Bisequent Calculi for Neutral Free Logic with Definite Descriptions

TL;DR

This work develops a uniform bisequent-calculus framework for neutral free logic (NFL) that incorporates identity and definite descriptions via the $ι$-operator. Building on prior work by Pavlović and Gratzl, the authors formulate a minimal theory of definite descriptions using a two-axiom scheme expressed without $\leftrightarrow$, and prove cut admissibility for the extended NFL/BSC system under strong and weak Kleene semantics. The resulting calculus features sound and invertible rules for propositional connectives, a quantified-deduction apparatus including the $E$ predicate and identity, and dedicated rules for $ιxφ$ and $DD$, with a detailed proof of cut-elimination (including complex triple-cut scenarios). The paper lays groundwork for deeper DD theories within NFL, discusses nesting restrictions, and points to future work on nested DD, alternate connectives, paraconsistent extensions, and automated proof-search using tableaux.

Abstract

We present a bisequent calculus (BSC) for the minimal theory of definite descriptions (DD) in the setting of neutral free logic, where formulae with non-denoting terms have no truth value. The treatment of quantifiers, atomic formulae and simple terms is based on the approach developed by Pavlović and Gratzl. We extend their results to the version with identity and definite descriptions. In particular, the admissibility of cut is proven for this extended system.

Figures (7)

• Figure 1: Rules of propositional $\bf K_3$
• Figure 2: Specific rules of propositional $\bf K_3^w$
• Figure 3: Rules for universal quantifier and existence predicate
• Figure 4: Rules for identity
• Figure 5: Rules for DD

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Neutral structure $\mathcal{S}_{nt}$
• Definition 2.4: Weak valuation $\mathcal{V}^w$
• Definition 2.5: Strong valuation $\mathcal{V}^s$
• Definition 2.6
• Lemma 1
• proof
• Definition 3.1
• Theorem 1: Soundness