Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fuzzy bi-Gödel modal logic and its paraconsistent relatives

Marta Bilkova, Sabine Frittella, Daniil Kozhemiachenko

TL;DR

An axiomatization of the fuzzy bi-Gödel modal logic formulated in the language containing $\triangle $ (Baaz Delta operator) and treating $-\!-\!

Abstract

We present the axiomatisation of the fuzzy bi-Gödel modal logic (formulated in the language containing $\triangle$ and treating the coimplication as a defined connective) and establish its PSpace-completeness. We also consider its paraconsistent relatives defined on fuzzy frames with two valuations $e_1$ and $e_2$ standing for the support of truth and falsity, respectively, and equipped with \emph{two fuzzy relations} $R^+$ and $R^-$ used to determine supports of truth and falsity of modal formulas. We establish embeddings of these paraconsistent logics into the fuzzy bi-Gödel modal logic and use them to prove their PSpace-completeness and obtain the characterisation of definable frames.

Fuzzy bi-Gödel modal logic and its paraconsistent relatives

TL;DR

An axiomatization of the fuzzy bi-Gödel modal logic formulated in the language containing (Baaz Delta operator) and treating $-\!-\!

Abstract

We present the axiomatisation of the fuzzy bi-Gödel modal logic (formulated in the language containing and treating the coimplication as a defined connective) and establish its PSpace-completeness. We also consider its paraconsistent relatives defined on fuzzy frames with two valuations and standing for the support of truth and falsity, respectively, and equipped with \emph{two fuzzy relations} and used to determine supports of truth and falsity of modal formulas. We establish embeddings of these paraconsistent logics into the fuzzy bi-Gödel modal logic and use them to prove their PSpace-completeness and obtain the characterisation of definable frames.
Paper Structure (16 sections, 32 theorems, 59 equations, 6 figures)

This paper contains 16 sections, 32 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. (bi-)Gödel modal logics
  3. Paraconsistent modal logics
  4. Logics in the paper
  5. Plan of the paper
  6. Preliminaries
  7. Propositional bi-Gödel logic
  8. $\mathbf{K}\mathsf{biG}$: language and semantics
  9. Axiomatisation of the fuzzy $\mathbf{K}\mathsf{biG}$
  10. $\mathbf{K}\mathsf{G}^2$ and $\mathsf{G}^{2}_{\blacksquare,\blacklozenge}$ --- paraconsistent relatives of $\mathbf{K}\mathsf{biG}$
  11. Semantics
  12. Embeddings into $\mathbf{K}\mathsf{biG}$
  13. Frame definability
  14. Transfer
  15. Complexity
  16. ...and 1 more sections

Key Result

Proposition 2.1

$\mathcal{H}\mathsf{G}\triangle$ is strongly complete: for any $\Gamma\cup\{\phi\}\subseteq\mathcal{L}_\triangle$, it holds that

Figures (6)

  • Figure 1: $[0,1]^{\Join}$: the truth order goes upwards and the information order goes rightwards.
  • Figure 2: Logics mentioned in the paper. $\mathsf{ff}$ stands for ‘permitting fuzzy frames’; $\pm$ for ‘permitting birelational frames’. Subscripts on arrows denote language expansions. $/$ stands for ‘or’ and comma for ‘and’. The logics we mainly focus on in this paper are put in frames.
  • Figure 3: A fuzzy model falsifying ${\sim}\triangle(\lozenge p\rightarrow\lozenge q)\rightarrow\lozenge{\sim}\triangle(p\rightarrow q)$ at $w_0$.
  • Figure 4: $(x,y)$ stands for $wR^+w'=x,wR^-w'=y$. $R^+$ (resp., $R^-$) is interpreted as the tourist's threshold of trust in positive (negative) statements by the friends.
  • Figure 5: All variables have the same values in all states exemplified by $p$.
  • ...and 1 more figures

Theorems & Definitions (85)

  • Definition 2.1
  • Definition 2.2: $\mathcal{H}\mathsf{G}\triangle$ --- Hilbert-style calculus for $\mathsf{biG}$
  • Remark 2.1
  • Proposition 2.1: Baaz1996
  • Definition 2.3: Frames
  • Definition 2.4: $\mathbf{K}\mathsf{biG}$ semantics
  • Proposition 2.2
  • proof
  • Definition 3.1: $\mathcal{H}{\mathbf{K}\mathsf{biG}}^{\mathsf{c}}$ --- Hilbert-style calculus for ${\mathbf{K}\mathsf{biG}}^{\mathsf{c}}$
  • Proposition 3.1
  • ...and 75 more