Relevant Connexive Logic

In this paper, a connexive extension of the Relevance logic R→ was presented. It is defined by means of a natural deduction system, and a deductively equivalent axiomatic system is presented too. The goal of such an extension is to produce a logic with stronger connection between the antecedent and the consequent of an implication

Peirce’s Triadic Logic and Its (Overlooked) Connexive Expansion

In this paper, we present two variants of Peirce’s Triadic Logic within a language containing only conjunction, disjunction, and negation. The peculiarity of our systems is that conjunction and disjunction are interpreted by means of Peirce’s mysterious binary operations Ψ and Φ from his ‘Logical Notebook’. We show that semantic conditions that can be extracted from the definitions of Ψ and Φ agree (in some sense) with the traditional view on the semantic conditions of conjunction and disjunction. Thus, we support the conjecture that Peirce’s special interest in these operations is due to the fact that he interpreted them as conjunction and disjunction, respectively. We also show that one of our systems may serve as a suitable base for an interesting implicative expansion, namely the connexive three-valued logic by Cooper. Sound and complete natural deduction calculi are presented for all systems examined in this paper

A Poly-Connexive Logic

The paper introduces a variant of connexive logic in which connexivity is extended from the interaction of negation with implication to the interaction of negation also with conjunction and disjunction. The logic is presented by two deductively equivalent methods: an axiomatic one and a natural-deduction one. Both are shown to be complete for a four-valued model theory

Bi-Classical Connexive Logic and its Modal Extension: Cut-elimination, completeness and duality

In this study, a new paraconsistent four-valued logic called bi-classical connexive logic (BCC) is introduced as a Gentzen-type sequent calculus. Cut-elimination and completeness theorems for BCC are proved, and it is shown to be decidable. Duality property for BCC is demonstrated as its characteristic property. This property does not hold for typical paraconsistent logics with an implication connective. The same results as those for BCC are also obtained for MBCC, a modal extension of BCC

De Finettian Logics of Indicative Conditionals Part I: Trivalent Semantics and Validity

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti (1936) and Reichenbach (1935, 1944). On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the predicament: one pairs de Finetti’s conditional (DF) with validity as the preservation of non-false values (TT-validity), but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table (CC) proposed by Cooper (Inquiry 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic 49, 245–260, 2008). In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties

A 4-valued logic of strong conditional

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions (Section 1), we argue that a stronger account of conditional can be obtained in two steps: firstly, by reminding its historical roots inside modal logic and set-theory (Section 2); secondly, by revising the meaning of logical values, thereby getting rid of the paradoxes of material implication whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections (Section 3). Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed through a broader definition of logical consequence that encompasses both a normal relation of truth propagation and a weaker relation of falsity non-propagation from premises to conclusion (Section 3)

Bayesian confirmation, connexivism and an unkindness of ravens

Bayesian confirmation theories (BCTs) might be the best standing theories of confirmation to date, but they are certainly not paradox-free. Here I recognize that BCTs’ appeal mainly comes from the fact that they capture some of our intuitions about confirmation better than those the- ories that came before them and that the superiority of BCTs is suffi- ciently justified by those advantages. Instead, I will focus on Sylvan and Nola’s claim that it is desirable that our best theory of confirmation be as paradox-free as possible. For this reason, I will show that, as they respond to different interests, the project of the BCTs is not incompatible with Sylvan and Nola’s project of a paradox-free confirmation logic. In fact, it will turn out that, provided we are ready to embrace some degree of non-classicality, both projects complement each other nicely

Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯ , the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson’s paraconsistent N4, this also supplements Nelson’s own proof of the Post consistency of N4♯ . Insofar as the present technique allows infinite models, this resolves Nelson’s concern that N4♯ is of interest only to those accepting that there are finitely many natural numbers