Some Concerns Regarding Ternary-relation Semantics and Truth-theoretic Semantics in General

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment

Propositional logic extended with a pedagogically useful relevant implication

First and foremost, this paper concerns the combination of classical propositional logic with a relevant implication. The proposed combination is simple and transparent from a proof theoretic point of view and at the same time extremely useful for relating formal logic to natural language sentences. A specific system will be presented and studied, also from a semantic point of view. The last sections of the paper contain more general considerations on combining classical propositional logic with a relevant logic that has all classical theorems as theorems

Measuring evidence: a probabilistic approach to an extension of Belnap-Dunn Logic

This paper introduces the logic of evidence and truth LETF as an extension of the Belnap-Dunn four-valued logic F DE. LETF is a slightly modified version of the logic LETJ, presented in Carnielli and Rodrigues (2017). While LETJ is equipped only with a classicality operator ○, LETF is equipped with a non-classicality operator ● as well, dual to ○. Both LETF and LETJ are logics of formal inconsistency and undeterminedness in which the operator ○ recovers classical logic for propositions in its scope. Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition α even if α is not true. As well as LETJ, LETF is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for LETF where statements P(α) and P(○α) express, respectively, the amount of evidence available for α and the degree to which the evidence for α is expected to behave classically - or non-classically for P(●α). A probabilistic scenario is paracomplete when P(α) + P(¬α) 1, and in both cases, P(○α) < 1. If P(○α) = 1, or P (●α) = 0, classical probability is recovered for α. The proposition ○α ∨ ●α, a theorem of LETF , partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theor

Theories of truth based on four-valued infectious logics

Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems