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This paper studies proof systems for the logics of super-strict implication ST2–ST5, which correspond to C.I. Lewis’ systems S2–S5 freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating STn in Sn and backsimulating Sn in STn, respectively(for n=2,...,5). Next, G3-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of G3-style calculi, that they are sound and complete, and (...) |
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The aim of this paper is to present a system of modal connexive logic based on a situation semantics. In general, modal connexive logics are extensions of standard modal logics that incorporate Aristotle’s and Boethius’ theses, that is the thesis that a sentence cannot imply its negation and the thesis that a sentence cannot imply a pair of contradictory sentences. A key problem in devising a connexive logic is to come up with a system that is both sufficiently strong to (...) |
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We present here some Boolean connexive logics (BCLs) that are intended to be connexive counterparts of selected Epstein’s content relationship logics (CRLs). The main motivation for analyzing such logics is to explain the notion of connexivity by means of the notion of content relationship. The article consists of two parts. In the first one, we focus on the syntactic analysis by means of axiomatic systems. The starting point for our syntactic considerations will be the smallest BCL and the smallest CRL. (...) |
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Over the past ten years, the community researching connexive logics is rapidly growing and a number of papers have been published. However, when it comes to the terminology used in connexive logic, it seems to be not without problems. In this introduction, we aim at making a contribution towards both unifying and reducing the terminology. We hope that this can help making it easier to survey and access the field from outside the community of connexive logicians. Along the way, we (...) |
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Motivated by supplying a new strategy for connexive logic and a better semantics for conditionals so that negating a conditional amounts to negating its consequent under the condition, we propose a new semantics for connexive conditional logic, by combining Kleene’s three-valued logic and a slight modification of Stalnaker’s semantics for conditionals. In the new semantics, selection functions for selecting closest worlds for evaluating conditionals can be undefined. Truth and falsity conditions for conditionals are then supplemented with a precondition that the (...) |
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This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ (...) |
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Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) |
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Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) |
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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 and \). 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 (...) |
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It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of CEM (...) |
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This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the modal cube. it is shown that all logics of super-strict implications are (...) |
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One of the main controversies of the Logic Schools of the 12th century centered on the question: What follows from the impossible? In this paper arguments for two diametrically opposed positions are examined. The author of the ‘Avranches Text’ who probably belonged to the school of the Parvipontani defended the view that from an impossible proposition everything follows (‘Ex impossibili quodlibet’). In particular he developed a proof to show that by means of so-called ‘disjunctive syllogism’ any arbitrary proposition B can (...) |
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The “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with (...) |
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In this paper, I investigate whether the intuitions that make connexive logics seem plausible might lie in pragmatic phenomena, rather than the semantics of conditional statements. I conclude that pragmatics indeed underwrites these intuitions, at least for indicative statements. Whether this has any effect on logic choice, however, heavily depends on one’s semantic theory of conditionals and on how one chooses to logically treat pragmatic failures. No categories |
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In this paper, I investigate whether the intuitions that make connexive logics seem plausible might lie in pragmatic phenomena, rather than the semantics of conditional statements. I conclude that pragmatics indeed underwrites these intuitions, at least for indicative statements. Whether this has any effect on logic choice (and what that effect might be), however, heavily depends on one’s semantic theory of conditionals and on how one chooses to logically treat pragmatic failures. |
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Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...) |
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We examine the relationship between the logics of nonsense of Bochvar and Halldén and the containment logics in the neighborhood of William Parry’s A I. We detail two strategies for manufacturing containment logics from nonsense logics—taking either connexive and paraconsistent fragments of such systems—and show how systems determined by these techniques have appeared as Frederick Johnson’s R C and Carlos Oller’s A L. In particular, we prove that Johnson’s system is precisely the intersection of Bochvar’s B 3 and Graham Priest’s (...) |
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Francez has suggested that connexivity can be predicated of connectives other than the conditional, in particular conjunction and disjunction. Since connexivity is not any connection between antecedents and consequents—there might be other connections among them, such as relevance—, my question here is whether Francez’s conjunction and disjunction can properly be called ‘connexive’. I analyze three ways in which those connectives may somehow inherit connexivity from the conditional by standing in certain relations to it. I will show that Francez’s connectives fail (...) |
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Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...) |
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