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Degree-preserving Godel logics with an involution: intermediate logics and (ideal) paraconsistency

M. E. Coniglio, F. Esteva, J. Gispert, L. Godo

TL;DR

The paper studies intermediate logics between the degree-preserving Gödel logic with involution $G^{\leq}_{\sim}$ and CPL, and between their finite-valued versions and CPL. It develops a matrix-semantics framework using order filters on $[0,1]_{G\sim}$ and analyzes $G^{\leq}_{\sim}$ and $G^{\leq}_{n\sim}$, showing paraconsistency with respect to $\sim$ but explosion for $\neg$, and lacking full deduction-detachment for $G_{\sim}$. The authors fully characterize ideal and saturated paraconsistent extensions: in the Gödel family, only $\mathsf{J}_3$ and $\mathsf{J}_4$ are ideal paraconsistent (with $\mathsf{J}_3\times\mathsf{J}_4$ as a saturated, non-ideal instance), while a large saturated family emerges in finite-valued Łukasiewicz logics. The work connects LFIs to paraconsistent fuzzy logics, and suggests avenues for extending the approach to other locally finite fuzzy logics such as NM, offering a detailed map of the interplay between degree-preserving semantics, paraconsistency, and ideal/saturated maximality. These results have implications for designing robust, information-tolerant logics in contexts combining vagueness and inconsistency, such as AI reasoning with uncertain or conflicting data.

Abstract

In this paper we study intermediate logics between the degree preserving companion of Godel fuzzy logic with an involution and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts. Although these degree-preserving Godel logics are explosive with respect to Godel negation, they are paraconsistent with respect to the involutive negation. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between the degree-preserving n-valued Godel fuzzy logic with an involution and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Lukasiewicz logics.

Degree-preserving Godel logics with an involution: intermediate logics and (ideal) paraconsistency

TL;DR

The paper studies intermediate logics between the degree-preserving Gödel logic with involution and CPL, and between their finite-valued versions and CPL. It develops a matrix-semantics framework using order filters on and analyzes and , showing paraconsistency with respect to but explosion for , and lacking full deduction-detachment for . The authors fully characterize ideal and saturated paraconsistent extensions: in the Gödel family, only and are ideal paraconsistent (with as a saturated, non-ideal instance), while a large saturated family emerges in finite-valued Łukasiewicz logics. The work connects LFIs to paraconsistent fuzzy logics, and suggests avenues for extending the approach to other locally finite fuzzy logics such as NM, offering a detailed map of the interplay between degree-preserving semantics, paraconsistency, and ideal/saturated maximality. These results have implications for designing robust, information-tolerant logics in contexts combining vagueness and inconsistency, such as AI reasoning with uncertain or conflicting data.

Abstract

In this paper we study intermediate logics between the degree preserving companion of Godel fuzzy logic with an involution and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts. Although these degree-preserving Godel logics are explosive with respect to Godel negation, they are paraconsistent with respect to the involutive negation. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between the degree-preserving n-valued Godel fuzzy logic with an involution and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Lukasiewicz logics.
Paper Structure (12 sections, 14 theorems, 23 equations, 2 figures)

This paper contains 12 sections, 14 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Truth-preserving Gödel logics
  4. Degree-preserving Gödel logics with involution
  5. Logics defined by matrices over $[0,1]_{G \sim}$ by means of order filters
  6. Logics between $\mathrm{G}^{\text{\footnotesize $\leq$}}_{n\sim}$ and CPL
  7. The intermediate logics of $\mathrm{G}^{\text{\footnotesize $\leq$}}_{n\sim}$ for $n> 4$
  8. Example: the case $n = 5$
  9. Ideal and saturated paraconsistent extensions of $\mathrm{G}^{\text{\footnotesize $\leq$}}_{n\sim}$
  10. Saturated paraconsistency and finite-valued Łukasiewicz logics
  11. A final remark: relationship to logics of formal inconsistency
  12. Conclusions

Key Result

Lemma 1

The logic $\vdash_1$ is finitary. Moreover, the logics $\vdash_{(1/2}, \vdash_{[1/2}$ and $\vdash_{(0}$ are equivalent, as deductive systems, to $\vdash_1$ and hence they are finitary as well. Therefore all these logics coincide with their finitary companions $\vdash^f_1, \vdash^f_{(1/2}, \vdash^f_{

Figures (2)

  • Figure 1: Graph of axiomatic extensions of $\rm{G}_{\sim}$.
  • Figure 2: Graph of logics over $[0, 1]_{\rm{G}_{\mathord{\sim}}}$ defined by order filters, where $1/2 < p < 1$ and $0 < n < 1/2$, where edges stand for inclusions (upward sense). The dashed edges denote that it is an open problem whether the connected logics are different.

Theorems & Definitions (36)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Corollary 1
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • ...and 26 more