Gödel Justification Logic: A Fuzzy Framework

• Gödel Justification Logic is a many-valued extension of propositional Gödel logic that internalizes graded justifications using explicit terms.
• It introduces a specialized syntax and axioms that enable fine-grained reasoning about partial evidences, diverging from classical realization theorems.
• The framework employs one-world and fuzzy Fitting models, with research focusing on completeness proofs and challenges in modal–justification correspondence.

Gödel justification logic (GJL) is the explicit justification analogue of modal logic based on Gödel fuzzy logic, internalizing justifications within the many-valued framework of [0,1]-valued t-norm logics. GJL extends propositional Gödel logic by introducing explicit justification terms and operators, thereby enabling fine-grained reasoning about partial, graded justifications. Classically, justification logics support the realization theorem linking modal theorems (□φ) to explicit justification assertions (t:φ). However, in the Gödel setting, this correspondence is fundamentally disrupted: several modal Gödel logics strictly contain the projections of their justification logic analogues, revealing deep differences arising from the many-valued semantics.

1. Language and Syntax

GJL is built over countable sets of propositional variables (Var = {p₀, p₁, ...}), justification variables (V = {x₀, x₁, ...}), and justification constants (C = {c₀, c₁, ...}). Justification terms (Jt) are constructed recursively by the grammar:

$$t ::= c \mid x \mid (t + t) \mid (t \cdot t) \mid {!}t \mid {?}t$$

where $+$ is the sum (alternative justifications), $\cdot$ is application (modus ponens for justifications), $!$ encodes positive introspection, and $?$ negative introspection (Pischke, 2019, Pischke, 2018).

Formulas are given by the grammar:

$$\varphi ::= \bot \mid p \mid (\varphi \wedge \varphi) \mid (\varphi \to \varphi) \mid t:\varphi$$

with $\neg\varphi$ as an abbreviation for $\varphi \to \bot$ (Pischke, 2019, Pischke, 2018, Ghari, 2014). The operators $\wedge$ and $\to$ denote, respectively, minimum and residuum operations in the Gödel algebra.

2. Proof Systems and Axiomatizations

The base Hilbert system $\mathbf{GJ}_0$ includes all axioms of propositional Gödel logic [Hájek 1998], namely:

• (A1) $$(\varphi \to \psi) \to ((\psi \to \chi) \to (\varphi \to \chi))$$
• (A2) $(\varphi \wedge \psi) \to \varphi$
• (A3) $(\varphi \wedge \psi) \to (\psi \wedge \varphi)$
• (A5a) $$(\varphi \to (\psi \to \chi)) \to ((\varphi \wedge \psi) \to \chi)$$
• (A5b) $$((\varphi \wedge \psi) \to \chi) \to (\varphi \to (\psi \to \chi))$$
• (A6) $$((\varphi \to \psi) \to \chi) \to (((\psi \to \varphi) \to \chi) \to \chi)$$
• (A7) $\bot \to \varphi$
• (G4) $\varphi \to (\varphi \wedge \varphi)$

Added are justification-specific axioms:

• (J) $$t:(\varphi \to \psi) \to (s:\varphi \to (t\cdot s):\psi)$$
• ($+$) $t:\varphi \to (t+s):\varphi$, $s:\varphi \to (t+s):\varphi$

Extensions $\mathbf{GJT}_0$, $\mathbf{GJ4}_0$, $\mathbf{GJ45}_0$, and $\mathbf{GLP}_0$ introduce:

• (F) $t:\varphi \to \varphi$ (factivity, T)
• (!): $t:\varphi \to !t: t:\varphi$ (positive introspection, 4)
• (?): $\neg t:\varphi\to ?t: \neg t:\varphi$ (negative introspection, 5)

The sole inference rule is modus ponens (MP) (Pischke, 2019, Pischke, 2018, Ghari, 2014).

3. Semantics: Gödel-Fuzzy Models and Evidence

Gödel justification logics admit two main semantic approaches: one-world (Gödel-Mkrtychev) models and multi-world (fuzzy Fitting) models. In both, truth values are taken from the real unit interval $[0,1]$.

• One-world (Gödel-Mkrtychev) models: A model $\mathcal{M} = \langle \mathcal{E}, e \rangle$, where $e: \text{Var}\to [0,1]$ and $$\mathcal{E}: \text{Jt} \times \mathcal{L}_J \to [0,1]$$ satisfy:
  • (App): $$\mathcal{E}(t, \varphi\to\psi)\odot\mathcal{E}(s,\varphi)\leq\mathcal{E}(t \cdot s, \psi)$$
  • (Sum): $$\mathcal{E}(t,\varphi)\oplus\mathcal{E}(s,\varphi) \leq \mathcal{E}(t+s,\varphi)$$ (with $\oplus = \max$)

Formula values: - $|\bot|_\mathcal{M} = 0$ - $|p|_\mathcal{M} = e(p)$ - $$|\varphi \wedge \psi|_\mathcal{M} = |\varphi|_\mathcal{M} \odot |\psi|_\mathcal{M}$$ - $$|\varphi\to\psi|_\mathcal{M} = |\varphi|_\mathcal{M} \Rightarrow |\psi|_\mathcal{M}$$, where $x\Rightarrow y = y$ if $x>y$, $=1$ otherwise - $|t:\varphi|_\mathcal{M} = \mathcal{E}(t,\varphi)$

Specialized semantic conditions model factivity, introspection, etc., for various GJL extensions (Pischke, 2019).

• Fuzzy Fitting Models: Quadruples $\mathfrak{M}=(W, R, \mathcal{E}, e)$; fuzzy accessibility $R:W\times W \to [0,1]$, evidence $$\mathcal{E}: W\times \text{Jt}\times\mathcal{L}_J\to[0,1]$$, and evaluations:

$$e(w, t:\varphi) = \mathcal{E}(w,t,\varphi) \odot \inf_{u \in W} (R(w,u) \Rightarrow e(u,\varphi))$$

(Pischke, 2018).

Soundness and completeness theorems establish a tight correspondence between axiom systems and the respective class of fuzzy models for each extension (Pischke, 2018, Pischke, 2019).

4. Forgetful Projection and the Non-Realization Phenomenon

The forgetful projection $\varphi \mapsto \varphi^\circ$ erases justifications from GJL formulas, mapping

• $p^\circ = p$
• $(t:\varphi)^\circ = \Box \varphi^\circ$
• standard connectives are preserved.

This sends theorems of GJL systems to theorems of corresponding Gödel modal logics (GML)—e.g., $\mathbf{GK}_\Box$, $\mathbf{GT}_\Box$—via:

$$\Gamma \vdash_{\mathbf{GJL}_{CS}} \varphi \implies \Gamma^\circ \vdash_{\mathbf{GML}_\Box} \varphi^\circ$$

(Pischke, 2019).

However, unlike the classical case, the converse inclusion fails. The non-realization theorem establishes the existence of modal Gödel logic theorems, specifically the axiom:

$$(Z)\qquad \neg\neg\Box\varphi \to \Box\neg\neg\varphi$$

that do not arise as projections of any GJL theorem. This is demonstrated by constructing $x$-rooted provability models with $x \in (0,1)$, exploiting intermediate truth values so that $Z$ fails to have a realizing justification formula (Pischke, 2019).

5. Model Construction and Completeness Results

Completeness for GJL systems is proved via canonical model constructions and star translations. Every strongly complete system has that, for any $\Gamma \cup \{\varphi\} \subseteq \mathcal{L}_J$,

$$\Gamma \vdash_{\mathbf{GJL}_{CS}} \varphi \iff \forall \mathfrak{M} \in \mathsf{GJL}_{CS}, \forall w \in W: \inf_{\psi \in \Gamma} e(w,\psi) \leq e(w,\varphi)$$

with accessibility-crisp models sufficing (Pischke, 2018). The construction utilizes star-translation, mapping $t:\psi$ to a new propositional variable, then transferring Gödel-propositional completeness to the explicit logic via canonical evaluation.

In the base setting, Ghari (Ghari, 2014) established soundness for GJL but left general completeness open; later work (Pischke, 2018) closed the gap for strong completeness in Hilbert systems matching various modal axiomatizations.

6. Relationship with Modal and Classical Justification Logics

GJL generalizes classical justification logics by replacing Boolean connectives and truth values with their Gödel-algebra analogues, internalizing graded justifications rather than binary ones. In the crisp case (truth values in $\{0,1\}$), GJL coincides with classical J; but in the fuzzy setting, the behavior diverges sharply.

A critical difference is that, while modal Gödel logics and their justification analogues are syntactically linked via the forgetful projection, the realization theorem fails in the Gödel context. The gap is semantically rooted: the key axiom (Z) is trivial in crisp models but non-trivial and necessary in fuzzy frames. Thus, the forgetful image of GJL is a strict fragment of the corresponding standard Gödel modal logic (Pischke, 2019). Alternative justification-style formalisms or modifications must be explored to achieve modal-justification correspondence in the fuzzy field.

7. Implications, Open Problems, and Directions

The non-realizability result for GJL answers negatively a longstanding question on the existence of justification logics matching fuzzy modal logics under forgetful projection [Ghari 2014, Pischke 2018]. Current research avenues include characterizing the maximal fragment of Gödel modal logic that is realizable by GJL (i.e., GT_□ minus axiom Z), and devising new frameworks where a full realization theorem may hold. Extensions for richer connectives, Pavelka-style truth constants, and the investigation of Baaz’s $\Delta$-operator also offer promising future directions (Pischke, 2018, Pischke, 2019).

GJL serves as a flexible bridge between justification logics and fuzzy frameworks, providing a context for expressing partial beliefs and graded evidences within explicit epistemic or proof-theoretic settings. However, the many-valued nature fundamentally alters the landscape of realization, completeness, and modal–justification correspondence, marking GJL as a distinct logic-theoretic subject within the landscape of contemporary logic research.