Natural Deduction for Assertibility and Deniability

TL;DR

This paper introduces a logical framework that distinguishes between extensional and intensional propositional connectives, providing a natural deduction system to analyze their assertibility and deniability properties.

Contribution

It develops a novel split of propositional connectives into extensional and intensional types, with corresponding natural deduction rules and semantic characterizations.

Findings

Semantic characterization of extensional connectives via possible worlds.

Direct assertibility and deniability conditions for intensional connectives.

A natural deduction system capturing the interaction of both connective types.

Abstract

In this paper we split every basic propositional connective into two versions, one is called extensional and the other one intensional. The extensional connectives are semantically characterized by standard truth conditions that are relative to possible worlds. Assertibility and deniability of sentences built out of atomic sentences by extensional connectives are defined in terms of the notion of truth. The intensional connectives are characterized directly by assertibility and deniability conditions without the notion of truth. We pay special attention to the deniability condition for intensional implication. We characterize the logic of this mixed language by a system of natural deduction that sheds some light on the inferential behaviour of these two kinds of connectives and on the way they can interact.