Reasoning From Uncertain Premises

Previous studies have shown that 1 participants are reluctant to accept a conclusion as certainly true when it is derived from a valid conditional argument that includes a doubtful premise, and 2 participants typically link the degree of uncertainty found in a given premise set to its conclusion. Two experiments were designed to further investigate these phenomena. Ninety adult participants in Experiment 1 were first asked to judge the validity of three conditional arguments Modus Ponens, Denial of the Antecedent, and Affirmation of the Consequent. They were then required to evaluate conclusion uncertainty as a function of two degrees of asserted uncertainty in the major conditional premise (If p then it is very probable that q and if p then it is not very probable that q) of the arguments from the first task that were otherwise unchanged. Results revealed an effect for asserted-uncertainty in two of the three argument forms. Marginal support was found for the hypothesis that perceived argument validity would be a predictor of performance. Experiment 2 investigated the way 40 adult participants combined two sources of asserted uncertainty, one in the major premise and another in the minor premise, when they had to score the uncertainty of the conclusion. The two most prominent kinds of responses were to choose the same likelihood as the weaker of the two expressed in the premises, or a lower one. However, the within-subject consistency was poor. Theoretical implications are discussed.     

Strategies during complex conditional inferences

In certain contexts reasoners reject instances of the valid Modus Ponens and Modus Tollens inference form in conditional arguments. Byrne (1989) observed this suppression effect when a conditional premise is accompanied by a conditional containing an additional requirement. In an earlier study, Rumain, Connell, and Braine (1983) observed suppression of the invalid inferences “the denial of the antecedent” and “the affirmation of the consequent” when a conditional premise is accompanied by a conditional containing an alternative requirement. Here we present three experiments showing that the results of Byrne (1989) and Rumain et al. (1983) are influenced by the answer procedure. When reasoners have to evaluate answer alternatives that only deal with the inferences that can be made with respect to the first conditional, then suppression is observed (Experiment 1). However, when reasoners are also given answer alternatives about the second conditional (Experiment 2) no suppression is observed. Moreover, contrary to the hypothesis of Byrne (1989), at least some of the reasoners do not combine the information of the two conditionals and do not give a conclusion based on the combined premise. Instead, we hypothesise that some of the reasoners have reasoned in two stages. In the first stage, they form a putative conclusion on the basis of the first conditional and the categorical premise, and in the second stage, they amend the putative conclusion in the light of the information in the second premise. This hypothesis was confirmed in Experiment 3. Finally, the results are discussed with respect to the mental model theory and reasoning research in general.     

有关条件推理中概率效应的实验研究

通过预备实验选取了四种不同条件概率的条件规则和四种不同前后件概率组合的条件规则作为实验材料,以大学生为被试,考察了两种概率因素(条件概率和前后件概率)对条件推理的演绎形式(MP、DA、AC、MT)以及变通形式(四卡问题)的影响。结果表明,两种概率因素对四种条件推理的影响都非常显著,研究进一步证实了人们对四种推理的认可程度主要与范畴前提的概率成正比的结论;但概率因素对四卡问题的解决影响不明显。     

A Probabilistic Interpretation of llIf-Then

This study proposes that subjects interpret thematic conditionals ('if p then q ') probabilistically in solving conditional reasoning problems. Experiment 1 found that subjects' correct responses increased with the perceived probability of q , given p for each of the four forms of conditional arguments: modus ponens (MP), modus tollens (MT), denial of the antecedent (DA), and affirmation of the consequent (AC). Experiment 2 ruled out two alternative explanations based on the comprehensibility of conditionals and on subjects interpreting conditionals as biconditionals. In Experiment 3, subjects solved two types of problems: (a) complete probabilistic problems, such as 'If p then q ; knowing p ; how probable is q ?', and (b) reduced probabilistic problems, such as 'Knowing p ; how probable is q ?' Two sources of information that determine the observable reasoning responses are identified. One source of information is based on one's general knowledge, and another is based on taking all premises into account.     

Induction as conditional probability judgment

Existing research on category-based induction has primarily focused on reasoning about blank properties, or predicates that are designed to elicit little prior knowledge. Here, we address reasoning about nonblank properties. We introduce a model of conditional probability that assumes that the conclusion prior probability is revised to the extent warranted by the evidence in the premise. The degree of revision is a function of the relevance of the premise category to the conclusion and the informativeness of the premise statement. An algebraic formulation with no free parameters accurately predicted conditional probabilities for single- and two-premise conditionals (Experiments 1 and 3), as well as problems involving negative evidence (Experiment 2).     

Bias in Conditional Inference: Implications for Mental Models and Mental Logic

Three experiments are reported in which subjects are given the opportunity to make any of the four inferences associated with conditional statements: modus ponens (MP), denial of the antecedent (DA), affirmation of the consequent (AC), and modus tollens (MT). The primary purpose of the research was to establish the generality and robustness of polarity biases that may be occasioned by systematic rotation of negative components in the conditional rules. In Experiments 1 & 2, three forms of conditionals were used: “if (not) p then (not) q”, “(not) p only if (not) q” and “(not) q if (not) p”. Experiment 1 used a conclusion evaluation task, whereas Experiment 2 used a conclusion production task. In Experiment 3, thematic conditionals were presented with and without a preceding scenario.

The biases investigated were (a) affirmative premise bias—the tendency to draw more inferences from affirmative premises and (b) negative conclusion bias—the tendency to draw more inferences with negative conclusions. The suggestive evidence for affirmative premise bias in the literature was not supported: very little evidence was found for it in the current experiments. Robust findings of negative conclusion bias were, however, found across the three experiments, although the bias was mostly restricted to DA and MT inferences. This suggests that the bias is best regarded as a difficulty with double negation.

The results are discussed with respect to both the mental logic and mental model accounts of propositional reasoning. Neither theory as currently formulated can explain all of our findings, although a plausible revision of each is considered.     

Causal and noncausal conditionals: An integrated model of interpretation and reasoning

We present an integrated model for the understanding of and the reasoning from conditional statements. Central assumptions from several approaches are integrated into a causal path model. According to the model, the cognitive availability of exceptions to a conditional reduces the subjective conditional probability of the consequent, given the antecedent. This conditional probability determines people's degree of belief in the conditional, which in turn affects their willingness to accept logically valid inferences. In addition to this indirect pathway, the model contains a direct pathway: Availability of exceptional situations directly reduces the endorsement of valid inferences. We tested the integrated model with three experiments using conditional statements embedded in pseudonaturalistic cover stories. An explicitly mentioned causal link between antecedent and consequent was either present (causal conditionals) or absent (arbitrary conditionals). The model was supported for the causal but not for the arbitrary conditional statements.     

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.     

Developmental and content effects in reasoning with causal conditionals

Two predictions derived from Markovits and Barrouillet's (2001) developmental model of conditional reasoning were tested in a study in which 72 twelve-year-olds, 80 fifteen-year-olds, and 104 adults received a paper-and-pencil test of conditional reasoning with causal premises ("if cause P then effect Q"). First, we predicted that conditional premises would induce more correct uncertainty responses to the Affirmation of the consequent and Denial of the antecedent forms when the antecedent term is weakly associated to the consequent than when the two are strongly associated and that this effect would decrease with age. Second, uncertainty responding to the Denial of the antecedent form ("P is not true") should be easier when the formulation of the minor premise invites retrieval of alternate antecedents ("if something other than P is true"). The results were consistent with the hypotheses and indicate the importance of retrieval processes in understanding developmental patterns in conditional reasoning with familiar premises.     

假言推理中的概率效应

近来Oaksford等人的研究表明,概率信息对假言推理的认知过程有着显著的影响,在一定程度上可以用它来解释和预测个体在推理中的行为反应。该文针对这种新的观点,就假言推理中所涉及的前后件概率信息、条件概率信息、连接概率信息、因果概率信息以及额外前提条件等对假言推理的影响分别做了简要的阐述,并介绍了一些相关的实验研究,最后作了总结和展望。     

Pronounced inferences: A study on inferential conditionals

An experimental study is reported which investigates the differences in interpretation between content conditionals (of various pragmatic types) and inferential conditionals. In a content conditional, the antecedent represents a requirement for the consequent to become true. In an inferential conditional, the antecedent functions as a premise and the consequent as the inferred conclusion from that premise. The linguistic difference between content and inferential conditionals is often neglected in reasoning experiments. This turns out to be unjustified, since we adduced evidence on the basis of a quantitative and a qualitative analysis that this difference has a manifest psychological relevance. For the inferential conditionals, participants appear to retrieve the order of events of the original content conditional on which it was based, before they start reasoning with it. The implications of this finding for reasoning research and linguistics will be discussed.     

Chronometric evidence for memory retrieval in causal conditional reasoning: The case of the association strength effect

According to many models of conditional reasoning, correct responses to the uncertain forms affirmation of consequent (AC) and denial of antecedent (DA) rely on the retrieval of an alternative antecedent from semantic memory. The main prediction issuing from this hypothesis of online retrieval is that the associative strength of the relevant information in long-term memory affects the latency of its retrieval and then of the correct response of uncertainty to AC and DA. This prediction was tested in 64 adults who performed a syllogism evaluation task with familiar causal conditional premises. As predicted, correct responses of uncertainty to AC and DA were not only more frequent but also faster when the antecedent term was weakly rather than strongly associated with the consequent. In line with the semantic framework, we also observed that this strength effect in response times was reversed for incorrect responses of certainty to AC.     

Indicatives,concessives, and evidential support

This paper discusses the issue of categorical acceptability of indicative and concessive conditionals. It presents experimental results in favour of two claims concerning the role of the evidential support relation for acceptability (or otherwise) of conditionals of both types. In particular, the results show that, contrary to fairly standard philosophical theorising, high probability of a conditional's consequent given its antecedent is necessary but not sufficient for the acceptability of that conditional, and that the antecedent being evidence for the consequent is a further acceptability condition. The results further show that the evidential support relation is crucial in differentiating between the acceptability of an indicative conditional and the acceptability of the corresponding concessive conditional: typically, the use of a concessive conditional signals that the corresponding conditional probability is high in spite of the fact that the antecedent is evidence against the consequent, or in any case is not evidence for the consequent.     

Conditionals As Representative Inferences

According to Adams (Inquiry 8:166–197, 1965), the acceptability of an indicative conditional goes with the conditional probability of the consequent given the antecedent. However, some conditionals seem to be inappropriate, although their corresponding conditional probability is high. These are cases with a missing link between antecedent and consequent. Other conditionals are appropriate even though the conditional probability is low. Finally, we have the so-called biscuit conditionals. In this paper we will generalize analyses of Douven (Synthese 164:19–44, 2008) and others to account for the appropriateness of conditionals in terms of evidential support. Our generalization involves making use of Value, or intensity. We will show how this generalization helps to account for biscuit conditionals and conditional threats and promises. Finally, a link is established between this analysis of conditionals and an analysis of generic sentences.

     

Reasoning from uncertain premises: Effects of expertise and conversational context

Four experiments investigated uncertainty about a premise in a deductive argument as a function of the expertise of the speaker and of the conversational context. The procedure mimicked everyday reasoning in that participants were not told that the premises were to be treated as certain. The results showed that the perceived likelihood of a conclusion was greater when the major or the minor premise was uttered by an expert rather than a novice (Experiment 1). The results also showed that uncertainty about the conclusion was higher when the major premise was uttered by a novice and an alternative premise by an expert, compared to when the major premise was uttered by an expert and the alternative by a novice (Experiment 2). Similarly, the believability of a conclusion was considerably lower when the minor premise was uttered by a novice and denied by an expert, as opposed to when an expert uttered the minor premise and a novice denied it (Experiment 3). Experiment 4 showed that the nature of the uncertainty induced by a denial of the minor premise depended on whether or not the context was a conversation. These results pose difficult problems for current theories of reasoning, as current theories are based on the results of experiments in which the premises are treated as certain. Our discussion of the results emphasises the importance of pragmatics in reasoning, namely, the role of general knowledge about the world in assessing the probability of a premise uttered by an expert or a novice and the role of interpretations of the premise based on pragmatic inferences in revising these initial probabilities.     

Kaufmann on the Probabilities of Conditionals

Kaufmann has recently argued that the thesis according to which the probability of an indicative conditional equals the conditional probability of the consequent given the antecedent under certain specifiable circumstances deviates from intuition. He presents a method for calculating the probability of a conditional that does seem to give the intuitively correct result under those circumstances. However, the present paper shows that Kaufmann’s method is inconsistent in that it may lead one to assign different probabilities to a single conditional at the same time.     

Reasoning with conditionals: Does every counterexample count? It’s frequency that counts

A series of experiments investigated what determines people's degree of belief in conditionals and their readiness to draw inferences from them. Information on the frequency of exceptions to conditional rules was contrasted with information about the number of different disabling conditions causing these exceptions. Experiments 1 and 2, using conditionals with arbitrary contents, revealed a strong effect of frequency information and no effect of disabling information. Experiment 3 established that, in the absence of frequency information, the disabling condition information used in Experiments 1 and 2 affected belief in the conditionals and inference acceptance, as has been found in many previous studies (Byrne, 1989; DeNeys, Schaeken, & d'Ydewalle, 2003b). Experiment 4 extended the results of Experiments 1 and 2 to everyday conditionals. The results show that belief in a conditional, and the confidence in inferences subsequently drawn from it, both depend on the subjective conditional probability of the consequent given the antecedent. This probability is estimated from the relative frequency of exceptions regardless of what causes them.     

A psychological study of unconnected conditionals

Unconnected conditionals, also called irrelevant conditionals, are sentences of form if A, C, whose antecedent and consequent bear no connection. According to the main theories of conditional reasoning, the truth or high probability of an antecedent and a consequent is sufficient to make true or highly probable the corresponding conditional. We tested this assumption and showed that it does not hold for unconnected conditionals. Furthermore, we investigated experimentally the factors which favour the endorsement of irrelevant conditional constructions and found that this rate increases when an analogy can be built between the antecedent and the consequent or when the conditional is asserted before its components.     

A Puzzle about Knowing Conditionals

We present a puzzle about knowledge, probability and conditionals. We show that in certain cases some basic and plausible principles governing our reasoning come into conflict. In particular, we show that there is a simple argument that a person may be in a position to know a conditional the consequent of which has a low probability conditional on its antecedent, contra Adams’ Thesis. We suggest that the puzzle motivates a very strong restriction on the inference of a conditional from a disjunction.