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 intersection of 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 compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation (...) property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. 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 logic. (shrink)

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

In this article we discuss a version of the Robinson property studied recently by Gyenis in, and we present a solution to one of his open problems. We say that a first‐order structure satisfies the Robinson property whenever the union of two non‐trivial partial n‐types over different finite sets is realizable if and only if they are not explicitly contradictory. In his article, Gyenis showed that a universal, homogeneous structure over a language that consists of at most binary relation symbols (...) satisfies the Robinson property if and only if its age has a generalized amalgamation property (the so called prescribed amalgamation property, PAP). We go further and define a series of new kinds of amalgamation properties, APn for any natural number n. Using these properties we can characterize all universal and homogeneous structures that satisfy the Robinson property independently from the arities of relation symbols of the language. (shrink)

We show that for infinite ordinals α the class of polyadic algebras of dimension α has the super amalgamation property.

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if is the free algebra in the given signature,, is in the subalgebra of generated by, is in the subalgebra of generated by and, then there exists an interpolant in the subalgebra generated by and such that. We call this a varying interpolation property because (...) the integer depends on the inequality. We also address cases where this interpolation property fails, but other weaker ones hold. One such interpolation theorem says that though an interpolant may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to making it smaller and the same number of existential quantifiers are applied to making it bigger. This number of quantifiers also varies; it depends on the inequality. Se.. (shrink)

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new (...) unpublished results concerning neat reducts and amalgamation are given. . Several counterexamples which convey the gist of techniques used in this area are presented two of which are new It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [31] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. This paper appears in two parts. The first part contains results on neat reducts. The present second part contains results relating the notion of neat embeddings to various amalgamation properties.1. (shrink)

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new (...) unpublished results concerning neat reducts and amalgamation are given. . Several counterexamples which convey the gist of techniques used in this area are presented two of which are new . It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [54] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. This paper appears in two parts. The present first part contains results on neat reducts. The second part contains results relating the notion of neat embeddings to various amalgamation properties.1. (shrink)