Wright's Strict Finitistic Logic in the Classical Metatheory: The Propositional Case

TL;DR

This paper formalizes Wright's strict finitistic logic within classical metatheory, providing semantics and proof systems, and shows it is weaker than classical logic but stronger than intermediate logics, with all classical theorems valid.

Contribution

It introduces a propositional strict finitistic logic with sound and complete semantics and proof calculus, bridging Wright's ideas with classical formalization.

Findings

The logic lacks excluded middle and Modus Ponens.

It is weaker than classical logic but stronger than intermediate logics.

All classical theorems are valid in this logic.

Abstract

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part of `strict finitistic logic' from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model.