Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented (...) following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism. (shrink)

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of (...) analysis is driven by exploiting the Curry–Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity. (shrink)

We study how to postpone the application of the reductio ad absurdum rule (RAA) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of RAA, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

We study how to postpone the application of the reductio ad absurdum rule ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of \, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

An anti-realist theory of meaning suitable for both logical and proper axioms is investigated. As opposed to other anti-realist accounts, like Dummett-Prawitz verificationism, the standard framework of classical logic is not called into question. In particular, semantical features are not limited solely to inferential ones, but also computational aspects play an essential role in the process of determination of meaning. In order to deal with such computational aspects, a relaxation of syntax is shown to be necessary. This leads to a (...) general kind of proof theory, where the objects of study are not typed objects like deductions, but rather untyped ones, in which formulas have been replaced by geometrical configurations. (shrink)

A combination of epistemic logic and dynamic logic of programs is presented. Although rich enough to formalize some simple game-theoretic scenarios, its axiomatization is problematic as it leads to the paradoxical conclusion that agents are omniscient. A cut-free labelled Gentzen-style proof system is then introduced where knowledge and action, as well as their combinations, are formulated as rules of inference, rather than axioms. This provides a logical framework for reasoning about games in a modular and systematic way, and to give (...) a step-by-step reconstruction of agents omniscience. In particular, its semantic assumptions are made explicit and a possible solution can be found in weakening the properties of the knowledge operator. (shrink)

A widely debated issue in philosophy of logic concerns the possibility of an inferentialist account of classical logic. Many proposals to show that classical logic satisfies the requirements of inferentialist semantics (such as harmony) demand to modify the ordinary natural deduction rules. In this paper, we try to explain why the ordinary natural deduction rules for classical logic are not harmonious and therefore not directly justifiable within an inferentialist framework. We show however that an indirect justification of classical logic, passing (...) through negative translation, can be acceptable from an inferentialist point of view. (shrink)

Cet article propose d’étudier les critiques avancées par Michael Dummett à l’encontre de l’argument consistant à faire appel à l’omniscience de Dieu pour justifier la thèse du réalisme sémantique (c’est-à-dire la thèse selon laquelle la signification et la vérité de nos énoncés seraient déterminées par une réalité indépendante de nous et de nos états cognitifs). Pour Dummett, un tel argument est en effet illégitime, car il contient une pétition de principe qui est à son tour induite par une erreur d’analyse (...) logico-linguistique. (shrink)

We investigate in intuitionistic first-order logic various principles of preference relations alternative to the standard ones based on the transitivity and completeness of weak preference. In particular, we suggest two ways in which completeness can be formulated while remaining faithful to the spirit of constructive reasoning, and we prove that the cotransitivity of the strict preference relation is a valid intuitionistic alternative to the transitivity of weak preference. Along the way, we also show that the acyclicity axiom is not finitely (...) axiomatizable in first-order logic. (shrink)