Meaning and proscription in formal logic. Variations on the propositional logic of William T. Parry
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 nonsense logic. In the next two chapters, the author elucidates further relationships between the notions of nonsense and containment. In Chapter 3, Kit Fine's analysis of Richard Angell's containment logic \(\mathsf{AC}\) is represented and, in Chapter 4, some containment logics are considered by developing a computational interpretation of conceptivist logics. All in all, this part of the book is largelly of a foundational and substantive character. The second part of the book, ``emphasizes formal and algebraic analysis the family of first-degree Parry logics intermediate between Parry's \(\mathsf{PAI}_{\mathrm{fde}}\) and Richard B. Angell's \(\mathsf{AC}\), with Chapters 5, 6, and 7 treating this family in the settings of many-valued semantics, Arieli/Avron-style logical bilattices, and Correia's 2004 semantics for \(\mathsf{AC}\)'' (p.\,13). More concretely, Chapter 5 deals with variations of Nuel Belnap's ``useful four-valued logic of how a computer should think''. Starting from the first-degree entailment system \(\mathsf{E}_{\mathrm{fde}}\), the author proceeds to consider Daniels-Priest system \(\mathsf{S}^\star_{\mathrm{fde}}\) and Angell's logic of analytic containment \(\mathsf{AC}\). Both systems are provided with suitable interpretation, which allows to apply a Gödel-Fine analysis to the logic of analytic containment. Furthermore, Chapter 6 extends the so-called bilattice and trilattice (and more generally, multilattice) semantics to both \(\mathsf{S}^\star_{\mathrm{fde}}\) and \(\mathsf{AC}\). It is claimed that an ``elegant connection between these systems and logical multilattices supports the fundamentality and naturalness of these logics and, additionally, allows us to extend epistemic interpretation of bilattices in the tradition of artificial intelligence to these systems'' (p\,133). Chapter 7 describes in detail Fabrice Correia's semantics for \(\mathsf{AC}\) which makes it possible to provide illuminating characterizations of a spectrum of propositional logics intermediate between AC and classical logic, Graham Priest's logic of paradox \(\mathsf{LP}\) among them. The book ends with some concluding remarks concerning the issues of refining the notion of content and calculi related to analytic implication.
scientific article; zbMATH DE number 6816296