Two Kinds of Consequential Implication

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by \(\rightarrow \) and \(\Rightarrow \)). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for \(\Rightarrow \), CI\(\Rightarrow \), is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for \(\rightarrow \), CI. The main drawback ot the weaker operator \(\Rightarrow \) is that it lacks unrestricted contraposition, but the final section of the paper argues that \(\Rightarrow \) has some properties which make it a valuable alternative to \(\rightarrow \), turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.     

Strong Boethius’ Thesis and Consequential Implication

The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying Aristotle's Thesis (p does not imply not p ) and Weak Boethius' Thesis (if p implies q, then p does not imply not q ). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of Strong Boethius' Thesis (that p implies q implies that p does not imply not q) is identified.     

Indicative conditionals,conditional probabilities,and the “defective truth-table”: A request for more experiments

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the “defective truth-table” in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces as valid numerous inference patterns that do seem to be part of ordinary usage. However, coupled with something the probability account gives us—namely that when conditional-free ? entails conditional-free ψ, “if ? then ψ” is a trivial, uninformative truth—we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional.     

Bivalence and subjunctive conditionals

Writers such as Stalnaker and Dummett have argued that specific features of subjunctive conditional statements undermine the principle of bivalence. This, paper is concerned with rebutting such claims. 1. It is shown how subjective conditionals pose a prima facie threat to bivalence, and how this threat can be dissolved by a distinction between the results of negating a subjective conditional and of negating its consequent. To make this distinction is to side with Lewis against Stalnaker in a dispute about possible worlds semantics for such conditionals, and reasons are given for doing so. 2. These arguments are extended to answer Dummett's claim that behaviourist and phenomenalist analyses in terms of subjunctive conditions violate bivalence. This answer is shown to be compatible with the principle that hypothetical statements are true only in virtue of categorical facts.     

Partitions and Conditionals

The literature on conditionals is rife with alternate formulations of the abstract semantics of conditional logic. Each formulation has its own advantages in terms of applications and generalizations; nevertheless, they are for the most part equivalent, in the sense that they underwrite the same range of logical systems. The purpose of the present note is to bring under this umbrella the partition semantics introduced by Brian Skyrms in (Skyrms, 1984).     

Conditional Heresies

The principles of Conditional Excluded Middle (CEM ) and Simplification of Disjunctive Antecedents (SDA) have received substantial attention in isolation. Both principles are plausible generalizations about natural language conditionals. There is however little discussion of their interaction. This paper aims to remedy this gap and explore the significance of having both principles constrain the logic of the conditional. Our negative finding is that, together with elementary logical assumptions, CEM and SDA yield a variety of implausible consequences. Despite these incompatibility results, we open up a narrow space to satisfy both. We show that, by simultaneously appealing to the alternative-introducing analysis of disjunction and to the theory of homogeneity presuppositions, we can satisfy both. Furthermore, the theory that validates both principles resembles a recent semantics that is defended by Santorio on independent grounds. The cost of this approach is that it must give up the transitivity of entailment: we suggest that this is a feature, not a bug, and connect it with recent developments of intransitive notions of entailment.     

The probability of causal conditionals

Conditionals in natural language are central to reasoning and decision making. A theoretical proposal called the Ramsey test implies the conditional probability hypothesis: that the subjective probability of a natural language conditional, P(if p then q), is the conditional subjective probability, P(q/p). We report three experiments on causal indicative conditionals and related counterfactuals that support this hypothesis. We measured the probabilities people assigned to truth table cases, P(pq), P(p notq), P( notpq) and P( notp notq). From these ratings, we computed three independent predictors, P(p), P(q/p) and P(q/ notp), that we then entered into a regression equation with judged P(if p then q) as the dependent variable. In line with the conditional probability hypothesis, P(q/p) was by far the strongest predictor in our experiments. This result is inconsistent with the claim that causal conditionals are the material conditionals of elementary logic. Instead, it supports the Ramsey test hypothesis, implying that common processes underlie the use of conditionals in reasoning and judgments of conditional probability in decision making.     

Wondering what might be

This paper explores the possibility of supplementing the suppositional view of indicative conditionals with a corresponding view of epistemic modals. The most striking feature of the suppositional view consists in its claim that indicative conditionals are to be evaluated by conditional probabilities. On the basis of a natural link between indicative conditionals and epistemic modals, a corresponding thesis about the probabilities of statements governed by epistemic modals can be derived. The paper proceeds by deriving further consequences of this thesis, in particular, the logic of epistemic modals and their logical interaction with indicative conditionals are studied.     

A Sequent Formulation of Conditional Logic Based on Belief Change Operations

Peter Gärdenfors has developed a semantics for conditional logic, based on the operations of expansion and revision applied to states of information. The account amounts to a formalisation of the Ramsey test for conditionals. A conditional A > B is declared accepted in a state of information K if B is accepted in the state of information which is the result of revising K with respect to A. While Gärdenfors's account takes the truth-functional part of the logic as given, the present paper proposes a semantics entirely based on epistemic states and operations on these states. The semantics is accompanied by a syntactic treatment of conditional logic which is formally similar to Gentzen's sequent formulation of natural deduction rules. Three of David Lewis's systems of conditional logic are represented. The formulations are attractive by virtue of their transparency and simplicity.     

Aristotle's Thesis between paraconsistency and modalization

If the arrow → stands for classical relevant implication, Aristotle's Thesis ¬(A→¬A) is inconsistent with the Law of Simplification (AB)→B accepted by relevantists, but yields an inconsistent non-trivial extension of the system of entailment E. Such paraconsistent extensions of relevant logics have been studied by R. Routley, C. Mortensen and R. Brady. After examining the semantics associated to such systems, it is stressed that there are nonclassical treatments of relevance which do not support Simplification. The paper aims at showing that Aristotle's Thesis may receive a sense if the arrow is defined as strict implication endowed with the proviso that the clauses of the conditional have the same modal status, i.e. the same position in the Aristotelian square. It is so grasped, in different form, the basic idea of relevant logic that the clauses of a true conditional should have something in common. It is proved that thanks to such definition of the arrow Aristotle's Thesis subjoined to the minimal normal system K yields a system equivalent to the deontic system KD.     

Conditionals,probability, and non-triviality

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities are sufficient to trivialize the semantics.     

From Aristotle's Syllogistic to Stoic Conditionals: Holzwege or Detectable Paths?

This paper is chiefly aimed at individuating some deep, but as yet almost unnoticed, similarities between Aristotle's syllogistic and the Stoic doctrine of conditionals, notably between Aristotle's metasyllogistic equimodality condition (as stated at APr. I 24, 41b27–31) and truth-conditions for third type (Chrysippean) conditionals (as they can be inferred from, say, S.E. P. II 111 and 189). In fact, as is shown in §1, Aristotle's condition amounts to introducing in his (propositional) metasyllogistic a non-truthfunctional implicational arrow '', the truth-conditions of which turn out to be logically equivalent to truth-conditions of third type conditionals, according to which only the impossible (and not the possible) follows from the impossible. Moreover, Aristotle is given precisely this non-Scotian conditional logic in two so far overlooked passages of (Latin and Hebraic translations of) Themistius' Paraphrasis of De Caelo (CAG V 4, 71.8–13 and 47.8–10 Landauer). Some further consequences of Aristotle's equimodality condition on his logic, and notably on his syllogistic (no matter whether modal or not), are pointed out and discussed at length. A (possibly Chrysippean) extension of Aristotle's condition is also discussed, along with a full characterization of truth-conditions of fourth type conditionals.     

Conditionals in reasoning

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent semantics and epistemic states are represented via preferential models.     

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.     

条件推理的语义关系表达形式效应

作者用以大学生为被试的实验考察, 基本条件句语义关系表达形式(充分关系、必要关系和析取关系表达形式)和作为心理模型外显建构的可能性判断任务对条件推理的可能影响。实验结果发现, 条件推理的语义关系表达形式效应显著, 条件句语义关系表达形式对被试条件推理有显著的影响, 条件推理成绩随条件句语义关系表达的外显程度增加而增加; 被试在可能性判断任务中对条件句所含心理模型的外显建构并没有明显改善条件推理的成绩; 在有可能性判断任务条件下, 被试外显心理模型建构的成绩变化模式并不能一致地预测条件推理成绩的变化模式。这些结果说明, 人们的条件推理可能并不是完全基于心理模型建构, 而是还受对条件句前后件之间语义关系理解的影响; 条件句表达形式中语义关系的外显内隐模式影响对条件句语义关系的知觉理解难易, 从而影响相应条件推理的成绩。     

Conditionals and Curry

Curry’s paradox for “if.. then..” concerns the paradoxical features of sentences of the form “If this very sentence is true, then 2 + 2 = 5”. Standard inference principles lead us to the conclusion that such conditionals have true consequents: so, for example, 2 + 2 = 5 after all. There has been a lot of technical work done on formal options for blocking Curry paradoxes while only compromising a little on the various central principles of logic and meaning that are under threat. Once we have a sense of the technical options, though, a philosophical choice remains. When dealing with puzzles in the logic of conditionals, a natural place to turn is independently motivated semantic theories of the behaviour of “if... then...”. This paper argues that a closest-worlds approach outlined in previous work offers a philosophically satisfying reason to deny conditional proof and so block the paradoxical Curry reasoning, and can give the verdict that standard Curry conditionals are false, along with related “contraction conditionals”.     

More Triviality

This paper uses the framework of Popper and Miller's work on axiom systems for conditional probabilities to explore Adams' thesis concerning the probabilities of conditionals. It is shown that even very weak axiom systems have only a very restricted set of models satisfying a natural generalisation of Adams' thesis, thereby casting severe doubt on the possibility of developing a non-Boolean semantics for conditionals consistent with it.     

On S

The sentential logic S extends classical logic by an implication-like connective. The logic was first presented by Chellas as the smallest system modelled by contraining the Stalnaker-Lewis semantics for counterfactual conditionals such that the conditional is effectively evaluated as in the ternary relations semantics for relevant logics. The resulting logic occupies a key position among modal and substructural logics. We prove completeness results and study conditions for proceeding from one family of logics to another.We are grateful to Peter Apostoli, Kosta Doen, and anonymous referees for their comments on an earlier version of this paper. A.F.'s work has been supported by a grant from the Volkswagen-Stiftung.Presented byJan Zygmunt     

A Critical Examination of the Historical Origins of Connexive Logic

It is often assumed that Aristotle, Boethius, Chrysippus, and other ancient logicians advocated a connexive conception of implication according to which no proposition entails, or is entailed by, its own negation. Thus Aristotle claimed that the proposition ‘if B is not great, B itself is great […] is impossible’. Similarly, Boethius maintained that two implications of the type ‘If p then r’ and ‘If p then not-r’ are incompatible. Furthermore, Chrysippus proclaimed a conditional to be ‘sound when the contradictory of its consequent is incompatible with its antecedent’, a view which, in the opinion of S. McCall, entails the aforementioned theses of Aristotle and Boethius. Now a critical examination of the historical sources shows that the ancient logicians most likely meant their theses as applicable only to ‘normal’ conditionals with antecedents which are not self-contradictory. The corresponding restrictions of Aristotle’s and Boethius’ theses to such self-consistent antecedents, however, turn out to be theorems of ordinary modal logic and thus don’t give rise to any non-classical system of genuinely connexive logic.