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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 degree-preserving Godel logics with involution and CPL, including finite-valued counterparts, where ordinary Godel negation remains explosive but involutive negation is paraconsistent. It introduces saturated paraconsistency, characterizes the ideal and saturated cases in the Godel family, and identifies many saturated cases for finite-valued Lukasiewicz logics.
Problem
The paper examines which intermediate degree-preserving fuzzy logics with involution are paraconsistent and how saturated paraconsistency relates to ideal paraconsistency.
Method
The paper introduces saturated paraconsistency and analyzes intermediate logics between degree-preserving finite-valued Godel logics with involution and CPL, with related analysis for finite-valued Lukasiewicz logics.
Results
Only three saturated paraconsistent logics occur between degree-preserving finite-valued Godel logics with involution and CPL: J3, J4, and J3 × J4; the first two are ideal.
Takeaways & Limitations
Saturated paraconsistency yields a weaker class than ideal paraconsistency and includes additional examples in the Godel and finite-valued Lukasiewicz settings.
Takeaways & Limitations
Future work is restricted to proposing analogous studies for other locally finite fuzzy logics, especially Nilpotent Minimum logic.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper motivates paraconsistent reasoning for contradictory information and questions whether maximality with respect to classical logic is necessary. It introduces saturated paraconsistency and studies such logics between degree-preserving fuzzy logics and CPL.
- Paraconsistent logics aim to tolerate contradictions without trivializing theories while still deriving useful conclusions.
- Ideal paraconsistency requires both maximal paraconsistency and maximality with respect to classical logic CPL.
- Critics argue that maximality with respect to CPL is not a natural requirement for information-oriented paraconsistent reasoning.
- Saturated paraconsistency weakens ideal paraconsistency by dropping maximality with respect to CPL.
- The paper finds only three saturated paraconsistent logics between degree-preserving finite-valued Godel logics with involution and CPL: J3, J4, and J3 × J4.
- For degree-preserving finite-valued Lukasiewicz logics, the paper identifies a large family of saturated paraconsistent logics that are not ideal.
2 Preliminaries
The preliminaries define Godel logic, its finite-valued and involutive extensions, and degree-preserving consequence. They establish that adding involutive negation preserves paraconsistency for that negation even though the systems remain explosive for Godel negation.
- Godel logic uses conjunction, implication, and bottom as primitives, with disjunction, negation, equivalence, and top defined from them.
- Godel logic is characterized by standard fuzzy semantics over [0, 1], using minimum for conjunction and the Godel implication.
- Its finite-valued counterparts use GVn and form a chain ending in G2 = CPL.
- The involutive extension G∼ interprets ∼ as an order-reversing involution, while its finite-valued counterpart Gn∼ uses the corresponding finite algebra.
- Degree-preserving consequence requires every designated lower bound a preserved from all premises to the conclusion across chains and evaluations.
- Although G∼ and its degree-preserving companion remain explosive for Godel negation, the degree-preserving companion is paraconsistent for involutive negation.
3 Logics defined by matrices over [0, 1]G∼by means of order filters
The section constructs matrix logics over the standard Godel algebra with involution using order filters, then classifies their equivalences, finitarity, inclusions, and paraconsistency. It also relates the resulting logic to degree-preserving Lukasiewicz logic.
- Matrix construction: Order filters of [0,1]G∼ are exactly F[a={x:x≥a} for a∈(0,1] and F(a={x:x>a} for a∈[0,1), defining the matrix family studied.The corresponding consequence relations and finitary companions are denoted separately.
- Equivalence and finitarity: The logics ⊢1, ⊢(1/2, ⊢[1/2, and ⊢(0 are equivalent and finitary.The equivalence is established through faithful interpretations using formula transformations and the ∆ operator.
- Relationships among logics: For all p,p′∈(1/2,1), ⊢[p=⊢[p′ and ⊢(p=⊢(p′; the analogous collapse holds for all n,n′∈(0,1/2).Automorphisms of [0,1]G∼ support the positive-region equivalences, with analogous arguments for the negative region.
- Relationships among logics: The positive-threshold logics are strictly below ⊢1, while ⊢[p and ⊢[1/2 are incomparable and ⊢[p and ⊢[n are incomparable.For example, φ⊢1∆φ but φ⊬[p∆φ for p<1.
- Paraconsistency classification: Only three different paraconsistent logics occur among the order-filter families: G[a∼ for a∈(0,1/2), G[1/2∼, and G(0∼; the remaining listed families are explosive.A matrix logic G[a∼ is paraconsistent only when a≤1/2, respectively a<1/2 for the strict filter.
- Connection with Lukasiewicz logic: The implication →FT is definable in [0,1]G∼ by x→FTy=∆(x→Gy)∧(∼x∨y), making FT interpretable in G(0.Conversely, ∆ and the Godel implication are definable in the matrix semantics for FT, yielding an equivalence up to language with FT0.
4 Logics between G ≤ n∼and CPL
The paper characterizes intermediate logics between degree-preserving finite-valued Gödel logics with involution and CPL using algebraic and matrix representations. It establishes structural results for finite algebras, analyzes paraconsistency and explosion, and identifies incomparability patterns among these logics.
- Finite-valued correspondences: G3∼ and G4∼ coincide with the 3- and 4-valued Łukasiewicz logics through termwise definability of their connectives.The paper explicitly defines Łukasiewicz implications in the corresponding Gödel algebras and shows the converse definability of Gödel connectives in finite MV-algebras.
- Finite-valued correspondences: For n > 4, Gn∼ is not equivalent to Ln, so its intermediate logics require separate analysis.The n = 3 and n = 4 correspondence does not extend to larger finite values.
- Algebraic and matrix characterization: Every finite Gn∼-algebra is a finite direct product of finite Gn∼-chains, enabling intermediate logics to be represented by products of matrix components with compatible lattice filters.This structural decomposition underlies the matrix characterization of the intermediate logics between G≤n∼ and CPL.
- Order structure: The logics L(M{t}) are pairwise incomparable and form the maximal elements of a meet-semilattice of logics defined by matrices over GVn∼.More generally, distinct equal-cardinality threshold sets yield incomparable logics, while intersections correspond to unions of threshold sets.
- Paraconsistency and explosion: L(T) derives a conclusion either when the premises meet a graded inconsistency condition or when the conclusion follows at every threshold in T.The first disjunct corresponds to premise evaluations falling below max T; the second uses the family of threshold-filter matrices.
- Examples: The product J3 × J4 is incomparable with both J3 and J4 because its theorems are the intersection of their theorem sets, while each has consequences absent from the other.The paper gives formulas witnessing that J3 and J4 derive different contradictions, both captured by their product.
- Paraconsistency and explosion: Logics extending Lexp validate explosion for the involutive negation, whereas those not extending Lexp are paraconsistent.The paper characterizes Lexp as the boundary separating explosive extensions from paraconsistent logics in this setting.
5 Ideal and saturated paraconsistent extensions
The section defines ideal and saturated paraconsistency and characterizes these extensions between degree-preserving finite-valued Gödel logics with involution and CPL. It identifies J3 and J4 as the only ideal cases, while J3 × J4 is saturated but not ideal.
- Definitions: Ideal paraconsistency requires paraconsistency, a deductive implication, containment in CPL, and maximality with respect to CPL.Saturated paraconsistency retains the other conditions but replaces maximality with the requirement that every proper extension is non-paraconsistent.
- Ideal extensions: J3 and J4 are ideal ∼-paraconsistent logics for the term-defined implication x ⇒ y := ¬x ∨ y.Both are established as ideal logics in the cited prior characterization.
- Definitions: The paper questions maximality with respect to CPL as necessary for an optimal paraconsistent logic and introduces saturated paraconsistency as a weaker notion.The motivation is that the remaining requirements of ideal paraconsistency are retained while CPL maximality is dropped.
- Saturated extensions: J3 × J4 is saturated ∼-paraconsistent but not ideal ∼-paraconsistent.It has no proper ∼-paraconsistent extension, but it is not maximal with respect to CPL because J2 × J3 is a proper nonclassical extension.
6 Saturated paraconsistency and finite-valued Lukasiewicz logics
The paper revisits finite-valued Łukasiewicz logics to characterize saturated paraconsistent extensions and distinguish them from ideal paraconsistent ones. It proves a broad family of saturated examples, including products associated with finite sets of primes.
- The study extends earlier work on ideal paraconsistency in finite-valued Łukasiewicz logics to the weaker notion of saturated paraconsistency.
- J3 and J4 are ideal ¬-paraconsistent matrix logics arising from the 3- and 4-valued Łukasiewicz chains.
- For prime q and 1 ≤ i < q with i/q ≤ 1/2, L_i^q is a (q + 1)-valued ideal ¬-paraconsistent logic.
- Among extensions of L_i^n, ideal ¬-paraconsistency is characterized by equality with L_j^q for a prime q dividing n and j/q ≤ 1/2.
- Every finite product built from the prime-valued components specified in Theorem 3 is saturated ¬-paraconsistent.
- These saturated logics are generally not ideal, while examples include arbitrary finite products indexed by prime numbers and the specific logic L_7^15.
7 A final remark: relationship to logics of formal inconsistency
The paper observes that the paraconsistent logics studied in both the Gödel-with-involution and finite-valued Łukasiewicz settings are logics of formal inconsistency. Their consistency operators are tied to the relevant negations.
- All paraconsistent logics based on Gödel fuzzy logic with involution and its finite-valued extensions are LFIs with respect to the involutive negation ∼.
- The finite-valued Łukasiewicz matrix logics and their products considered here are LFIs with respect to Łukasiewicz negation ¬.
- For Gödel-based expansions, the consistency operator is given by ◦φ = ∆(¬φ ∨ φ) when the Baaz–Monteiro operator is definable.
8 Conclusions
The paper studies degree-preserving Gödel logics with involution and their finite-valued counterparts as intermediate logics below CPL. It introduces saturated paraconsistency, characterizes the relevant Gödel cases, and finds many finite-valued Łukasiewicz examples.
- Degree-preserving Gödel logics with involution are analyzed because they are ∼-paraconsistent, unlike the corresponding Gödel logics without degree preservation.
- Between G_n^≤∼ and CPL, exactly three saturated paraconsistent logics occur: J3, J4, and J3 × J4.
- J3 and J4 are ideal paraconsistent, whereas J3 × J4 is saturated but not ideal.
- The finite-valued Łukasiewicz analysis identifies additional saturated paraconsistent logics that are not ideal.