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Galois Connections in Categorial Type Logic
, 2001
"... The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and ..."
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Cited by 12 (5 self)
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The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and prooftheoretic groundwork. In x3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quanti er expressions, and to describe a typology of polarity items.
Taming displayed tense logics using nested sequents with deep inference
 In Martin Giese and Arild Waaler, editors, Proceedings of TABLEAUX, volume 5607 of Lecture Notes in Computer Science
, 2009
"... Abstract. We consider two sequent calculi for tense logic in which the syntactic judgements are nested sequents, i.e., a tree of traditional onesided sequents built from multisets of formulae. Our first calculus SKt is a variant of Kashima’s calculus for Kt, which can also be seen as a display calcu ..."
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Cited by 10 (4 self)
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Abstract. We consider two sequent calculi for tense logic in which the syntactic judgements are nested sequents, i.e., a tree of traditional onesided sequents built from multisets of formulae. Our first calculus SKt is a variant of Kashima’s calculus for Kt, which can also be seen as a display calculus, and uses “shallow ” inference whereby inference rules are only applied to the toplevel nodes in the nested structures. The rules of SKt include certain structural rules, called “display postulates”, which are used to bring a node to the top level and thus in effect allow inference rules to be applied to an arbitrary node in a nested sequent. The cut elimination proof for SKt uses a proof substitution technique similar to that used in cut elimination for display logics. We then consider another, more natural, calculus DKt which contains no structural rules (and no display postulates), but which uses deepinference to apply inference rules directly at any node in a nested sequent. This calculus corresponds to Kashima’s S2Kt, but with all structural rules absorbed into logical rules. We show that SKt and DKt are equivalent, that is, any cutfree proof of SKt can be transformed into a cutfree proof of DKt, and vice versa. We consider two extensions of tense logic, Kt.S4 and S5, and show that this equivalence between shallow and deepsequent systems also holds. Since deepsequent systems contain no structural rules, proof search in the calculi is easier than in the shallow calculi. We outline such a procedure for the deepsequent system DKt and its S4 extension. 1
Deep inference in Biintuitionistic logic
 In Int Workshop on Logic, Language, Information and Computation, WoLLIC 2009, LNAI 5514
, 2009
"... Abstract. Biintuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cutelimination in biintuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calcu ..."
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Cited by 5 (1 self)
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Abstract. Biintuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cutelimination in biintuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calculus, have been proposed to address this problem. In this paper, we present a new extended sequent calculus DBiInt for biintuitionistic logic which uses nested sequents and “deep inference”, i.e., inference rules can be applied at any level in the nested sequent. We show that DBiInt can simulate our previous “shallow ” sequent calculus LBiInt. In particular, we show that deep inference can simulate the residuation rules in the displaylike shallow calculus LBiInt. We also consider proof search and give a simple restriction of DBiInt which allows terminating proof search. Thus our work is another step towards addressing the broader problem of proof search in display logic. 1
Scope Ambiguities through the mirror
 John Bejamins Publishing Company. vol
"... In this paper we look at the interpretation of Quantifier Phrases from the perspective of Symmetric Categorial Grammar. We show how the apparent mismatch between the syntactic and semantic behaviour of these expressions can be resolved in a typelogical system equipped with two Merge relations: one f ..."
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In this paper we look at the interpretation of Quantifier Phrases from the perspective of Symmetric Categorial Grammar. We show how the apparent mismatch between the syntactic and semantic behaviour of these expressions can be resolved in a typelogical system equipped with two Merge relations: one for syntactic units, and one for the evaluation contexts of the semantic values associated with these syntactic units. Keywords:
Galois Connections in Categorial Type Logic
"... The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In §2 we do the basic modeltheoretic and ..."
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The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In §2 we do the basic modeltheoretic and prooftheoretic groundwork. In §3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quantifier expressions, and to describe a typology of polarity items. 1
Galois Connections in Categorial Type Logic Carlos Areces a;1;2 Raffaella Bernardi b;3 Michael Moortgat b;4
"... Abstract The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheo ..."
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Abstract The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and prooftheoretic groundwork. In x3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quantifier expressions, and to describe a typology of polarity items. 1 Introduction Categorial type logic provides a vocabulary of logical constants for the assembly of form and meaning in natural language. The binary product operator ffl captures the composition of grammatical parts, the residual implications = and n express incompleteness with respect to the composition relation. In [11,18] the categorial vocabulary has been extended with a pair of unary operators, 3 and its residual 2#. This addition greatly increases the analytical power of the categorial type language. In combination with the binary connectives, the unary operators can be used as licensing features, providing lexically anchored control over structural reasoning. But already in the grammatical base logic NL(3), the unary constants yield refinements of type assignments that would be overgenerating without the 3; 2 # decoration. The 3; 2 # connectives form a residuated pair, which means they are orderpreserving operators with respect to the derivability relation. The algebraic structure of the base logic also provides room for a pair of orderreversing
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"... The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In §2 we do the basic modeltheoretic and ..."
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The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In §2 we do the basic modeltheoretic and prooftheoretic groundwork. In §3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quantifier expressions, and to describe a typology of polarity items. 1