Conditionals,Context, and the Suppression Effect

Modus ponens is the argument from premises of the form If A, then B and A to the conclusion B (e.g., from If it rained, Alicia got wet and It rained to Alicia got wet). Nearly all participants agree that the modus ponens conclusion logically follows when the argument appears in this Basic form. However, adding a further premise (e.g., If she forgot her umbrella, Alicia got wet) can lower participants’ rate of agreement—an effect called suppression. We propose a theory of suppression that draws on contemporary ideas about conditional sentences in linguistics and philosophy. Semantically, the theory assumes that people interpret an indicative conditional as a context‐sensitive strict conditional: true if and only if its consequent is true in each of a contextually determined set of situations in which its antecedent is true. Pragmatically, the theory claims that context changes in response to new assertions, including new conditional premises. Thus, the conclusion of a modus ponens argument may no longer be accepted in the changed context. Psychologically, the theory describes people as capable of reasoning about broad classes of possible situations, ordered by typicality, without having to reason about individual possible worlds. The theory accounts for the main suppression phenomena, and it generates some novel predictions that new experiments confirm.     

Developmental trends in everyday conditional reasoning: the retrieval and inhibition interplay

Two experiments examined developmental patterns in children’s conditional reasoning with everyday causal conditionals. In Experiment 1, a group of pre-, early, young, and late adolescents generated counterexamples for a set of conditionals to validate developmental claims about the counterexample retrieval capacity. In Experiment 2, participants in the same age range were presented with a conditional reasoning task with similar conditionals. Experiment 1 established that counterexample retrieval increased from preadolescence to late adolescence. Experiment 2 showed that acceptance rates of the invalid affirmation of the consequent inference gradually decreased in the same age range. Acceptance rates of the valid modus ponens inference showed a U-shaped pattern. After an initial drop from preadolescence to early adolescence, modus ponens acceptance ratings increased again after the onset of adolescence. Findings support the claim that the development of everyday conditional reasoning can be characterized as an interplay between the development of a counterexample retrieval and inhibition process.     

Conditional reasoning and causation

An experiment was conducted to investigate the relative contributions of syntactic form and content to conditional reasoning. The content domain chosen was that of causation. Conditional statements that described causal relationships (if mean value of cause, then mean value of effect) were embedded in simple arguments whose entailments are governed by the rules of truth-functional logic (i.e., modus ponens, modus tollens, denying the antecedent, and affirming the consequent). The causal statements differed in terms of the number of alternative causes and disabling conditions that characterized the causal relationship. (A disabling condition is an event that prevents an effect from occurring even though a relevant cause is present). Subjects were required to judge whether or not each argument's conclusion could be accepted. Judgements were found to vary systematically with the number of alternative causes and disabling conditions. Conclusions of arguments based on conditionals with few alternative causes or disabling conditions were found to be more acceptable than conclusions based on those with many.     

Inference suppression and semantic memory retrieval: Every counterexample counts

Reasoning with conditionals involving causal content is known to be affected by retrieval of counterexamples from semantic memory. In this study we examined the characteristics of this search process in everyday conditional reasoning. In Experiment 1 we manipulated the number (zero to four) of explicitly presented counterexamples (alternative causes or disabling conditions) for causal conditionals. In Experiment 2, using a generation pretest, we measured the number of counterexamples participants could retrieve for a set of causal conditionals. One month after the pretest, participants were presented a reasoning task with the same conditionals. The experiments indicated that acceptance of modus ponens linearly decreased with every additionally retrieved disabler, whereas affirmation of the consequent acceptance linearly decreased as a function of the number of retrieved alternatives. Results for denial of the antecedent and modus tollens were less clear. The findings show that the search process does not necessarily stop after retrieval of a single counterexample and that every additional counterexample has an impact on the inference acceptance.     

Advice conditionals about tips and warnings: interpretations and inferences

Two experiments examine how people interpret and reason about advice conditionals, such as tips, for example, “if you study more your grades will improve”, and warnings, for example, “if you stop exercising you will gain weight”. Experiment 1 showed that when participants reason about whether a tip or warning could be true in different situations, their judgments correspond to a biconditional or conditional interpretation on about half of all trials, but to an enabling or tautology interpretation on many others. Experiment 2 showed that participants make few modus ponens and tollens inferences from tips and warnings, and more modus ponens inferences from tips than warnings. The implications for alternative theories are discussed.     

When will is not the same as should: The role of modals in reasoning with legal conditionals

Naïve reasoners reject logically valid conclusions from conditional rules if they can think of exceptions in which the antecedent is true, but the consequent is not. However, when reasoning with legal conditionals (e.g., “If a person kills another human, then this person should be punished for manslaughter”) people hardly consider exceptions but evaluate conclusions depending on their own sense of justice. We show that participants’ reluctance to consider exceptions in legal reasoning depends on the modal auxiliary used. In two experiments we phrased legal conditionals either with the modal “should” (i.e., “ . . . then this person should be punished”), or with “will” (i.e., “ . . . then this person will be punished”) and presented them as modus ponens or modus tollens inferences. Participants had to decide whether the offender should or will be punished (modus ponens) or whether the offender indeed committed the offence (modus tollens). For modus ponens inferences phrased with “should” we replicate previous findings showing that participants select conclusions on the basis of their own sense of justice (Experiments 1 and 2). Yet, when the legal conditional is phrased with the modal “will” this effect is attenuated (Experiments 1 and 2), and exceptions are considered (Experiment 1). The modal auxiliary did not affect modus tollens inferences.     

Individual differences in conditional reasoning: A dual-process account

Dual-process theories of conditional reasoning predict that relationships among four basic logical forms, and to intellectual ability and thinking predictions, are most evident when conflict arises between experiential and analytic processing (e.g., Stanovich & West, ). To test these predictions, 210 undergraduates were presented with conditionals for which the consequents were either weakly or strongly associated with alternative antecedents (i.e., WA and SA problems, respectively). Consistent with predictions, modus ponens inferences were not related to inferences on the uncertain forms (affirmation of the consequent, denial of the antecedent). On WA problems, modus tollens, affirmation of the consequent, and denial of the antecedent were related to each other and to verbal ability. Modus ponens was linked to verbal ability only when disabling conditions were activated. In accord with the predictions of Stanovich and West (), on most problems, thinking dispositions predicted variance in inferences independently from verbal ability. We argue that a largely automatic experiential processing system governs performance on modus ponens, unless disablers are activated. Consciously controlled analytic processing predominates on the uncertain forms and, under some conditions, on modus tollens.     

The emergence of inferential rules: The use of pragmatic reasoning schemas by preschoolers

This study contrasts the pragmatic view with the natural logic view regarding the origin of inferential rules in conditional reasoning. The pragmatic view proposes that pragmatic rules emerge first, and the generalizations of these produce formal rules. In contrast, the natural logic view proposes that the formal rules emerge first and serve as a core that is then supplemented by pragmatic rules. In an experiment, scenarios involving conditional rules in different contexts, permission and arbitrary, were administered to independent groups of preschool children. To rule out the matching bias [Evans, J. St. B. T., & Lynch, J. S. (1973). Matching bias in the selection task. Br J Psychol 64, 391–397] as a possible explanation of reasoning performance, children were given conditional rules with a negated consequent. The results show that in the arbitrary context modus tollens (MT) was unavailable, and the use of modus ponens (MP) was unstable. In contrast, children in the permission context reliably used both MP and MT. In addition, they realized that a conditional rule does not imply a definite answer when the consequent holds. These findings suggest that, in their explicit forms, pragmatic rules emerge earlier than formal rules and in particular, even as basic a rule as MP is generalized from a context-specific form to a context-general one in preschool children.     

A successive-conditionalization approach to disjunctive and syllogistic reasoning

With p and q each standing for a familiar event, a disjunctive statement, “either p or q”, seems quite different from its material conditional, “if not p then q”. The notions of sufficiency and necessity seem specific to conditional statements. It is surprising, however, to find that perceived sufficiency and necessity affect disjunctive reasoning in the way they affect conditional reasoning. With B and C each standing for a category name, a universal statement, “all B are C”, seems stronger than its logically equivalent conditional statement, “if B then C”. However, the effects of perceived sufficiency or necessity were found to be as pronounced in conditional reasoning as in syllogistic reasoning. Furthermore, two experiments also showed that (a) MP (modus ponens)-comparable disjunctive reasoning was as difficult as MT (modus tollens)-comparable disjunctive reasoning, and that (b) MT-comparable syllogisms were easier to solve than MT problems in conditional reasoning.     

Causal conditional reasoning and semantic memory retrieval: a test of the semantic memory framework

This study tested and refined a framework that proposes a mechanism for retrieving alternative causes and disabling conditions (Cummins, 1995) during reasoning. Experiment 1 examined the relation between different factors affecting retrieval. The test revealed high correlations between the number of possible alternative causes or disabling conditions and their strength of association and plausibility. Experiment 2 explored the hypothesis that due to a more extended search process, conditional inferences would last longer when many alternative causes or disabling conditions were available. Affirmation of the consequent (AC) and modus ponens (MP) latencies showed the hypothesized pattern. Denial of the antecedent (DA) and modus tollens (MT) inferences did not show latency effects. The experiment also identified an effect of the number of disabling conditions on AC and DA acceptance. Experiment 3 measured efficiency of disabler retrieval by a limited time, disabler generation task. As predicted, better disabler retrieval was related to lower acceptance of the MP and MT inferences.     

Hypothetical and certain inferences from conditional arguments read in texts

In 3 experiments, affirmative and hypothetical probes were presented after narrative texts containing conditional arguments. According to the data, readers represented modus ponens deductions as certain, except when it was only a weakly necessary cause of a given effect. They represented any logically invalid inferences resulting from affirming the conditional consequent as hypothetical, except when it was the effect of strongly sufficient cause. Accordingly, readers must be processing conditional syntax as an asymmetric constraint. However, the underlying causal knowledge can be sufficient either to discredit or warrant the inferences. Thus according to the theory of natural logic, readers can draw formal deductions and be convinced of their necessity. This provides further evidence that readers can represent their inferences as hypothetical (N. Campion, 2004).     

One's Modus Ponens: Modality,Coherence and Logic

Recently, there has been a shift away from traditional truth‐conditional accounts of meaning towards non‐truth‐conditional ones, e.g., expressivism, relativism and certain forms of dynamic semantics. Fueling this trend is some puzzling behavior of modal discourse. One particularly surprising manifestation of such behavior is the alleged failure of some of the most entrenched classical rules of inference; viz., modus ponens and modus tollens. These revisionary, non‐truth‐conditional accounts tout these failures, and the alleged tension between the behavior of modal vocabulary and classical logic, as data in support of their departure from tradition, since the revisionary semantics invalidate some of these patterns. I, instead, offer a semantics for modality with the resources to accommodate the puzzling data while preserving classical logic, thus affirming the tradition that modals express ordinary truth‐conditional content. My account shows that the real lesson of the apparent counterexamples is not the one the critics draw, but rather one they missed: namely, that there are linguistic mechanisms, reflected in the logical form, that affect the interpretation of modal language in a context in a systematic and precise way, which have to be captured by any adequate semantic account of the interaction between discourse context and modal vocabulary. The semantic theory I develop specifies these mechanisms and captures precisely how they affect the interpretation of modals in a context, and do so in a way that both explains the appearance of the putative counterexamples and preserves classical logic.     

Inferences from counterfactual threats and promises

We examine how people understand and reason from counterfactual threats, for example, "if you had hit your sister, I would have grounded you" and counterfactual promises, for example, "if you had tidied your room, I would have given you ice-cream." The first experiment shows that people consider counterfactual threats, but not counterfactual promises, to have the illocutionary force of an inducement. They also make the immediate inference that the action mentioned in the "if" part of the counterfactual threat and promise did not occur. The second experiment shows that people make more negative inferences (modus tollens and denial of the antecedent) than affirmative inferences (modus ponens and affirmation of the consequent) from counterfactual threats and promises, unlike indicative threats and promises. We discuss the implications of the results for theories of the mental representations and cognitive processes that underlie conditional inducements.     

Being in a position to know

The concept of being in a position to know is an increasingly popular member of the epistemologist’s toolkit. Some have used it as a basis for an account of propositional justification. Others, following Timothy Williamson, have used it as a vehicle for articulating interesting luminosity and anti-luminosity theses. It is tempting to think that while knowledge itself does not obey any closure principles, being in a position to know does. For example, if one knows both p and ‘If p then q’, but one dies or gets distracted before being able to perform a modus ponens on these items of knowledge and for that reason one does not know q, one is still plausibly in a position to know q. It is also tempting to suppose that, while one does not know all logical truths, one is nevertheless in a position to know every logical truth. Putting these temptations together, we get the view that being in a position to know has a normal modal logic. A recent literature has begun to investigate whether it is a good idea to give in to these twin temptations—in particular the first one. That literature assumes very naturally that one is in a position to know everything one knows and that one is not in a position to know things that one cannot know. It has succeeded in showing that, given the modest closure condition that knowledge is closed under conjunction elimination (or ‘distributes over conjunction’), being a position to know cannot satisfy the so-called K axiom (closure of being in a position to know under modus ponens) of normal modal logics. In this paper, we explore the question of the normality of the logic of being in a position to know in a more far-reaching and systematic way. Assuming that being in a position to know entails the possibility of knowing and that knowing entails being in a position to know, we can demonstrate radical failures of normality without assuming any closure principles at all for knowledge. (However, as we will indicate, we get further problems if we assume that knowledge is closed under conjunction introduction.) Moreover, the failure of normality cannot be laid at the door of the K axiom for knowledge, since the standard principle NEC of necessitation also fails for being in a position to know. After laying out and explaining our results, we briefly survey the coherent options that remain.

     

Reasoning with conditionals: a test of formal models of four theories

The four dominant theories of reasoning from conditionals are translated into formal models: The theory of mental models (Johnson-Laird, P. N., & Byrne, R. M. J. (2002). Conditionals: a theory of meaning, pragmatics, and inference. Psychological Review, 109, 646-678), the suppositional theory (Evans, J. S. B. T., & Over, D. E. (2004). If. Oxford: Oxford University Press), a dual-process variant of the model theory (Verschueren, N., Schaeken, W., & d'Ydewalle, G. (2005). A dual-process specification of causal conditional reasoning. Thinking &Reasoning, 11, 278-293), and the probabilistic theory (Oaksford, M., Chater, N., & Larkin, J. (2000). Probabilities and polarity biases in conditional inference. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26, 883-899). The first three theories are formalized as multinomial models. The models are applied to the frequencies of patterns of acceptance or rejection across the four basic inferences modus ponens, acceptance of the consequent, denial of the antecedent, and modus tollens. Model fits are assessed for two large data sets, one representing reasoning with abstract, basic conditionals, the other reflecting reasoning with pseudo-realistic causal and non-causal conditionals. The best account of the data was provided by a modified version of the mental-model theory, augmented by directionality, and by the dual-process model.     

Conditional reasoning and conditionalization

In solving conditional reasoning problems, reasoners are assumed to compute the probability of the conclusion, conditionalizing first on the categorical premise, giving the knowledge-based component, and conditionalizing then on the conditional-statement premise, from which the assumption-based component is derived. Because reasoners find it difficult to compute the second-step conditionalization except when the conditional-statement premise is found to be related to the result of the first-step conditionalization as for modus ponens or, possibly, for modus tollens, the knowledge-based component generally dominates reasoning performance. After representing all the possible cases in which conditional-argument forms may appear, this approach was found to be consistent with the results from the 3 experiments reported in this study, whereas 2 alternative hypotheses account for only some of the results.     

Vann McGee's counterexample to modus ponens

Conclusion After rejecting some unjustified criticism of McGee's claim about modus ponens, I argued for two conditionals: If the Adams-Appiah theory of conditionals is correct, then modus ponens is saved from McGee's counterexample. If the Lewis-Jackson theory is correct, the same holds. Furthermore, I raised some doubts whether the antecedents of these conditionals are true. But, if one of them were, then, by modus ponens, modus ponens would be generally valid.     

Numerical evaluation of the validity of experimental proofs in biology

Summary This paper suggests a method to calculate a degree of validity for the proof of a statement which is derived from empirical statements by means of logic conclusions. The empirical statements are assumed not to be completely valid or their validity to be doubtful. The suggested rules are consistent with two-valued logic, yield decreasing validities with increasing number of applications of modus ponens and obey the law of the excluded middle. The actual calculation of validity values, the relation of the suggested method to some truth tables of multi-valued logic and to fuzzy logic are discussed.     

Suppressing valid inferences with conditionals

Three experiments are reported which show that in certain contexts subjects reject instances of the valid modus ponens and modus tollens inference form in conditional arguments. For example, when a conditional premise, such as: If she meets her friend then she will go to a play, is accompanied by a conditional containing an additional requirement: If she has enough money then she will go to a play, subjects reject the inference from the categorical premise: She meets her friend, to the conclusion: She will go to a play. Other contexts suppress the conditional fallacies. The first experiment demonstrates the effects of context on conditional reasoning. The second experiment shows that the inference suppression disappears when the categorical premise refers to both of the antecedents, such as: She meets her friend and she has enough money. In this case, subjects make both the valid inferences and the fallacies, regardless of the contextual information. The third experiment establishes that when subjects are given general information about the duration of a situation in which a conditional inducement was uttered, such as: If you shout then I will shoot you, they reject both the valid inferences and the fallacies. The results suggest that the interpretation of premises plays an even more central role in reasoning than has previously been admitted.     

What we know and what to do

This paper discusses an important puzzle about the semantics of indicative conditionals and deontic necessity modals (should, ought, etc.): the Miner Puzzle (Parfit, ms; Kolodny and MacFarlane, J Philos 107:115–143, 2010). Rejecting modus ponens for the indicative conditional, as others have proposed, seems to solve a version of the puzzle, but is actually orthogonal to the puzzle itself. In fact, I prove that the puzzle arises for a variety of sophisticated analyses of the truth-conditions of indicative conditionals. A comprehensive solution requires rethinking the relationship between relevant information (what we know) and practical rankings of possibilities and actions (what to do). I argue that (i) relevant information determines whether considerations of value may be treated as reasons for actions that realize them and against actions that don’t, (ii) incorporating this normative fact requires a revision of the standard ordering semantics for weak (but not for strong) deontic necessity modals, and (iii) an off-the-shelf semantics for weak deontic necessity modals, due to von Fintel and Iatridou, which distinguishes “basic” and “higher-order” ordering sources, and interprets weak deontic necessity modals relative to both, is well-suited to this task. The prominence of normative considerations in our proposal suggests a more general methodological lesson: formal semantic analysis of natural language modals expressing normative concepts demands that close attention be paid to the nature of the underlying normative phenomena.