Logical Pluralists maintain that there is more than one genuine/true logical consequence relation. This paper seeks to understand what the position could amount to and some of the challenges faced by its formulation and defence. I consider in detail Beall and Restall’s Logical Pluralism—which seeks to accommodate radically different logics by stressing the way that they each fit a general form, the Generalised Tarski Thesis (GTT)—arguing against the claim that different instances of GTT are admissible precisifications of logical consequence. I then consider what it is to endorse a logic within a pluralist framework and criticise the options Beall and Restall entertain. A case study involving many-valued logics is examined. I next turn to issues of the applications of different logics and questions of which logic a pluralist should use in particular contexts. A dilemma regarding the applicability of admissible logics is tackled and it is argued that application is a red herring in relation to both understanding and defending a plausible form of logical pluralism. In the final section, I consider other ways to be and not to be a logical pluralist by examining analogous positions in debates over religious pluralism: this, I maintain, illustrates further limitations and challenges for a very general logical pluralism. Certain less wide-ranging pluralist positions are more plausible in both cases, I suggest, but assessment of those positions needs to be undertaken on a case-by-case basis.
Logic isn’t Normative
Some writers object to logical pluralism on the grounds that logic is normative. The rough idea is that the relation of logical consequence has consequences for what we ought to think and how we ought to reason, so that pluralism about the consequence relation would result in an incoherent or unattractive pluralism about those things. In this paper I argue that logic isn’t normative. I distinguish three different ways in which a theory – such as a logical theory – can be entangled with the normative and argue that logic is only entangled in the weakest of these ways, one which requires it to have no normativity of its own. I use this view to show what is wrong with three different arguments for the conclusion that logic is normative.
The Dialogical Approach to Paraconsistency
Being a pragmatic and not a referential approach to semantics, the dialogical formulation of paraconsistency allows the following semantic idea to be expressed within a semi-formal system: In an argumentation it sometimes makes sense to distinguish between the contradiction of one of the argumentation partners with himself (internal contradiction) and the contradiction between the partners (external contradiction). The idea is that external contradiction may involve different semantic contexts in which, say A and not A have been asserted. The dialogical approach suggests a way of studying the dynamic process of contradictions through which the two contexts evolve for the sake of argumentation into one system containing both contexts. More technically, we show a new, dialogical, way to build paraconsistent systems for propositional and first-order logic with classical and intuitionistic features (i.e. paraconsistency both with and without tertium non-datur) and present their corresponding tableaux.
The Semantics of First Degree Entailment
From the introduction: “we argue that the semantics of the first degree paradox-free implication system FD supports the claim it is superior to strict implication as an analysis of entailment at the first degree level. The semantics also reveals that Disjunctive Syllogism, […] far from being a paradigmatic entailment, is invalid, and allows the illegitimate suppression of tautologies”
Logical Nihilism: Could there be no Logic?
Logical nihilism can be understood as the view that there are no laws of logic. This paper presents both a counterexample-based argument in favor of logical nihilism, and a way to resist it by using Lakatos’ method of lemma incorporation. The price to pay is the loss of absolute generality.
Classical Logic
Summary: This article provides the basics of a typical logic, sometimes called ‘classical elementary logic’ or ‘classical first-order logic’, in a rigorous yet accessible manner. Section 2 develops a formal language, with a syntax and grammar. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. Section 4 provides a model-theoretic semantics. Section 5 turns to the relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. The authors show that an argument is derivable only if it is valid (soundness). Then they establish a converse: that an argument is valid only if it is derivable (completeness). They also briefly indicate other features of the logic, some of which are corollaries to soundness and completeness. The final section, Section 6, is devoted to a brief examination of the philosophical position that classical logic is ‘the one right logic’.
Logical Consequence
Description: This article is a short overview of philosophical and formal issues in the treatment and analysis of logical consequence. The purpose of the paper is to provide a brief introduction to the central issues surrounding two questions: (1) that of the nature of logical consequence and (2) that of the extension of logical consequence. It puts special emphasis in the role played by formal systems in the investigation of logical consequence.
Models and Modality
Abstract: This paper examines the connection between model-theoretic truth and necessary truth. It is argued that though the model-theoretic truths of some standard languages are demonstrably “necessary” (in a precise sense), the widespread view of model-theoretic truth as providing a general guarantee of necessity is mistaken. Several arguments to the contrary are criticized.
Models and Logical Consequence
Abstract: This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical consequence.
Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro’s blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed.