of forward or backward inferences that participants make depends on the conditional form used. A new extension of the semantic hypothesis (Oberauer & Wilhelm, 2000) is presented to explain the directionality effect in double conditionals with different directionality. This hypothesis claims that the directional effect depends on which term plays the role of relatum. It also makes several novel claims which have been confirmed in three experiments: Experiments 1 and 2 showed there were more forward than backward inferences when the end-term that played the role of relatum was in the first premise, experiment 1: t (45) = 2.73, p < .01, experiment 2: t (38) = 12.06, p < .05, but there were more backward than forward inferences when the end-term that played the role of relatum was in the second premise, experiment 1: t (45) = 2.84, p < .01, experiment 2: t (38) = 2.21, p < .04. Experiment 3 showed that there was no directional effect when both end-terms played the role of relatum, t (34) = 1.39, p = .17, or when both middle-terms (or neither of the end-terms) played the role of relatum, t (34) = .78, p = .44. These experiments confirmed the predictions of the new extension of the semantic hypothesis.

The influence of verbal mood in exceptive conditional reasoning: indicative versus subjunctive.

BACKGROUND: We report the results of two experiments that examine the mental representations underlying the comprehension and reasoning stages of negative exceptive conditionals requiring the subjunctive ('B a menos que A' and 'B a no ser que A' = 'B unless A') and the indicative mood ('B excepto si A' and 'B salvo si A' = 'B except if A'). METHOD: A truth table task was employed to infer the mental representation that people have in mind when they reason with negative exceptive conditionals. RESULTS: Both experiments showed that participants selected the possibility 'B & not-A' more frequently than the possibility 'not-B & A' when the conditional required the indicative mood, but they selected the possibilities 'B & not-A' and 'not-B & A' equally frequently when the conditional required the subjunctive mood. CONCLUSIONS: Exceptive conditionals in the subjunctive mood lead people to think in terms of dual possibilities, whereas the indicative mood leads people to consider just one possibility.

People think about what is true for conditionals, not what is false: only true possibilities prime the comprehension of "if".

We report the results of two priming experiments that examine the comprehension of conditionals--for example, "if there are apples then there are oranges"--and biconditionals--for example, "if and only if there are apples then there are oranges". The first experiment showed that participants read a biconditional faster when it was primed by a true possibility, "there were apples and there were oranges" than when it was primed by a false possibility, "there were no apples and there were oranges"; a conditional was primed equally by both possibilities. The second experiment showed that participants read the negated-antecedent conjunction faster when it was primed by a conditional than when it was primed by a biconditional; the affirmative conjunction was primed equally by both connectives. The experiments show that (a) when people understand "if A then B", they access the true possibilities, "A and B", and "not-A and B", and (b) when they understand "if and only if A then B" they access "A and B", but they do not access "not-A and B". We discuss their implications for current theories of reasoning.

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.