This paper discusses some key connexive principles construed as principles about reasons, that is, as principles that express logical properties of sentences of the form ‘p is a reason for q’. Its main goal is to show how the theory of reasons outlined by Crupi and Iacona, which is based on their evidential account of conditionals, yields a formal treatment of such sentences that validates a restricted version of the principles discussed, overcoming some limitations that affect most extant accounts of (...) conditionals. (shrink)

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second question should be (...) answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’. (shrink)

Buridan endorses the basic idea that q follows from p iff it is impossible that p is true but q is false. Since he also accepts the law that, if p is impossible, the conjunction (p ∧ q) must be impossible, he comes to regard the principle ‘Ex impossibili quodlibet’ (EIQ) as basically correct. However, his logic is based on a ‘nominalist’ view according to which propositions are tokens of spoken, written or thought language existing in space of time, and (...) the truth of such a proposition presupposes that it exists. This conception prompted Buridan to modify the definition of ‘consequence’, but none of his attempts was fully successful. Yet, a large part of his theory of consequences is based on the usual account of necessary truth-conservation. As regards principle EIQ, Buridan distinguishes two variants of impossibility. Proposition p is possible iff the state of affairs as described by p is possible; this doesn’t guarantee, however, that p is ‘possibly true’. E.g. ‘No proposition is negative’ is considered by Buridan as describing a possible state of affairs, but this proposition is not possibly true, because in order to be true, it would have to exist, and as soon as it comes into existence, it becomes false. Accordingly, EIQ requires that the antecedent p is really impossible and not only impossibly true. (shrink)

Buridan endorsed the principles that any impossible, and a fortiori any self-contradictory, proposition entails each proposition. These principles are usually referred to as ‘Ex impossibili quodlibet’ (EIQ) and ‘Ex contradictione quodlibet’ (ECQ). Buridan further considered the instance ECCQ according to which any proposition follows from the conjunction of two contradictory propositions. Buridan showed how ECCQ can be proven by means the usual laws of conjunction and disjunction. Furthermore, he discovered that EIQ can be derived from ECCQ by means of the (...) principle that if a conclusion q follows from two premises, one of which is necessary, then q follows from the other premise alone. Buridan thought that this principle would be provable; but his proof turned out to be circular since it presupposed EIQ. Nevertheless, EIQ is validated by the definition of a ‘bona consequentia’. Somewhat more exactly, according to Buridan one may distinguish between inferences which are formally valid, ‘materially’ valid, and valid ‘as of now’. It is shown that formally valid inferences satisfy ECQ while ‘materially’ valid inferences satisfy EIQ. Furthermore, inferences which are valid ‘as of now’ satisfy the principle ‘Ex falso quodlibet’ saying that any false proposition ‘as of now’ entails every proposition. (shrink)

Since McCall (1966), the heterodox principle of propositional logic that it is impossible for a proposition to be entailed by its own negation—in symbols, ¬(¬φ→φ)—has gone by the name of Aristotle’s thesis, since Aristotle apparently endorses it in Prior Analytics 2.4, 57b3–14. Scholars have contested whether Aristotle did endorse his eponymous thesis, whether he could do so consistently, and for what purpose he endorsed it if he did. In this article, I reconstruct Aristotle’s argument from this passage and show that (...) he accepts this thesis. Further, I show that the argument he gives is, making plausible assumptions, a correct proof in a consistent fragmentary nonclassical metalogic for a metatheorem he previously states concerning his assertoric syllogistic. In this way, Aristotle’s argument emerges as a fascinating case study in the use of a nonclassical metalogic to prove a result about a nonclassical object system. (shrink)

Straipsnyje parodoma, kad daugelis Viduramžių logikų manė, kad kai kurie „dėsniai“ galioja tik esant tam tikroms sąlygoms. Pavyzdžiui, Aristotelio, Boecijaus ir Abelardo ginti vadinamosios jungčių logikos (connexive logic) baziniai principai galioja tik galimų ar neprieštaringų antecedentų arba nebūtinų ar kontingentiškų konsekventų atžvilgiu. Panašus apribojimas galioja „dėsniui“ (kurį galimai gynė Chrisipas), kad kiekvienas teiginys yra suderinamas su pačiu savimi.

The term ‘hyperconnexive logic’ (or ‘hyperconnexivity’ in general) in relation to a certain logical system was coined by Sylvan to indicate that not only do Boethius’ theses hold in such a system, but also their converses. The plausibility of the latter was questioned by some connexive logicians. Without going into the discussion regarding the plausibility of hyperconnexivity and the converses of Boethius’ theses, this paper proposes a quite simple way to escape the hyperconnexivity within the semantic framework of Wansing-style constructive (...) connexive logics. In particular, we present a working method for escaping hyperconnexivity of constructive connexive logic $${{\textbf{C}}}$$, discuss the problem that creates an obstacle to using the same method in the case of logic $${{\textbf{C3}}}$$ and provide a possible solution to this problem that allows us to construct a logical theory which is similar to $${{\textbf{C3}}}$$ and free from hyperconnexivity. All new logics introduced in this paper are equipped with sound and complete Hilbert-style calculi, and their relationships with other well-known connexive logics are discussed. (shrink)

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 (...) fulfill some specific connexive theses and sufficiently well-motivated from a semantical point of view. The approach proposed here tries to address this problem by defining an appropriate connexive relation in terms of more basic notions. The result is a well-motivated system of modal connexive logic that nicely fits in both with the traditional ideas concerning the connexive conditional and with the current developments in connexive logic. (shrink)

This paper scrutinizes Everett Nelson's conception of consistency by comparing it with the “standard” account of C. I. Lewis. This conflict surprisingly resembles a related controversy between the ancient logicians Chrysippus and Diodorus. Nelson's intuitions behind his peculiar conception of consistency are analysed and certain features of his logical system are critically examined. In particular, his objections against the law of the transitivity of implication and against the laws of conjunction and disjunction have to be discussed. Although Nelson's considerations contain (...) interesting ideas in favour of a relevantist conception of logic, they fail to convincingly support the hardcore connexive thesis that even a selfcontradictory proposition doesn't entail its own negation. (shrink)

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 (...) will make clear the context to which the papers in this special issue on Frontiers of Connexive Logic belong and contribute. (shrink)

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 (...) selection function is defined. Based on the new semantics, we define six conditional logics by different classes of models, using two notions of validity, one preserving truth and the other preserving tolerant truth. We prove soundness and completeness of these logics and compare them with some conditional logics in the literature. Our partial function plus precondition strategy may be used in other semantic frameworks to generate more connexive logics. (shrink)

In his seminal paper ‘Truth’, M. Dummett considered negated conditional statements as one of the main motivations for introducing a three-valued logical framework. He left a sketch of an implication connective that, as we observe, shares some intuitions with Wansing-style account for connexivity. In this article, we discuss Dummett’s ‘unfinished’ implication and suggest two possible reconstructions of it. One of them collapses into implication from W. Cooper’s ‘Logic of Ordinary Discourse’ \(\textbf{OL}\) and J. Cantwell’s ‘Logic of Conditional Negation’ \(\textbf{CN}\), whereas (...) the other turns out to be previously unknown implication connective and can be used to obtain a novel logical system, entitled here as \(\textbf{cRM}_\textbf{3}\). As to the technical results, we introduce a sound and complete axiomatic proof-system for \(\textbf{cRM}_\textbf{3}\) and present a theorem for the semantic embedding of \(\textbf{CN}\) into \(\textbf{cRM}_\textbf{3}\). (shrink)