Logicality and Meaning

The Bulletin of Symbolic Logic

We ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in $L_{\infty \infty }$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty \infty }$ by a “tamer” logic.

Journal of Philosophical Logic, 2014

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed.

Notre Dame Journal of …, 1992

Quantificational accounts of logical truth and logical consequence aim to reduce these modal concepts to the nonmodal one of generality. A logical truth, for example, is said to be an instance of a "maximally general" statement, a statement whose terms other than variables are "logical constants." These accounts used to be the objects of severe criticism by philosophers like Ramsey and Wittgenstein. In recent work, Etchemendy has claimed that the currently standard model-theoretic account of the logical properties is a quantificational account and that it fails for reasons similar to the ones provided by Ramsey and Wittgenstein. He claims that it would fail even if it were propped up by a sensible account of what makes a term a logical constant. In this paper I examine to what extent the model-theoretic account is a quantificational one, and I defend it against Etchemendy's criticisms.

1985

model theory is the attempt to systematize the study of logics by studying the relationships between them and between various of their properties. The perspective taken in abstract model theory is discussed in Section 2 of Chapter I. The basic definitions and results of the subject were presented in Part A. Other results are scattered throughout the book. This final part of the book is devoted to more advanced topics in abstract model theory.

Several philosophers of science construe models of scientic theories as set-theoretic structures. Some of them moreover claim that models should not be construed as structures in the sense of model theory because the latter are languagedependent. I argue that if we are ready to construe models as set-theoretic structures (strict semantic view), we could equally well construe them as model-theoretic structures of higher-order logic (liberal semantic view). I show that every family of set-theoretic structures has an associated language of higher-order logic and an up to signature isomorphism unique model-theoretic counterpart, which is able to serve the same purposes. This allows to carry over every syntactic criterion of equivalence for theories in the sense of the liberal semantic view to theories in the sense of the strict semantic view. Taken together, these results suggest that the recent dispute about the semantic view and its relation to the syntactic view can be resolved.

Synthese, 2017

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension-it is a logic of cardinality (or, more accurately, of "isomorphism type"). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less "intensional" it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things "in the world". Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus's principles of explicit and implicit extensionality, we are able to characterize purely logical languages as "sub-extensional", namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.

It is shown that the classes of Routley-Meyer models which are axiomatizable by a theory in a propositional relevant language with fusion and the Ackermann constant can be characterized by their closure under certain model-theoretic operations involving prime filter extensions, relevant directed bisimulations and disjoint unions.

Eprint Arxiv Cs 0403002, 2004

Stable model semantics has become a very popular approach for the management of negation in logic programming. This approach relies mainly on the closed world assumption to complete the available knowledge and its formulation has its basis in the so-called Gelfond-Lifschitz transformation. The primary goal of this work is to present an alternative and epistemic-based characterization of stable model semantics, to the Gelfond-Lifschitz transformation. In particular, we show that stable model semantics can be defined entirely as an extension of the Kripke-Kleene semantics. Indeed, we show that the closed world assumption can be seen as an additional source of `falsehood' to be added cumulatively to the Kripke-Kleene semantics. Our approach is purely algebraic and can abstract from the particular formalism of choice as it is based on monotone operators (under the knowledge order) over bilattices only.

Synthese, 2017

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality (or, more accurately, of “isomorphism type”). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1985