Synonymy And Intra-Theoretical Pluralism

Proof-theoretic semantics is a well-established inferentialist theory of meaning that develops ideas proposed by Prawitz and Dummett. The main aim of this theory is to find a foundation of logic based on some aspects of the linguistic use of the logical terms, as opposed to the regular foundation offered by a model-theoretic approach à la Tarski, in which the denotation of non-linguistic entities is central. Traditionally, intuitionistic logic is considered justified in proof-theoretic semantics (although some doubts are sometimes raised regarding ex falso quodlibet). Even though this approach to semantics has greatly progressed in the last decades, it remains nonetheless controversial the existence of a justification of classical logic that suits its restraints. In this thesis I examine various proposals that try to give such a justification and propose a new one greatly inspired by one of Peter Milne’s papers. The conclusion is, to some extent, open since a reformulation of some notions of proof-theoretic semantics is needed in order to justify classical logic. I conclude the thesis with a general defence of logical pluralism and a description of the kind of pluralism that can be applied to our reformulation of proof-theoretic semantics.

Journal of Indian Philosophy, 1992

Outstanding contributions to logic, 2024

The topic of identity of proofs was put on the agenda of general (or structural) proof theory at an early stage. The relevant question is: When are the differences between two distinct proofs (understood as linguistic entities, proof figures) of one and the same formula so inessential that it is justified to identify the two proofs? The paper addresses another question: When are the differences between two distinct formulas so inessential that these formulas admit of identical proofs? The question appears to be especially natural if the idea of working with more than one kind of derivations is taken seriously. If a distinction is drawn between proofs and disproofs (or refutations) as primitive entities, it is quite conceivable that a proof of one formula amounts to a disproof of another formula, and vice versa. A notion of inherited identity of derivations is introduced for derivations in a cut-free sequent system for Almukdad and Nelson's constructive paraconsistent logic N4 with strong negation. The notion is obtained by identifying sequent rules the application of which has no effect on the identity of derivations. Then the notion of inherited identity is used to define a bilateralist notion of synonymy between formulas, which is a relation drawing more fine-grained distinctions between formulas than strong equivalence.

Inquiry, 2020

According to the form of logical pluralism elaborated by Beall and Restall there is more than one relation of logical consequence. Since they take the relation of logical consequence to reside at the very heart of a logical system, different relations of logical consequence yield different logics. In this paper, we are especially interested in understanding what are the consequences of endorsing Beall and Restall’s version of logical pluralism vis-à-vis the normative guidance that logic is taken to provide to reasoners. In particular, the aim of this paper is threefold. First, in sections 2 and 3, we offer an exegesis of Beall and Restall’s logical pluralism as a thesis of semantic indeterminacy of our concept of logical consequence—i.e. understood as indeterminacy logical pluralism. Second, in sections 4 and 5, we elaborate and critically scrutinise three models of semantic indeterminacy that we think are fit to capture Beall and Restall’s indeterminacy logical pluralism. Third, in section 6, following Beall and Restall’s assumption that the notion of logical consequence has normative significance for deductive reasoning, we raise a series of normative problems for indeterminacy logical pluralism. The overall conclusion that we aim to establish is that Beall and Restall’s indeterminate logical pluralism cannot offer an adequate account of the normative guidance that logic is taken to provide us with in ordinary contexts of reasoning.

Bulletin of the Section of Logic

We consider an approach to propositional synonymy in proof-theoretic semantics that is defined with respect to a bilateral G3-style sequent calculus \(\mathtt{SC2Int}\) for the bi-intuitionistic logic \(\mathtt{2Int}\). A distinctive feature of \(\mathtt{SC2Int}\) is that it makes use of two kind of sequents, one representing proofs, the other representing refutations. The structural rules of \(\mathtt{SC2Int}\), in particular its cut-rules, are shown to be admissible. Next, interaction rules are defined that allow transitions from proofs to refutations, and vice versa, mediated through two different negation connectives, the well-known implies-falsity negation and the less well-known co-implies-truth negation of \(\mathtt{2Int}\). By assuming that the interaction rules have no impact on the identity of derivations, the concept of inherited identity between derivations in \(\mathtt{SC2Int}\) is introduced and the notions of positive and negative synonymy of formulas are defined. Sev...

Erkenntnis, 2013

In this paper one prominent version of logical pluralism is the main target of further questions. J.C. Beall and Greg Restall have with their book Logical Pluralism (Beall/Restall 2006) elaborated on their previous statements on logical pluralism. 1 Their view of logical pluralism is centred on ways of understanding logical consequence. This essay therefore tries to come to grips with their doctrine of logical pluralism by highlighting some points that might be made clearer, and questioning the force of some of Beall's and Restall's central arguments. Beall and Restall claim 'that there is more than one genuine deductive consequence relation' (3). According to them there are different and incompatible ways to spell out logical consequence, none of which can be singled out as the 'true logic'. They found this claim on the supposed observation that 'the pre-theoretic notion of logical consequence is not formally defined, and it does not have sharp edges' (28), as well as on observing the existence of a multitude of formal systems. This pluralism applies as well to logics within a language (a linguistic framework), as there are 'different accounts of deductive logical consequence (for the same language)' (29). And thus they claim that 'there are at least two relation of logical consequence (in English)' (31).

2014

This paper addresses the problem of revisionism in Logic from the standpoint of monism and logical pluralism. The underlying problem can be summarized succinctly: "Is there only one correct logic?" I begin with a brief critical summary of this debate, and then introduce the issue by presenting basic formal distinctions between logics outlining rivalry. After this exposition, I turn to our major concern: the argument that there is no actual rivalry between apparently rival logical systems. While ensuring a sort of territorial immunity for both the monist and the Carnapian pluralist, this position renders the debate about revisionism sterile. I hope to show that the reasons offered to support this argument are challenged by Gentzen's sequent calculus. While offering a conceptually richer framework for the study of non-classical logics, the sequent calculus allows us to give an account of one crucial aspect of the logical practice: plurality of logical systems. Finally, I will sketch how one may think of logical pluralism in such a framework.

In [5], Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic (without invoking the role of classical negation in the completeness proof). However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics.

To appear in J. of Logic and Computation, 2014

I argue for a version of logical pluralism based on the plurality of legitimate formalizations of the logical vocabulary. In particular, I argue that the apparent rivalry between classical and relevant logic can be resolved, given that both logics capture and formalize normative and legitimate senses of logical consequence: classical logic encodes “follows from” as truth preservation and captures the truth conditions of the logical constants, while relevant logic encodes a notion of “follows from” which, apart from preserving truth, avoids the violation of certain Gricean maxims and captures a different inferential role for the same logical constants, enriching their meaning pragmatically.