A Class of Fuzzy Modal Propositional Logic Systems with Three Kinds of Negation

2016 ◽  
pp. 101-111
Author(s):  
Cheng Chen ◽  
Li-Juan Zhang ◽  
Zheng-Hua Pan
Keyword(s):  
Propositional Logic ◽  
Logic Systems ◽  
Modal Propositional Logic

UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS

2014 ◽  
Vol 7 (3) ◽  
pp. 455-483 ◽  
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.

scholarly journals Relative Necessity and Propositional Quantification

2019 ◽  
Vol 49 (4) ◽  
pp. 703-726
Author(s):  
Alexander Roberts
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
Recent Article ◽  
Symbolic Logic ◽  
Philosophical Logic ◽  
Alternative Account ◽  
Standard Account ◽  
Current Article ◽  
Modal Propositional Logic ◽  
Propositional Quantification

AbstractFollowing Smiley’s (The Journal of Symbolic Logic, 28, 113–134 1963) influential proposal, it has become standard practice to characterise notions of relative necessity in terms of simple strict conditionals. However, Humberstone (Reports on Mathematical Logic, 13, 33–42 1981) and others have highlighted various flaws with Smiley’s now standard account of relative necessity. In their recent article, Hale and Leech (Journal of Philosophical Logic, 46, 1–26 2017) propose a novel account of relative necessity designed to overcome the problems facing the standard account. Nevertheless, the current article argues that Hale & Leech’s account suffers from its own defects, some of which Hale & Leech are aware of but underplay. To supplement this criticism, the article offers an alternative account of relative necessity which overcomes these defects. This alternative account is developed in a quantified modal propositional logic and is shown model-theoretically to meet several desiderata of an account of relative necessity.

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