Developmental changes in conditional reasoning: Linguistic or logical?

This study investigated reasoning with abstract conditional sentences as a function of age. Subjects from third-grade to eleventh-grade were required to evaluate the conclusion of several conditional arguments. The results confirmed the previously established finding that performance improves with age, particularly between 11 and 15 years. This finding could be interpreted to indicate that individuals become more logical as they get older. However, another possible interpretation is that the meaning of a conditional sentence like pthenq for naive Ss may not always be given by the truth function pq true, $\text{p}\text{q}$ false, $\text{p}\text{q}$ true, and $\text{p}\text{q}$ true. Further analysis suggested that at 9 years individuals treat the connective in the sentence, ifpthenq, as if it were either a conjunctive or a biconditional, that the conjunctive meaning disappears with increasing age and after 13 years is gradually superceded by the conditional.     

Comparing the meanings of “if” and “all”

In this study, we compared the everyday meanings of conditionals (“if p then q”) and universally quantified statements (“all p are q”) when applied to sets of elements. The interpretation of conditionals was predicted to be directly related to the conditional probability, such that P(“if p then q”) = P(q|p). Quantified statements were assumed to have two interpretations. According to an instance-focused interpretation, quantified statements are equivalent to conditionals, such that P(“all p are q”) = P(q|p). According to a set-focused interpretation, “all p are q” is true if and only if every instance in set p is an instance of q, so that the statement would be accepted when P(q|p) = 1 and rejected when this probability was below 1. We predicted an instance-focused interpretation of “all” when the relation between p and q expressed a general law for an infinite set of elements. A set-focused interpretation of “all” was predicted when the relation between p and q expressed a coincidence among the elements of a finite set. Participants were given short context stories providing information about the frequency of co-occurrence of cases of p, q, not-p, and not-q in a population. They were then asked to estimate the probability that a statement (conditional or quantified) would be true for a random sample taken from that population. The probability estimates for conditionals were in accordance with an instance-focused interpretation, whereas the estimates for quantified statements showed features of a set-focused interpretation. The type of the relation between p and q had no effect on this outcome.     

Language and reasoning: a study of temporal factors

The paper is concerned with the testing of psycholinguistic hypotheses by the use of deductive reasoning tasks. After reviewing some of the problems of interpretation which have arisen with particular reference to conditional rules, an experiment is presented which measures comprehension and verification latencies in addition to response frequencies in a truth table evaluation task.The experiment tests a psycholinguistic hypothesis concerning the different usage of the logically equivalent forms of sentence: If p then q and p only if q with respect to the temporal order of the events p and q. It is proposed that the former sentence is more natural when the event p precedes the event q in time, and the latter more natural when the opposite temporal relation holds.Although significant support is found for the hypothesis in the analysis of the latency data, it is only distinguished from an alternative explanation by detailed analysis of response frequencies, thus indicating the general usefulness of the paradigm adopted.     

Blockage Contraction

Blockage contraction is an operation of belief contraction that acts directly on the outcome set, i.e. the set of logically closed subsets of the original belief set K that are potential contraction outcomes. Blocking is represented by a binary relation on the outcome set. If a potential outcome X blocks another potential outcome Y, and X does not imply the sentence p to be contracted, then Y?≠?K ÷ p. The contraction outcome K ÷ p is equal to the (unique) inclusion-maximal unblocked element of the outcome set that does not imply p. Conditions on the blocking relation are specified that ensure the existence of such a unique inclusion-maximal set for all sentences p. Blockage contraction is axiomatically characterized and its relations to AGM-style operations are investigated. In a finite-based framework, every transitively relational partial meet contraction is also a blockage contraction.     

Specified Meet Contraction

Specified meet contraction is the operation ?\div defined by the identity K ?p = K   ~  f(p)K \div p = K \,{\sim}\, f(p) where ∼ is full meet contraction and f is a sentential selector, a function from sentences to sentences. With suitable conditions on the sentential selector, specified meet contraction coincides with the partial meet contractions that yield a finite-based contraction outcome if the original belief set is finite-based. In terms of cognitive realism, specified meet contraction has an advantage over partial meet contraction in that the selection mechanism operates on sentences rather than on temporary infinite structures (remainders) that are cognitively inaccessible. Specified meet contraction provides a versatile framework in which other types of contraction, such as severe withdrawal and base-generated contraction, can be expressed with suitably chosen properties of the sentential selector.     

Nontransitive measurable utility

A new theory of preferences under risk is presented that does not use the transitivity and independence axioms of the von Neumann-Morgenstern linear utility theory. Utilities in the new theory are unique up to a similarity transformation (ratio scale measurement). They key to this generalization of the traditional linear theory lies in its representation of binary preferences by a bivariate rather than univariate real valued function. Linear theory obtains a linear function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. The new theory obtains a skew-symmetric bilinear function φ on P × P for which φ(p, q) > 0 if and only if p is preferred to q. Continuity, dominance, and symmetry axioms are shown to be necessary and sufficient for the new representation.     

Occasion setting is specific to the CS-US association

In Experiment 1, rats were trained on a discrimination in which one occasion setter, A, signaled that one cue (conditioned stimulus, CS), x, would be followed by one outcome, p (unconditioned stimulus, US), and a second CS, y, by a different outcome, q (x → p and y → q); a second occasion setter, B signalled the reverse CS-outcome relations (x → q and y → p). In a subsequent stage, the animals were divided into two groups, and trained as before, except that both A and B were presented in compound with a novel occasion setter, C. For Group S (same) the CS-outcome relations following A and B were identical to those in the pretraining stage, whereas in Group D (different) they were reversed. In a subsequent test, stimulus C was shown to be a more effective occasion setter in Group D than in Group S. In Experiment 2, rats were trained on a negative occasion-setting discrimination in which CS x signaled outcome p, and y outcome q; when x and y were signaled by the occasion setter A then no outcome followed (x → p, y → q, A:x-, and A:y-). In a subsequent stage, A was now trained as a positive occasion setter, signaling reinforcement of x and y. In Group S, x and y signaled the same outcomes as in the prior training stage (x-, y-, A:x → p, and A:y → q), whereas in Group D they signaled the opposite outcomes (x, y, A:x → q, and A:y → p); more efficient test performance was seen in the latter group. These results suggest that the each occasion setter conveyed information about the specific combination of CS and US paired in its presence (i.e., x → p and y → q, or x → no p and y → no q). These results are consistent with the suggestion that occasion setters operate, at least in part, on a specific CS-US association.     

Piaget's logical formalism for formal operations: An interpretation in context

Two interpretations of the equations used by B. Inhelder and J. Piaget in The growth of logical thinking from childhood to adolescence (London: Routledge & Kegan Paul, 1958) are discussed, with implication as the central example. The expression (p ? q) (p implies q) is said to be equal to $\text{(p · q)}\text{V}\text{(}\text{p}\text{· q)}\text{V}\text{(}\text{p}\text{·}\text{q}\text{)}$ (i.e.,p and q, or not p and q, or neither p nor q). According to one, (p ? q) asserts the existence of each case mentioned. According to the other, (p ? q) only asserts that current knowledge allows the possibility of each of the cases. Neither interpretation makes sense of all the relevant passages. Both can be combined in a consistent interpretation when attention is paid to the functional context of the subject's use of logical operations in this book: the “operations” describe the knowledge states which the subject can differentiate and relate, on his way to solving Inhelder's tasks. The logical notation used to represent these states is not a representational format attributed to the subject and manipulated by his cognitive processes, but part of a structuralist account of the subject's reasoning capacities.     

Clarity about concessive knowledge attributions: reply to Dodd

Recently, Dylan Dodd (this Journal) has tried to clear up what he takes to be some of the many confusions surrounding concessive knowledge attributions (CKAs)??i.e., utterances of the form ??S knows that p, but it??s possible that q?? (where q entails not-p) (Rysiew, Noûs 35(4): 477?C514, 2001). Here, we respond to the criticisms Dodd offers of the account of the semantics and the sometime-infelicity of CKAs we have given (Dougherty and Rysiew, Philosophy and Phenomenological Research 78(1): 121?C132, 2009), showing both how Dodd misunderstands certain central features of that view and how the latter can, pace Dodd, be naturally extended to explain the oddity of those ??For all I know?? statements to which Dodd draws attention.     

Force irradiation effect of kinesiotaping on contralateral muscle activation

We aimed to determine the force irradiation effect of kinesiotaping (KT) on contralateral muscle activity during unilateral muscle contraction. Forty healthy (26 females, 14 males) subjects were divided into two groups: KT and control groups. KT was applied on the biceps brachii at the contralateral limb (non-dominant limb) in the KT group, whereas no taping was applied to the control group. All participants performed unilateral isometric, concentric, and eccentric contractions with their dominant upper limbs (exercised limb) by means of an isokinetic dynamometer, while the contralateral limb was in the resting condition, neutral position, and motionless during the testing procedure. During the exercise, contralateral biceps brachii muscle activity was recorded by surface electromyography (EMG). To quantify the muscle activation, EMG signals were expressed as a percentage of the maximal isometric voluntary contraction, which is referred to as %EMG_max. The KT group showed significantly higher %EMG_max in the biceps brachii compared to the control group at the contralateral limb during the isometric, concentric, and eccentric contractions (p = 0.035, p = 0.046, and p = 0.002, respectively) The median values of the contralateral muscle activity were 2.74 %EMG_max and 6.62 %EMG_max during the isometric contraction for the control and KT groups, respectively (p = 0.035). During the concentric contraction, the median values of the contralateral muscle activity were 1.61 %EMG_max and 9.39 %EMG_max for the control and KT groups, respectively (p = 0.046). The median values of the contralateral muscle activity were 4.49 %EMG_max and 22.89 %EMG_max for the eccentric contraction for the control and KT groups, respectively (p = 0.002). In conclusion, KT application on the contralateral limb increased the contralateral muscle activation in the biceps brachii during the unilateral isometric, concentric, and eccentric contractions.     

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.     

Rethinking Gibbard’s Riverboat Argument

According to the Principle of Conditional Non-Contradiction (CNC), conditionals of the form “If p, q” and “If p, not q” cannot both be true, unless p is inconsistent. This principle is widely regarded as an adequacy constraint on any semantics that attributes truth conditions to conditionals. Gibbard has presented an example of a pair of conditionals that, in the context he describes, appear to violate CNC. He concluded from this that conditionals lack truth conditions. We argue that this conclusion is rash by proposing a new diagnosis of what is going on in Gibbard’s argument. We also provide empirical evidence in support of our proposal.     

Reasoning as we read: Establishing the probability of causal conditionals

Indicative conditionals of the form if p then q (e.g., if student tuition fees rise, then applications for university places will fall) invite consideration of a hypothetical event (e.g., tuition fees rising) and of one of its possible consequences (e.g., applications falling). Since a rise in tuition fees is an uncertain event with equally uncertain consequences, a reader may believe the statement to a greater or lesser extent. As a conditional is read, the earliest point at which this probabilistic evaluation can take place is as the consequent clause is wrapped up (e.g., as the critical word fall is read in the example above). Wrap-up processing occurs at the end of the clause, as it is evaluated and integrated into the evolving discourse representation. Five sources of probability may plausibly influence the evaluation of a conditional as it is wrapped up; these are P(p), P(q), P(pq), P(q|p), and P(not-p or q). A total of 128 conditionals were constructed, with these probabilities calculated for each item in a pretest. The conditionals were then embedded in vignettes and read by 36 participants on a word-by-word basis. Using linear mixed-effects modeling, we found that wrap-up reading times were predicted by pretest ratings of P(p) and P(q|p). There was no influence of P(q), P(pq), or P(not-p or q) on wrap-up reading times. Our findings are consistent with the suppositional theory of conditionals proposed by Evans and Over (2004) but do not support the mental-models theory advanced by Johnson-Laird and Byrne (2002).     

Kant,the Third Antinomy and Transcendental Arguments

In this paper, I consider whether a reading of Kant's solution to the Third Antinomy can offer material for devising a new model of transcendental argument. The problem that this form of argument is meant to address is an antinomy between two apparently contradictory claims, q and ¬q, where we seem equally justified in holding both. The model has the following form: p; q is a necessary condition of p; the only justification we have for q is that it is a necessary condition of p; p is justified only in domain X (where X is a domain of objects of cognition); therefore, q is justified only in domain X. Because the argument shows that our justification for q is valid only in X, it also establishes that there is conceptual space to hold ¬q outside of X.     

Reliabilism, bootstrapping, and epistemic circularity

Pretheoretically we hold that we cannot gain justification or knowledge through an epistemically circular reasoning process. Epistemically circular reasoning occurs when a subject forms the belief that p on the basis of an argument A, where at least one of the premises of A already presupposes the truth of p. It has often been argued that process reliabilism does not rule out that this kind of reasoning leads to justification or knowledge (cf. the so-called bootstrapping-problem or the easy-knowledge-problem). For some philosophers, this is a reason to reject reliabilism. Those who try to defend reliabilism have two basic options: (I) accept that reliabilism does not rule out circular reasoning (or bootstrapping), but argue that this kind of reasoning is not as epistemically “bad” as it seems, or (II) hold on to the view that circular reasoning (or bootstrapping) is epistemically “bad”, but deny that reliabilism really allows this kind of reasoning. Option (I) has been spelled out in several ways, all of which have found to be problematic. Option (II) has not been discussed very widely. Vogel (J Philos 97:602–623, 2000) considers and quickly dismisses it on the basis of three reasons. Weisberg (Philos Phenomenol Res 81:525–548, 2010) has shown in detail that one of these reasons is unconvincing. In this paper I argue that the other two reasons are unconvincing as well and that therefore option (II) might in fact be a more promising starting point to defend reliabilism than option (I).     

Balance,positivity, and agreement in the Jordan Paradigm: A defense of balance theory

It was argued that Heider's p?o?x triad can best be conceptualized according to a three- factor analysis of variance model in which the p?o, p?x, and o?x bands are all factors. From this perspective the balance effect is the triple interaction, the positivity effect a main effect for the p?o factor, and the agreement effect a p?x by o?x interaction. Although the existence of the latter two effects has previously been regarded as damaging to balance theory, it was shown that these effects could be interpreted from a balance perspective and that balance theory could be used to generate supportable propositions regarding these effects. Thus in agreement with a unit-relation interpretation it was shown, in accordance with balance theory, that positivity effects are obtained when the subject, or p, assumes future contact with o, that reverse positivity effects are obtained when the subject anticipates breaking off contact with o, and that no positivity effect is obtained when there is absolutely no contact, past or future. It was also demonstrated in an experiment involving the p?o?q triad that, in accordance with balance theory, positivity effects may be produced by the assumption or inference of same-sign reciprocation in sentiment. Evidence for two balance processes underlying agreement effects was also found. One of these processes was based on the assumption that the subject would have or reveal psycho-logical reasons for the disagreement and thus produce imbalance. Consistent with this interpretation it was found that the agreement effect was significantly larger when future contact with discussion of x was assumed than when future contact without discussion of x was assumed. The other, or unit-relation interpretation was supported by evidence indicating that the breaking off of contact resulted in a reversed agreement effect. In general, it was argued that balance theory did quite well in such phenomenological investigations when attention was not narrowly restricted to the three-sign pattern but considered other potential cognitive bands.     

Obligation as Optimal Goal Satisfaction

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation O q is a logical consequence of the obligations O p and O (p → q). In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation O p is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.     

Belief contraction without recovery

The postulate of recovery is commonly regarded to be the intuitively least compelling of the six basic Gärdenfors postulates for belief contraction. We replace recovery by the seemingly much weaker postulate of core-retainment, which ensures that if x is excluded from K when p is contracted, then x plays some role for the fact that K implies p. Surprisingly enough, core-retainment together with four of the other Gärdenfors postulates implies recovery for logically closed belief sets. Reasonable contraction operators without recovery do not seem to be possible for such sets. Instead, however, they can be obtained for non-closed belief bases. Some results on partial meet contractions on belief bases are given, including an axiomatic characterization and a non-vacuous extension of the AGM closure condition.     

Quantifying disablers in reasoning with universal and existential rules

People accept conclusions of valid conditional inferences (e.g., if p then q, p therefore q) less, the more disablers (circumstances that prevent q to happen although p is true) exist. We investigated whether rules that through their phrasing exclude disablers evoke higher acceptance ratings than rules that do not exclude disablers. In three experiments we re-phrased content-rich conditionals from the literature as either universal or existential rules and embedded these rules in Modus Ponens and Modus Tollens inferences. In Experiments 2 and 3, we also used abstract rules. The acceptance of conclusions increased when the rule was phrased with “all” instead of “some” and the number of disablers had a higher impact on existential rules than on universal rules. Further, the effect of quantifier was more pronounced for abstract rules and when tested within subjects. We discuss the relevance of phrasing, quantifiers and knowledge on reasoning.