This paper presents a simple but, by my lights, effective argument for a subclassical account of logic—an account according to which logical consequence is (properly) weaker than the standard, so‐called classical account. Alas, the vast bulk of the paper is setup. Because of the many conflicting uses of ‘logic’ the paper begins, following a disclaimer on logic and inference, by fixing the sense of ‘logic’ in question, and then proceeds to rehearse both the target subclassical account of logic and its (...) well‐known relative (viz., classical logic). With background in place the simple argument—which takes up less than five pages—is advanced. My hope is that the minimal presentation will help to get ‘the simple argument’ plainly on the table, and that subsequent debate can move us forward. (shrink)

This paper aims to empirically explore the state of practical applications of paraconsistent logics. To this end, we performed an exploratory literature review, analysing papers published between the years 2015 and 2018. Paraconsistent formalisms based on annotated logics are practically the sole type of approach we found to be applied in engineering applications. The engineering problems solved by paraconsistent approaches were mainly in the fields of signal and image processing and decision support. The results of our exploratory review indicate that (...) recent developments in the theory of paraconsistency have not yet been adopted for applications. (shrink)

This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn’s weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication and its neighbors. The algebraic structures corresponding to (...) the systems are then defined, and completeness results are provided. We next consider fuzzy systems satisfying new relevance principles introduced by Yang. We show that the weakening-free uninorm systems and some extensions and neighbors of R satisfy the new relevance principles. (shrink)

The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of (...) the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s). (shrink)

The aim of this paper is to give a general solution to the paradoxes of the material conditional, including the paradoxes generated by embedded conditionals. The solution consists in a pragmatic reinterpretation of the formal languages of classical logic LK and relevant logic LR as presented in Paoli. In particular I argue that the material conditional in the classical logic LK captures the truth conditions of “if...then”, but ignores certain pragmatic enrichments that are associated to it, while relevant logic LR (...) can give a systematic diagnostic of the cases in which a conditional is pragmatically enriched and those cases in which it is not. This diagnostic shows the reason why the paradoxes seem unacceptable and why they are, nevertheless, truth-preserving. This reinterpretation will also cover the solution that Paoli gives to McGee’s paradoxes of the Modus Ponens. (shrink)

This paper presents a sequent calculus for the positive relevant logic with necessity and a proof that it admits the elimination of cut.

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) (...) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...) the logics of lattices with weak negation and modal operators. (shrink)

KR is Anderson and Belnap’s relevance logic R with the addition of the axiom of EFQ: \ \rightarrow q\). Since KR is relevantistic as to implication but classical as to negation, it has been dubbed, among many others, a ‘classical relevance logic.’ For KR, there have been known so far just two pretabular normal extensions. For these pretabular logics, no simple axiomatizations have yet been presented. In this paper, we offer some and show that they do the job. We also (...) introduce some Routley–Meyer semantic conditions for these axioms. All these may facilitate realizing our desire to discover other classical pretabular extensions over KR, if such extensions exist. (shrink)