Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

AbstractWe show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm, but a proper subset of the set of theorems of RMIm. Hence it still has the variable-sharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm, in the second - the system SRm). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.

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Inter-Model Connectives and Substructural Logics

Recent Trends in Philosophical Logic - Trends in Logic ◽

10.1007/978-3-319-06080-4_14 ◽

2014 ◽

pp. 195-209

Author(s):

Igor Sedlár

Keyword(s):

Substructural Logics

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UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS

The Review of Symbolic Logic ◽

10.1017/s175502031400015x ◽

2014 ◽

Vol 7 (3) ◽

pp. 455-483 ◽

Cited By ~ 2

Author(s):

MAJID ALIZADEH ◽

FARZANEH DERAKHSHAN ◽

HIROAKIRA ONO

Keyword(s):

Propositional Logic ◽

Predicate Logic ◽

Interpolation Property ◽

Substructural Logics ◽

Substructural Logic ◽

Sequent Calculi ◽

Structural Rule ◽

Modal Propositional Logic ◽

Uniform Interpolation ◽

Classical Predicate

AbstractUniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4, which have Craig’s interpolation property but do not have uniform interpolation property. The situation is even worse for predicate logics, as classical predicate logic does not have uniform interpolation property as pointed out by L. Henkin.In this paper, uniform interpolation property of basic substructural logics is studied by applying the proof-theoretic method introduced by A. Pitts (Pitts, 1992). It is shown that uniform interpolation property holds even for their predicate extensions, as long as they can be formalized by sequent calculi without contraction rules. For instance, uniform interpolation property of full Lambek predicate calculus, i.e., the substructural logic without any structural rule, and of both linear and affine predicate logics without exponentials are proved.

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SUBSTRUCTURAL INQUISITIVE LOGICS

The Review of Symbolic Logic ◽

10.1017/s1755020319000017 ◽

2019 ◽

Vol 12 (2) ◽

pp. 296-330 ◽

Cited By ~ 2

Author(s):

VÍT PUNČOCHÁŘ

Keyword(s):

Fuzzy Logic ◽

General Theory ◽

Propositional Logic ◽

Substructural Logics ◽

Substructural Logic ◽

Fuzzy Logics ◽

Inquisitive Semantics ◽

Semantic Framework ◽

Noncommutative Version ◽

Image Position

AbstractThis paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

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