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Back and Forth Between Modal Logic and Classical Logic
, 1994
"... Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. ..."
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Cited by 34 (3 self)
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Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth 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, kvariable 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 Avalues 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...
Changing Preferences
, 1995
"... 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 ..."
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Cited by 8 (6 self)
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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 highlighting 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...
The Categorial FineStructure of Natural Language
, 2003
"... 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, i ..."
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Cited by 3 (1 self)
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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 examplebased 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 typetheoretic 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.
The Range Of Modal Logic  an essay in the memory of George Gargov
 TO APPEAR IN D. VAKARELOV, ED., MEMORIAL ISSUE FOR GEORGE GARGOV, "JOURNAL OF APPLIED NONCLASSICAL LOGICS"
, 1997
"... 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 ..."
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Cited by 2 (0 self)
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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.
Categorial Grammar at a CrossRoads
, 2003
"... 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 algebrasarrow models, and vect ..."
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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 algebrasarrow models, and vector spaces. This is the crossroads of our title, where open directions abound.
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF PREDICATE LOGIC 1. MODAL LOGIC AND CLASSICAL LOGIC
"... Modal Logic is traditionally concerned with the intensional operators “possibly ” and “necessary”, whose intuitive correspondence with the standard quantifiers “there exists ” and “for all ” comes out clearly in the usual Kripke semantics. This observation underlies the wellknown translation from p ..."
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Modal Logic is traditionally concerned with the intensional operators “possibly ” and “necessary”, whose intuitive correspondence with the standard quantifiers “there exists ” and “for all ” comes out clearly in the usual Kripke semantics. This observation underlies the wellknown translation from propositional modal logic with operators ♦ and ¤, possibly indexed, into the firstorder language over possible worlds models (van Benthem 1976, 1984). In this way, modal formalisms correspond to fragments of a full firstorder (or sometimes higherorder) language over these models, which are both expressively perspicuous and deductively tractable. In this paper, by the “modal fragment ” of predicate logic we understand the set of all firstorder formulas obtainable as translations of basic (poly)modal formulas. As the modal fragment is merely a notational variant of the basic modal language, we will often refer to the two interchangeably. Basic modal logic shares several nice properties with full predicate logic, namely, finite axiomatizability, Craig Interpolation and Beth Definability, as well as modeltheoretic preservation results such as the Loś–Tarski Theorem characterizing those formulas that are preserved under taking submodels. In addition, basic modal logic has some nice properties not shared with predicate logic as a whole: e.g., its axiomatization does not need side conditions on free or bound variables – and most evidently: basic modal logic is decidable. We shall concentrate on this list in what follows, in the hope that it forms a representative sample. Our aim is to find natural fragments of predicate logic extending the modal one which inherit the abovementioned nice properties. This quest has two virtues. It forces us to understand why basic modal logic has these nice properties. And it points the way to new insights concerning predicate logic itself. Note that this study takes place over the universe of all models, without special restrictions on accessibility relations. This is the domain of the “minimal modal logic”, which
unknown title
, 2003
"... Abstract 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, ..."
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Abstract 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 crossroads of our title, where open directions abound. 1 From categorial proof theory to categorial model theory Categorial grammars are driven by resource logics in a proof format (van Benthem 1991, Buszkowski 1997, Moortgat 1997). Thus, they revolve around derivation and computation, with the CurryHoward Gestalt switch taking proofs to typetheoretic denotations for the expression analyzed. But over the past decades, categorial logics have also been analyzed modeltheoretically in modal logics with standard possible worldsstyle models (cf. Kurtonina 1995). Thus, e.g., a categorial product A•B is ‘true ’ of some object s iff s is a concatenation, or some suitable semantic merge of two objects t, u satisfying A, B, respectively. This is a standard binary modality, which needs a ternary accessibility relation R for its abstract truth condition:
unknown title
"... Abstract 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 con ..."
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Abstract 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 examplebased 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 typetheoretic 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. 1 From Montague Grammar to Categorial Grammar Logic and linguistics have had lively connections from Antiquity right until today (GAMUT 1991). A recurrent theme in this history is the categorial structure of language and ontology, from Aristotle's grammatical categories to Russell's theory of types in the foundations of mathematics. Further bridges were thrown as logic and