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Topological semantics for a non-self-extensional LFI

Esha Jain, Sankha S. Basu

TL;DR

This work introduces $vd$, a non-self-extensional Logic of Formal Inconsistency (LFI) with a Hilbert-style calculus and a topological semantics. It defines a signature including a unary consistency operator and a bottom element, derives a classical-like negation via a tilde, and ensures the logic is Tarskian, finitary, and structural, yet non-self-extensional. The topological semantics assigns interpretations to the connectives, with disjunction only partially determined, and proves soundness and completeness via a canonical construction built from $\alpha$-saturated theories and Kuratowski closure operators. The completeness proof proceeds through a canonical model, Kuratowski-type closure, and Lindenbaum-style arguments, culminating in a full semantic justification for $vd$. The results illuminate how topological semantics can capture paraconsistent, non-self-extensional logics and set the stage for further exploration of LFIs and related systems.

Abstract

In this article, we have introduced a Logic of Formal Inconsistency (LFI) that we call $\vd$. This logic is non-self-extensional, i.e., the replacement property, or the rule for substitution of equivalents, does not hold. A Hilbert-style presentation for the logic has been provided. Then, a topological semantics for $\vd$ has been described, subsequent to which we have established the Soundness and Completeness results for it with respect to this semantics.

Topological semantics for a non-self-extensional LFI

TL;DR

This work introduces , a non-self-extensional Logic of Formal Inconsistency (LFI) with a Hilbert-style calculus and a topological semantics. It defines a signature including a unary consistency operator and a bottom element, derives a classical-like negation via a tilde, and ensures the logic is Tarskian, finitary, and structural, yet non-self-extensional. The topological semantics assigns interpretations to the connectives, with disjunction only partially determined, and proves soundness and completeness via a canonical construction built from -saturated theories and Kuratowski closure operators. The completeness proof proceeds through a canonical model, Kuratowski-type closure, and Lindenbaum-style arguments, culminating in a full semantic justification for . The results illuminate how topological semantics can capture paraconsistent, non-self-extensional logics and set the stage for further exploration of LFIs and related systems.

Abstract

In this article, we have introduced a Logic of Formal Inconsistency (LFI) that we call . This logic is non-self-extensional, i.e., the replacement property, or the rule for substitution of equivalents, does not hold. A Hilbert-style presentation for the logic has been provided. Then, a topological semantics for has been described, subsequent to which we have established the Soundness and Completeness results for it with respect to this semantics.

Paper Structure

This paper contains 10 sections, 19 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. The logic vd
  3. Topological semantics
  4. Soundness
  5. vD: a non-self-extensional LFI
  6. Completeness
  7. Kuratowski operators
  8. a-saturated sets
  9. Canonical model and completeness
  10. Conclusion

Key Result

Theorem 2.7

The Deduction theorem holds in $\mathbf{vD}=\langle\mathcal{L},\vdash_{\mathbf{vD}}\rangle$, i.e., for any $\Gamma\cup\{\alpha,\beta\}\subseteq\mathcal{L}$, $\Gamma\cup\{\alpha\}\vdash_{\mathbf{vD}}\beta$ iff $\Gamma\vdash_{\mathbf{vD}}\alpha\longrightarrow\beta$.

Theorems & Definitions (56)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Theorem 2.8
  • proof
  • ...and 46 more