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...

Abstract This is an exploratory document for a new research line in logical semantics which is emerging from several current developments in computer science. Standard logic employs 'flat' unstructured sets of statements for its theories and unstructured classes of models for its semantic universes. Nowadays, however, there is an incipient literature on structured universes of models as well as structured theories, both employing 'preference relations' of some sort. The purpose of this brief report is (1) to propose a more systematic framework for this trend, while also connecting it up with some historical predecessors, (2) to design some new logical systems bringing preferences out explicitly, thereby high-lighting the theoretical properties of 'preferential reasoning' while raising some new kinds of technical question for further research, and (3) to link up with some current computational ideas (emerging also in the field of linguistics), by bringing in the general dynamic logic o...

Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper considers, in a light example-based manner, where this elegant logical paradigm stands when confronted with the wear and tear of reality. Starting from a brief history of the Lambek tradition since the 1980s, we discuss three main issues: (a) the fit of the lambda calculus engine to characteristic semantic structures in natural language, (b) the coexistence of the original type-theoretic and more recent modal interpretations of categorial logics, and (c) the place of categorial grammars in the complex total architecture of natural language, which involves - amongst others - mixtures of interpretation and inference.

George Gargov was an active pioneer in the 'Sofia School' of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that 'modal logic' as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet.

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Categorial grammars are driven by substructural logics. These are fragments of modal logics for the structures the grammar deals with. We will discuss modal language as a means of access to families of relevant structures: formal languages, type hierarchies, relation algebras--arrow models, and vector spaces. This is the cross-roads of our title, where open directions abound.