Model Theory. That is, we have a non-empty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard Back-and-Forth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. (A complete argument is in [16].) An outline is reproduced here, for convenience. First, k-variable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism I 2 I which is defined on the A-values for all variables x 1 ; : : : ; x k , that M;A j= OE iff N; I ffi A j= OE: The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be restricted to size at most k \Gamma 1, whence a matching choice for the witness can be made on the opposite side. This proves "only if". Next, "if" has a proof analogous to...

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone’s inaccessibility logic. Viewing first order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases. In 1957, W. Craig proved the interpolation theorem for first order logic [Cra57]. Comer [Com69] showed that the property fails for all finite variable fragments except the onevariable fragment. The n-variable fragment of first order logic –for short Ln – contains all first order formulas using just n variables and containing only predicate symbols of arity not higher that n (we assume the language has only variables as terms.) Here we will show that the axiom which makes the quantifiers commute can be seen as the reason for this failure. Since Craig’s paper, interpolation has become one of the standard properties that one investigates when designing a logic, though it hasn’t received the status of a completeness or a decidability theorem. One of the main reasons why a logic should have interpolation is because of “modular theory building”. As we will see below interpolation in modal logic is equivalent to the following property (which is the semantical version of Robinson’s consistency lemma.)

Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...

The guarded fragment (GF) was introduced in [1] as a fragment of first order logic which combines a great expressive power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. While GF has been established as a particularly well-behaved fragment of first order logic in many respects, interpolation fails in restriction to GF, [9]. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. The Beth property for GF is here established on the basis of a limited form of interpolation, which more closely resembles the interpolation property that is usually studied in modal logics. ¿From this we obtain that, more specifically, even every n-variable guarded fragment with up to nary

We give an algebraic version of rst order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 (a concrete version of which is given by Theorems 2.8 and 4.2), while its logical form is Corollary 5.2. For rst order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case | instead of nitely axiomatizing the corresponding class of all representable algebras | we nitely axiomatize only the equational theory of that class. See Subsection 6.1, especially Remark 6.6 there. The proof of Theorem 1.1 is elaborated in Sections 3 and 4. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 4.2 is a purely semigroup theoretic result. Cf. also \Further main results" in the

Connections between Algebraic Logic and (ordinary) Logic. Algebraic counterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections.

Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RRA: for instance, one obtains finite axiomatizability, decidability and amalgamation by relativization. The properties of the class obtained by relativizing RRA depend on the kind of element with which is relativized. We give a systematic account of all interesting choices of relativizing RRA, and show that relativizing with transitive elements forms the borderline where all above mentioned three properties switch from negative to positive. In algebraic logic, relativized cylindric and relation algebras have been studied relatively deeply (cf. e.g., Henkin et al. (Henkin et al., 1981), Maddux (Maddux, 1982) and Resek-- Thompson (Resek and Thompson, 1991)). The emphasis, however was different from the perspective we will take here. As the name "relativization" indicates, the non--...

The language of relation algebras is expanded with variables denoting individual elements in the domain and with the ↓ binder from hybrid logic. Every elementary property of binary relations is expressible in the resulting language, something which fails for the relation algebraic language. That the new language is natural for speaking about binary relations is indicated by the fact that both Craig’s Interpolation, and Beth’s Definability theorems hold for its set of validities. The paper contains a number of worked out examples.

In the present paper we will show that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. Furthermore, we will give an equally general characterization of the weak Beth definability property, hereby giving a solution to Problem 14 in [Sa 88]. Finally, we will present two counterexamples showing that two of the more striking assumptions imposed on the logics in our characterizations are necessary. Supported by a grant from the NWO. 1 Contents 1 Introduction 2 2 Preliminaries 6 2.1 Definitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Preliminary Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 3 An Algebraic Characterization of the (strong) Beth Definability Property 16 4 An Algebraic Characterization of the Weak Beth Definability Property 23 5 Discussion 28 A Appendix 36 1 Introduction Abstract Algebraic Logic The aim of this paper is to give an algebraic chara...

In this paper it will be shown that the Beth definability property corresponds to surjectiveness of epimorphisms in abstract algebraic logic. This generalizes a result by I. N'emeti (cf. [HMT85, Theorem 5.6.10]). Moreover, an equally general characterization of the weak Beth property will be given. This gives a solution to Problem 14 in [Sai90]. Finally, the characterization of the projective Beth property for varieties of modal algebras by L. Maksimova (see [Mak97]) will be shown to hold for the larger class of semantically algebraizable logics. 1 Introduction Abstract algebraic logic is typically concerned with equivalence theorems of the following kind, Logic L has property P () Alg(L) has property alg(P ), where Alg(L) denotes the class of algebras associated to the logic L in some canonical way, on which we come to speak later. The main motivation for making these efforts is that it shows us that algebra and logic are, so to speak, nothing but two sides of the same coin; they stu...