Beth's definability theorem in relevant logics

Journal of Logic and Computation, 2009

This article develops a comprehensive study of various types of interpolation properties and Beth definability properties (BDPs) for substructural logics, and their algebraic characterizations through amalgamation properties (APs) and epimorphisms surjectivity. In general, substructural logics are algebraizable but lack many of the basic logical properties that modal and superintuitionistic logics enjoy [Gabbay and Maksimova (2005, Oxford Logic Guides, Vol. 46)]. In this case, careful examination is necessary to see how these logical and algebraic properties are related. To describe these relations exactly, many variants of interpolation properties and BDPs, and also corresponding algebraic properties, are introduced. Because of their generality, the results reported here hold not only for substructural logics, but can also be extended to a more general setting such as abstract algebraic logic [

Lecture Notes in Computer Science, 2019

We investigate interpolation and Beth definability in default logics. To this end, we start by defining a general framework which is sufficiently abstract to encompass most of the usual definitions of a default logic. In this framework a default logic DL is built on a base, monotonic, logic L. We then investigate the question of when interpolation and Beth definability results transfer from L to DL. This investigation needs suitable notions of interpolation and Beth definability for default logics. We show both positive and negative general results: depending on how DL is defined and of the kind of interpolation/Beth definability involved, the property might or might not transfer from L to DL.

Studies in Universal Logic, 2015

We prove that n-variable logics do not have the weak Beth definability property, for all n ≥ 3. This was known for n = 3 (Ildikó Sain and András Simon [19]), and for n ≥ 5 (Ian Hodkinson, ). Neither of the previous proofs works for n = 4. In this paper we settle the case of n = 4, and we give a uniform, simpler proof for all n ≥ 3. The case for n = 2 is still open.

1997

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a | M |= ϕ(a)} defined by some formula ϕ) is invariant under all automorphisms of M. The same is of course true for subsets of M n defined by formulas with n free variables. Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M , in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L. Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models. The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models. * Both authors acknowledge support from the Netherlands Science Organisation (NWO).

Israel Journal of Mathematics, 1973

We prove that the Beth definability theorem fails for a comprehensive class of first-order logics with cardinality quantifiers. In particular, we give a counterexample to Beth's theorem for L(Q), which is finitary first-order logic (with identity) augmented with the quantifier "there exists uncountably many". 0. Introduction The Beth definability theorem is a basic theorem about L-finitary predicate calculus with identity. It asserts the natural closure condition on a logic: that implicit definitions made in the logic can be replaced by explicit ones. For which natural logics extending L that are currently under investigation, does Beth's theorem hold? Barwise [1] shows that Beth's theorem holds for the first-order logic based on any admissible subset of HC. (Actually the Craig interpolation theorem is proved; any logic obeying Craig's theorem also obeys Beth's.) Gregory [3], using results of Morley, proves that Beth's theorem fails for any logic betweenSe~,2o ~ and 5coco ,. Malitz [6] proves that Beth's theorem fails for any logic between ~,i~,, and ~o oo. This paper is devoted to counterexamples for Beth's theorem in first order logics based on cardinality quantifiers.For each ordinal a, let L(Q~) be finitary first order logic with id mtity and the additional quantifier (Q~x) with the interpretation "there are at least r many". Let L(Q) be finitary first order logic with identity t This research was partially supported by NSF GP29254.

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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.

Bulletin of the Section of Logic

In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi$ formulated in the intersectionof the vocabularies of \(\phi$ and \(\psi\), such that \(\mathfrak{M}\models \phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We discuss examples and show that while the \(n\)-variable fragment of first order logic and difference logic have no Craig interpolation, they both have the modelwise interpolation property. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the...

In this article we study interpolation properties for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the -modality easily follow from this result. Furthermore, these properties are extended to the system ILP. Failure of arrow interpolation for ILW is established by providing an explicit counterexample. The related issues of Beth de nability and xed points are also addressed. It will be shown that for a general class of logics the Beth property and the xed point property are interderivable. This in particular yields alternative proofs for the xed point theorem for IL (cf. de Jongh and Visser 1991) and the Beth theorem for all provability logics (cf. Maksimova 1989). Moreover, it entails that all extensions of IL have the Beth property.