Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic

Studia Logica ◽  
2013 ◽  
Vol 102 (3) ◽  
pp. 479-497 ◽  
Author(s):  
Michael Kaminski ◽  
Nissim Francez
Keyword(s):  
Propositional Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Classical Propositional Logic

scholarly journals THE UBIQUITY OF CONSERVATIVE TRANSLATIONS

2012 ◽  
Vol 5 (4) ◽  
pp. 666-678 ◽  
Author(s):  
EMIL JEŘÁBEK
Keyword(s):  
Propositional Logic ◽  
Broad Class ◽  
Paraconsistent Logic ◽  
Lambek Calculus ◽  
Classical Propositional Logic ◽  
Consequence Relation ◽  
Deductive Systems ◽  
Nonclassical Logics ◽  
Conservative Translations

AbstractWe study the notion of conservative translation between logics introduced by (Feitosa & D’Ottaviano2001). We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.

scholarly journals Deduction in Non-Fregean Propositional Logic SCI

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 115 ◽  
Author(s):  
Joanna Golińska-Pilarek ◽  
Magdalena Welle
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Classical Propositional Logic ◽  
Tableau System ◽  
Soundness And Completeness

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS

2010 ◽  
Vol 3 (1) ◽  
pp. 41-70 ◽  
Author(s):  
ROGER D. MADDUX
Keyword(s):  
Propositional Logic ◽  
Relevance Logic ◽  
Binary Relations ◽  
Classical Propositional Logic

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

scholarly journals The Method of Socratic Proofs Meets Correspondence Analysis

2019 ◽  
Vol 48 (2) ◽  
pp. 99-116
Author(s):  
Dorota Leszczyńska-Jasion ◽  
Yaroslav Petrukhin ◽  
Vasilyi Shangin
Keyword(s):  
Correspondence Analysis ◽  
Boolean Functions ◽  
Propositional Logic ◽  
Sequent Calculus ◽  
Sequent Calculi ◽  
Classical Propositional Logic ◽  
Logic Of Questions ◽  
Proof Systems

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic (i.e. pertaining to the logic of questions) calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for the method of Socratic proofs. Correspondence analysis is Kooi and Tamminga's technique for designing proof systems. In this paper it is used to consider sequent calculi with non-branching (the only exception being the rule of cut), invertible rules for the negation fragment of classical propositional logic and its extensions by binary Boolean functions.

Update Semantics in Communicated Information System

2011 ◽  
Vol 403-408 ◽  
pp. 1460-1465
Author(s):  
Guang Ming Chen ◽  
Xiao Wu Li
Keyword(s):  
Information Systems ◽  
Propositional Logic ◽  
General Purpose ◽  
Main Task ◽  
Information Communication ◽  
Classical Propositional Logic ◽  
Update Semantics ◽  
Essential Form ◽  
Soundness And Completeness ◽  
Multi Agents

An approach, which is called Communicated Information Systems, is introduced to describe the information available in a number of agents and specify the information communication among the agents. The systems are extensions of classical propositional logic in multi-agents context, providing with us a way by which not only the agent’s own information, but the information from other agents may be applied to agent’s reasoning as well. Communication rules, which are defined in the most essential form, can be regarded as the base to characterize some interesting cognitive proporties of agents. Since the corresponding communication rules can be chosen for different applications, the approach is general purpose one. The other main task is that the soundness and completeness of the Communicated Information Systems for the update semantics have been proved in the paper.

Relational semantics and a relational proof system for full Lambek calculus

1998 ◽  
Vol 63 (2) ◽  
pp. 623-637 ◽  
Author(s):  
Wendy MacCaull
Keyword(s):  
Proof Theory ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Complete Proof ◽  
Proof Search ◽  
Proof Tree ◽  
Relational Logic ◽  
Completeness Theorems

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

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