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On Three-Valued Dependence-Like Logics
Yaroslav Petrukhin
TL;DR
The paper examines how Epstein-style subject-matter semantics relates to variable-inclusion conditions in infectious logics. It introduces two three-valued dependence-like logics and provides sound, complete, cut-free bisequent calculi for them.
Problem
Method
Results
Takeaways & Limitations
Kd3 and LPd offer a simple many-valued setting for viewing variable-inclusion conditions as variants of Epstein-style subject-matter restrictions, though they are not dependence logics in Epstein’s original sense.
Source: §4 · p. 8
Abstract
from arXiv · showhide
In this paper, we investigate connections between dependence logics, viewed as a subclass of relating logics, and three-valued logics. More specifically, we identify common features of Epstein's subject-matter semantics and the variable-inclusion conditions characteristic of some infectious many-valued logics. Inspired by Del Cerro and Lugardon's sequent calculi for dependence logics, in which classical connectives are combined with connectives satisfying Epstein-style conditions, we introduce two three-valued dependence-like logics that combine classical conjunction and disjunction with infectious negation and implication. We also provide sound, complete, and cut-free bisequent calculi for these logics.
1 Introduction
The introduction connects Epstein-style dependence semantics with variable-inclusion conditions in infectious logics. It proposes two three-valued logics combining classical conjunction and disjunction with non-classical implication and negation.
1 Introduction
- The proposal builds on connections among dependence logics, variable-inclusion logics, and Del Cerro and Lugardon’s sequent-calculus approach.Dependence logics are described as a special class of relating logics, and DAI was studied in connection with infectious logics.Source: §1 · pp. 1–2
- Under the Proscriptive Principle, classical entailment yields entailment in weak Kleene logic Kw3.The principle requires every variable in the conclusion to occur in the premises and is presented as a possible specification of Epstein-style subject-matter conditions.Source: §1 · pp. 1–2
- The converse variable-inclusion condition yields paraconsistent weak Kleene entailment PWK whenever the premises classically entail the conclusion.This converse requires all premise variables to occur in the conclusion; Epstein also considered an implication with the subject-matter inclusion direction reversed.Source: §1 · p. 2
- The paper proposes two three-valued logics with classical conjunction and disjunction and non-classical implication and negation.One uses infectious implication and negation from Kw3, connected with the Proscriptive Principle; the other uses them from PWK, connected with its converse.Source: §1 · p. 2
2 From semantics of dependence logics to semantics of our logics
The section connects variable-inclusion restrictions in infectious logics with subject-matter conditions in dependence logics, then develops two three-valued dependence-like logics and examines their properties.
2 From semantics of dependence logics to semantics of our logics
- Variable-inclusion restrictions in infectious logics can be characterized through non-infectious entailment subject to variable conditions, which the paper relates to Epstein’s subject-matter conditions.The authors describe infectious logics as relatives of dependence logics, not dependence logics themselves.Source: §2 · p. 4
- The paper considers both classical negation with infectious implication and a second option that makes negation infectious as well.It also allows connectives that need not match their classical counterparts on {0, 1}, provided their designatedness follows the corresponding classical connective’s output.Source: §2 · pp. 4–5
- The two logics pair classical conjunction and disjunction with infectious negation and implication, drawing on weak Kleene–Bochvar connectives.The paper focuses on logics whose conjunction and disjunction are minimum and maximum, and whose implication and negation are weak Kleene–Bochvar.Source: §2 · pp. 4–5
- Weak Kleene–Bochvar implication is the only three-valued infectious implication in the section’s comparison that coincides with classical implication on {0, 1}.The shared weak Kleene–Bochvar negation is likewise the only three-valued infectious negation that coincides with classical negation on {0, 1}.Source: §2 · p. 5
- Kd3 and LPd use the same connectives but differ in designated values, defining entailment by preservation of 1 and non-falsity, respectively.Kd3 designates 1; LPd designates 1 and 1/2.Source: §2 · p. 5
- LPd is paraconsistent and has the same tautologies as classical logic, whereas Kd3 has no tautologies.The section also reports that K3 differs from Kd3 and LP differs from LPd in their consequence relations.Source: §2 · p. 6
3 Proof theory for Kd 3 and LPd
The section presents bisequent calculi for Kd3 and LPd, defining their proof structures and logical rules. It establishes key metatheoretic properties, including soundness and completeness, admissibility results, and height-preserving invertibility.
3 Proof theory for Kd 3 and LPd
- A bisequent consists of two sequents, and a bisequent is axiomatic when one of three specified formula intersections is nonempty.The specified intersections are Γ ∩Σ, Γ ∩∆, and Π ∩Σ.Source: §3 · p. 7
- The calculi have the subformula property, and their rules are explicit, separated, symmetric, context-independent, and permutable.Context-independence means the rules impose no side conditions.Source: §3 · p. 7
- For finite Γ and formula A, Kd3 entailment holds exactly when the corresponding bisequent is derivable.The correspondence is Γ |=Kd3 A iff ⊢Kd3 Γ ⇒A | ⇒.Source: §3 · p. 8
- The calculus admits both displayed cut rules.The rules cut a shared formula across either component of the bisequent.Source: §3 · p. 8
- Weakening and contraction are height-preserving admissible, and every premise of a logical rule is derivable at no greater height than its conclusion.The latter property is height-preserving invertibility.Source: §3 · p. 8
4 Conclusion
The conclusion characterizes Kd3 and LPd as dependence-like three-valued logics built from classical conjunction and disjunction with weak Kleene–Bochvar negation and implication. It reports sound, complete, cut-free bisequent calculi while distinguishing these logics from Epstein’s dependence logics.
4 Conclusion
- Kd3 and LPd combine K3 and LP conjunction and disjunction with weak Kleene–Bochvar implication and negation.The paper presents variable-inclusion conditions as a variant of Epstein-style subject-matter restrictions.Source: §4 · p. 8
- The authors call the logics dependence-like because they are not dependence logics in Epstein’s original sense.They provide sound, complete, and cut-free bisequent calculi for both using the uniform framework in.Source: §4 · p. 8
- Extending Kd3 and LPd with quantifiers and definite descriptions is identified as a possible direction for further research.Source: §4 · p. 8