Meaning and proscription in formal logic. Variations on the propositional logic of William T. Parry
type: Item
instance of: scholarly article
title: Meaning and proscription in formal logic. Variations on the propositional logic of William T. Parry
zbMATH Open document ID: 1441.03002
DOI: 10.1007/978-3-319-70821-8
author: Thomas Macaulay Ferguson
published in: Trends in Logic -- Studia Logica Library
publication date: 2017-12-06T00:00:00Z
review text: This book presents an in-depth inquiry into a wealth of topics connected to Parry's logic of analytic implication, and more broadly, to a family of systems that can generally be characterized as ``containment logics''. In the focus of consideration is the so-called ``proscriptive principle'', saying the ``no formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent'' (p.\,4). Ferguson observes that on a propositional level this principle can be implemented in two different ways, either with respect to an object level implication or a consequence relation. Let \(\mathbf{At}(A) = \{p_0, p_1, \ldots\}\) be a denumerable set of atomic formulae. Then a logical system \(\mathsf{L}\) is \(\rightarrow\)-Parry if it enjoys the property that \(\vdash_\mathsf{L} A \rightarrow B\) only if \(\mathbf{At}(B) \subseteq \mathbf{At}(A)\), and a it is \(\vdash\)-Parry if it enjoys the property that \( \varGamma\vdash_\mathsf{L} A\) only if \(\mathbf{At}(A) \subseteq \textbf{At}[\varGamma]\). The book covers a broad range of topics related either to the \(\rightarrow\)-Parry or \(\vdash\)-Parry systems. Except of the introductory and concluding chapters, there are six main chapters, which are roughly divided into two parts. Chapters 2, 3 and 4 constitute the first part, which is devoted to the proscriptive principle itself and to its comprehensive consideration. Namely, Chapter 2 explicates certain relationships between the logics of nonsense of Dmitri Bochvar and Sören Halldén and the containment logics in the neighborhood of Parry's analytic implication. In particular, Friederick Johnson's containment logic \(\mathsf{RC}\) is considered, as well as a four-valued first-degree logic \(\mathsf{S}_{\mathrm{fde}}\) introduced independently by Harry Deutsch and Carlos Oller. It is shown that \(\mathsf{S}_{\mathrm{fde}}\) and \(\mathsf{RC}\) can be interpreted as the Parry-type subsystems of Bochvar's no…
reviewed by: Yaroslav Shramko
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03B47
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B53
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03B60
Mathematics Subject Classification ID: 03A05
zbMATH DE Number: 6816296
zbMATH Keywords: Analytic implication
zbMATH Keywords: containment logic
zbMATH Keywords: proscription principle
zbMATH Keywords: first-degree entailment
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-70821-8
OpenAlex ID: W2780986050
description: scientific article; zbMATH DE number 6816296