Logic may be simple. Logic, congruence and algebra

This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra)

Relevant Arithmetic and Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture

Normalisation for Some Quite Interesting Many-Valued Logics

In this paper, we consider a set of quite interesting three- and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a detailed version of Prawitz’s proof of Nelson’s logic N4 and its extension by intuitionist negation

Non-Boolean classical relevant logics I

Under embargo until: 2020-12-13Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing.acceptedVersio

Two-valued Logics for Transparent Truth Theory

It is part of the current wisdom that the Liar and similar semantic paradoxes can be taken care of by the use of certain non-classical multivalued logics. It is also well-known that some of these logics can be characterized by means of two-valued semantics. An immediate consequence of this is that there are two-valued logics that support a transparent truth predicate. In this paper I want to suggest that these logics are not just interesting from a formal point of view but also from a philosophical perspective. In particular, I will argue that the two-valued presentation of these logics has a number of advantages over the more usual presentations.Fil: Rosenblatt, Lucas Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; Argentin