Some Problems for Proof-Theoretic Semantics (original) (raw)

The original sin of proof-theoretic semantics

Synthese

Proof-theoretic semantics is an alternative to model-theoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association of a consequence relation to a logical calculus is not as straightforward as it may seem. We show that these facts are problematic for the proof-theoretic project but the problems can be solved. Our thesis is that the consequence relation relevant for proof-theoretic semantics is the one given by the sequent-to-sequent derivability relation in Gentzen systems.

Proof-Theoretic Semantics for natural language

Topoi, 2021

The paper has two parts: 1. A brief exposition of proof-theoretic semantics (PTS), not necessarily in connection to natural language (NL). 2. A review, with a contrastive flavour, of some of the applications of PTS to NL with an indication of advantages of PTS as a theory of meaning for NL.

An interactive approach to proof-theoretic semantics

2015

In truth-functional semantics for propositional logics, categoricity and compositionality are unproblematic. This is not the case for proof-theoretic semantics, where failures of both occur for the semantics determined by monological entailment structures for classical and intuitionistic logic. This is problematic for inferentialists, where the meaning of logical constants is supposed to be determined by their rules. Recent attempts to overcome these issues have primarily considered symmetric entailment structures, but these are tricky to interpret. Here, I instead consider an entailment structure that combines provability with the dual notion of disproof (or refutation). This is interpreted as a dialogue structure between the roles of prover and denier, where an assertion of a statement involves a commitment to its defence, and a denial of the statement involves a commitment to its challenge. The interaction between the two is constitutive of a proof-theoretic semantics capable of ...

Pluralism in Proof-Theoretic Semantics

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.

The granularity of meaning in proof-theoretic semantics

Lecture notes in computational linguistics, Springer 2014

Abstract: The paper compares two conceptions of meaning both for logic and for natural language: — Model-theoretic semantics (MTS), basing meaning on reference and truth conditions (in arbitrary models). — Proof-theoretic semantics (PTS), basing meaning on canonical derivations in meaning-conferring natural-deduction proof-systems. It is shown that PTS induces a much finer granularity on meanings, in particular distinguishing the meanings of logically equivalent sentences. A certain coarsening by means of equating meanings based on identical grounds for assertion is proposed, useful in certain contexts. Key-words and phrases: proof-theoretic semantics, granularity of meaning, grounds for assertion"

Proof-Theoretic Semantic Values for Logical Constants

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Frege's Context Principle, by taking "contributions" to sentential meanings as determined by the functionargument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The compositionality of the resulting attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal.