Necessity as justified truth

TL;DR

This paper develops a logic that defines necessity through justifications, extending existing logics with algebraic semantics, and shows how modal logics S4 and S5 can be captured within this framework.

Contribution

It introduces a justification-based logic for necessity, extending Suszko's non-Fregean logic with algebraic semantics and modal principles, unifying justification and modal logics.

Findings

Necessity is definable via justifications within the logic.

Soundness and completeness of the system are established.

Modal logics S4 and S5 are characterized by additional semantic constraints.

Abstract

We present a logic for the reasoning about necessity and justifications which is independent from relational semantics. We choose the concept of justification -- coming from a class of "Justification Logics" (Artemov 2008, Fitting 2009) -- as the primitive notion on which the concept of necessity is based. Our axiomatization extends Suszko's non-Fregean logic SCI (Brown, Suszko 1972) by basic axioms from Justification Logic, axioms for quantification over propositions and over justifications, and some further principles. The core axiom is: $\varphi$ is necessarily true iff there is a justification for $\varphi$. That is, necessity is first-order definable by means of justifications. Instead of defining purely algebraic models in the style of (Brown, Suszko 1972) we extend the semantics investigated in (Lewitzka 2012) by some algebraic structure for dealing with justifications and prove soundness and completeness of our deductive system. Moreover, we are able to restore the modal logic principle of Necessitation if we add the axiom schema $\square\varphi \to \square\square\varphi$ and a rule of Axiom Necessitation to our system. As a main result, we show that the modal logics S4 and S5 can be captured by our semantics if we impose the corresponding modal logic principles as additional semantic constraints. This will follow from proof-theoretic considerations and from our completeness theorems. For the system S4 we present also a purely model-theoretic proof.