Ternary relations and relevant semantics
type: Item
instance of: scholarly article
title: Ternary relations and relevant semantics
zbMATH Open document ID: 1060.03039
author: Robert K. Meyer
published in: Annals of Pure and Applied Logic
publication date: 2004-08-06T00:00:00Z
review text: This is a largely expository paper on the family of relevant logics. Meyer gives some history and also shows that all these relevant logics can be motivated by the properties of a ternary relation \(R\), where \(RXYZ\) is \(X\circ Y \subseteq Z\) and \[ X\circ Y=\{ B: \exists A(A\to B\in X \text{ and }A\in Y)\}. \] The ``worlds'' \(X, Y\ldots \) over which such binary and ternary relations usually range in semantics, are here ``demythologized'' as theories. Theories are sets of formulas subject to certain truth conditions and may represent (relevant) logics. These theories can be seen as the unions of the types of combinators. In the case of some theories the simple types of some combinators are sufficient. In these cases the theory is in fact a model of the corresponding combinatory logic, where application is interpreted as fusion \((o)\). Meyer's ``Key to the Universe'' is the fact that \(B\wedge T\), a theory based on intersection types, models SK (i.e. full) combinatory logic. The following illustrates the key to the universe syntactically: We list two combinators, their ``fusion fact'' (i.e. their reduction rule) and their ``implication fact'' (i.e. their type): \[ \begin{alignedat}{3} &S &\qquad ABC&\leq AC(BC) &\qquad A\to (B\to C)&\leq (A\to B)\to (A\to C)\\ W&B &\qquad AB&\leq A(AB) &\qquad (B\to C)&\wedge (A\to B) \leq A\to C. \end{alignedat} \] If \(L\) is any logic extending \(B[\to, \wedge , 0]\), then a fusion fact is a theorem scheme of \(L\) iff the corresponding implication fact is a theorem scheme.
reviewed by: Martin W. Bunder
Mathematics Subject Classification ID: 03B47
Mathematics Subject Classification ID: 03B40
Mathematics Subject Classification ID: 08A02
Mathematics Subject Classification ID: 03-02
zbMATH DE Number: 2083198
zbMATH Keywords: ternary relation
zbMATH Keywords: bunch
zbMATH Keywords: fusion
zbMATH Keywords: combinators
zbMATH Keywords: combinatory logic
zbMATH Keywords: key2u theorem
zbMATH Keywords: relevant entailment
zbMATH Keywords: relevant logic
MaRDI profile type: Publication
full work available at URL: j.apal.2003.11.015
OpenAlex ID: W1996113466
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DOI: 10.1016/J.APAL.2003.11.015
description: scientific article; zbMATH DE number 2083198