On the Computational Meaning of Axioms

Abstract

An anti-realist theory of meaning suitable for both logical and proper axioms is investigated. As opposed to other anti-realist accounts, like DummettPrawitz 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where the objects of study are not typed objects like deductions, but rather untyped ones, in which formulas have been replaced by geometrical configurations.

Perspectives on Interrogative Models of Inquiry

“…In Naibo et al (2015) it is claimed, indeed, that the untyped theory framework is not compatible with a verificationist approach, and this claim is based on the treatment of the notion of proper axiom. In Naibo et al (2015) it is claimed, indeed, that the untyped theory framework is not compatible with a verificationist approach, and this claim is based on the treatment of the notion of proper axiom.…”

“…In Naibo et al (2015) the notion of untyped proof theory is introduced in order to describe a general framework for dealing with an abstract mathematical (and in particular, geometrical) notion of proof from which it is possible to generate a class of deductive systems suitable both for logical and proper mathematical theories. More precisely, an untyped proof theory represents a very abstract model of computation, based only on two fundamental notions, that of execution and that of termination.…”

“…24 As explained in Naibo et al (2015), the approach undertaken by untyped proof theory differs from the usual techniques adopted in standard proof theory, as it considers a set of objects -the paraproofs -which is much larger than the set of proofs. 24 As explained in Naibo et al (2015), the approach undertaken by untyped proof theory differs from the usual techniques adopted in standard proof theory, as it considers a set of objects -the paraproofs -which is much larger than the set of proofs.…”

From abstraction and indiscernibility to classification and types:

“…4 The methodology is for example used in [Girard, 2001], [Krivine, 2001], [Hyland-Schalk, 2003]. For a more theoretical/conceptual focus on the methodology itself : [Naibo-Petrolo-Seiller, 2016]. 5 The notion of orthogonality can be seen as a formulation of Garett Birkhoff's polarities.…”

A correspondence between maximal abelian sub-algebras and linear logic fragments

“…(or rather proof structures) (Naibo et al 2016), ludics is built upon an abstraction of (focalised) MALL sequent calculus derivations (with a modified axiom rule (Curien 2006;Naibo et al 2016)).…”