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.

THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS

2018 ◽  
Vol 13 (1) ◽  
pp. 141-205 ◽  
Author(s):  
MARKO MALINK ◽  
ANUBAV VASUDEVAN
Keyword(s):  
Propositional Logic ◽  
17Th Century ◽  
Gottfried Wilhelm Leibniz ◽  
Natural System ◽  
Relevance Logic ◽  
Classical Propositional Logic ◽  
Categorical Logic ◽  
Reductio Ad Absurdum ◽  
Image Position ◽  
Hypothetical Syllogism

AbstractGreek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$.

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.

scholarly journals POSITIVE FRAGMENTS OF RELEVANCE LOGIC AND ALGEBRAS OF BINARY RELATIONS

2010 ◽  
Vol 4 (1) ◽  
pp. 81-105 ◽  
Author(s):  
ROBIN HIRSCH ◽  
SZABOLCS MIKULÁS
Keyword(s):  
Equational Theory ◽  
Relevance Logic ◽  
Binary Relations ◽  
Finitely Based ◽  
Similarity Type ◽  
Relation Composition

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to 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.

Sign in / Sign up

Export Citation Format

Share Document