A content semantics for quantified relevant logics. II
type: Item
instance of: scholarly article
title: A content semantics for quantified relevant logics. II
zbMATH Open document ID: 0696.03003
DOI: 10.1007/BF02770515
author: Ross T. Brady
published in: Studia Logica
publication date: 1989-01-01T00:00:00Z
review text: This is the sequel to Part I [ibid. 47, 111-127 (1988; Zbl 0666.03017)]. As promised there, additional postulates are given here to specify the content semantics (an algebraic style semantics) of almost any first- order extension of the system BBQ which one might care to consider, e.g., TWQ, RWQ, TQ, RQ and KQ (classical first-order logic). (See the author [Logique Anal., Nouv. Sér. 27, 355-377 (1984; Zbl 0559.03011)] for a larger list of suitable extensions.) EWQ and EQ are the only significant systems notably absent. Generalised consistency and completeness proofs are given to cover all of the systems included. The author then goes on to produce ``reduced'' content semantics. (See \textit{R. Routley}, \textit{R. K. Meyer}, \textit{V. Plumwood} and the author [Relevant logics and their rivals (1982; Zbl 0579.03011)] for the original notion of reduced modelling and its motivation.) The point here is to give models certain plausible properties which they lack otherwise, to wit, primeness and saturation properties for the truth filter: If \(A\vee B\) is an element of the truth filter, then so is either A or B ; and if (\(\exists x)A\) is an element of the truth filter, then so is the result of substituting some object of the domain for x in A. To this end the original system BBQ and its extensions are modified by adding the ``meta-rules'': MR1. If \(A\Rightarrow B\), then \(C\vee A\Rightarrow C\vee B,\) MR2. If \(A\Rightarrow B\), then (\(\exists x)A\Rightarrow (\exists x)B,\) yielding the system \(BB^ dQ\) and its extensions. The model structures for these systems are just like the originals except for the addition of these two postulates: (p6c) The union of c and d is in T only if either c is or d is. (p11b) The generalised union of \(\{F^ ns^ b/k(i_ 1,...,i_ n):\) b is in \(D\}\) is in T only if \(F^ ns^ b/k(i_ 1,...,i_ n)\) is in T for some b in D. Soundness and completeness are then proved as required.
reviewed by: Steve Giambrone
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03C90
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 4139710
zbMATH Keywords: quantified relevant logics
zbMATH Keywords: algebraic semantics
zbMATH Keywords: reduced models
zbMATH Keywords: first- order extensions of BBQ
zbMATH Keywords: content semantics
MaRDI profile type: Publication
cites work: Q3347856
cites work: On 2nd order intuitionistic propositional calculus with full comprehension
cites work: Q3701398
full work available at URL: https://doi.org/10.1007/bf02770515
OpenAlex ID: W2022800703
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description: scientific article; zbMATH DE number 4139710