An analytic calculus for intuitionistic belief

TL;DR

This paper develops a natural deduction system for intuitionistic belief, proving its normalization, decidability, and key properties using proof-theoretic methods, thus advancing the formal understanding of intuitionistic epistemic logic.

Contribution

It introduces and analyzes a natural deduction calculus for intuitionistic belief, establishing normalization, subformula property, and other fundamental proof-theoretic results.

Findings

Normalization theorem proved for the calculus

Decidability and consistency established

Disjunction properties and reflection rule admissibility demonstrated

Abstract

Intuitionistic belief has been axiomatized by Artemov and Protopopescu as an extension of intuitionistic propositional logic by means of the distributivity scheme K, and of co-reflection $A\rightarrow\Box A$. This way, belief is interpreted as a result of verification, and it fits an extended Brouwer-Heyting-Kolmogorov interpretation for intuitionistic propositional logic with an epistemic modality. In the present paper, structural properties of a natural deduction system $\mathsf{IEL}^{-}$ for intuitionistic belief are investigated. The focus is on the analyticity of the calculus, so that the normalization theorem and the subformula property are proven firstly. From these, decidability and consistency of the logic follow as corollaries. Finally, disjunction properties, $\Box$-primality, and admissibility of reflection rule are established by using purely proof-theoretic methods.