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Proof theory in the abstract
 Ann. Pure Appl. Logic
, 2002
"... with great affection and respect, this small tribute to his influence. 1 Background In the Introduction to the recent text Troelstra and Schwichtenberg [44], the authors contrast structural proof theory on the one hand with interpretational proof theory on the other. They write thus. Structural proo ..."
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with great affection and respect, this small tribute to his influence. 1 Background In the Introduction to the recent text Troelstra and Schwichtenberg [44], the authors contrast structural proof theory on the one hand with interpretational proof theory on the other. They write thus. Structural proof theory is based on a combinatorial analysis of the structure of formal proofs; the central methods are cut elimination
Meeting a modality? Restricted permutation for the Lambek calculus. ' OTS Working Paper, Onderzoekinstituut voor Taal en Spraak, Universiteit
, 1993
"... This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzenstyle proof theory in which there are constraints on the application of some structural rules. In particular, we address the question how to add an operator to the Lambek Calculus in order to giv ..."
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This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzenstyle proof theory in which there are constraints on the application of some structural rules. In particular, we address the question how to add an operator to the Lambek Calculus in order to give it a restricted access to the rule of Permutation, an extension which is partly motivated by linguistic applications. In line with tradition, we use the operator (∇) as a label telling us how the marked formula may be used, qua structural rules. New in our approach is that we do not see ∇ as a modality. Rather, we treat a formula ∇A as the meet of A with a special type Q. In this way we can make the specific structural behaviour of marked formulas more explicit. We define a minimal proof calculus for the system and prove some nice properties of it, like cutelimination, decidability an an embedding result. The main motivation for our approach however is that we can supply the proof system with an intuitive semantics.
Meeting strength in substructural logics
 Logic Group Preprint Series 38, Dept. of
, 1993
"... This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzenstyle proof theory in which there is only a limited possibility to use structural rules. Following the literature, we use an operator to mark formulas to which the extra structural rules may be a ..."
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This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzenstyle proof theory in which there is only a limited possibility to use structural rules. Following the literature, we use an operator to mark formulas to which the extra structural rules may be applied. New in our approach is that we do not see this ∇ as a modality, but rather as the meet of the marked formula with a special type Q. In this way we can make the specific structural behaviour of marked formulas more explicit. The main motivation for our approach is that we can provide a nice, intuitive semantics for hybrid substructural logics. Soundness and completeness for this semantics are proved; besides this we consider some prooftheoretical aspects like cutelimination and embeddings of the ‘strong’ system in the hybrid one.
Dialectica and Chu Constructions: Cousins?
 In this Volume
, 2006
"... This note investigates two generic constructions used to produce categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The constructions have the same objects, but are rather di#erent in other ways. We discuss similarities and di#erences and prove ..."
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This note investigates two generic constructions used to produce categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The constructions have the same objects, but are rather di#erent in other ways. We discuss similarities and di#erences and prove that the dialectica construction can be done over a symmetric monoidal closed basis. We also point out several interesting open problems concerning the Dialectica construction.
A lambdaCalculus for the Lambek Calculus
, 1996
"... :The Lambek Calculus was introduced by Lambek [2] in order to obtain effective rules to allow forming sentences and distinguishing sentences from nonsentences. Two primitives types are defined in this calculus and from these types is allowed to construct compound types to the words of the sentence. ..."
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:The Lambek Calculus was introduced by Lambek [2] in order to obtain effective rules to allow forming sentences and distinguishing sentences from nonsentences. Two primitives types are defined in this calculus and from these types is allowed to construct compound types to the words of the sentence. We construct a calculus for the Lambek Calculus, CurryHoward isomorphic to the sequent calculus. Some prooftheoretical results and a denotational semantic for this calculus, showing its adequacy with regard to the calculus, are shown. A phase semantics, based on Girard's phase semantics for the Linear Logic, searching to implement a verificationistic approach for meaning in construtivism, is also shown for this calculus. Keywords: Lambek Sintactic Calculus, Calculus, Denotational Semantic, Phase Semantic, Linear Logic. Resumo: O Lambek Calculus foi introduzido por Lambek [2] com o objetivo de obter regras efetivas que permitam formar sentencas e distinguir sentencas de n~aosentencas...
Characterizing Logic Grammars: A Substructural Logic Approach
, 1996
"... this paper. A substructural logic sequent calculus proof system is given which is shown to be equivalent to SDGs for parsing problems, in the sense that a string of terminal symbols is accepted by a grammar if and only if the corresponding sequent is derivable in the calculus. One calculus is given ..."
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this paper. A substructural logic sequent calculus proof system is given which is shown to be equivalent to SDGs for parsing problems, in the sense that a string of terminal symbols is accepted by a grammar if and only if the corresponding sequent is derivable in the calculus. One calculus is given for each of the two major interpretations of SDGs; the two calculi differ by only a small restriction in one rule. Since SDGs encompass other major grammar formalisms, including DCGs, the calculi serve to characterize those formalisms as well. / It is the authors' wish that no agency should ever derive military benefit from the publication of this paper. Authors who cite this work in support of their own are requested to qualify similarly the availability of these results.
A Concrete Categorical Model for the Lambek Calculus
, 1995
"... : There are concrete constructions like the universe of functions and function composition [Almeida93], which determine a denotational semantics for the Lambek Calculus [Lambek58] and can be regarded as realizing the dynamic features of it. On the other hand, nonsymmetric monoidal biclosed categor ..."
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: There are concrete constructions like the universe of functions and function composition [Almeida93], which determine a denotational semantics for the Lambek Calculus [Lambek58] and can be regarded as realizing the dynamic features of it. On the other hand, nonsymmetric monoidal biclosed categories determine an abstract model for the calculus [De Paiva91]. We present a categorical/denotational semantics for the Lambek Calculus, indeed for a typed version CurryHoward isomorphic to it. The main novelty of our approach is an abstract noncommutative construction, socalled sequential product, with right and left adjoints which is defined through a hierarchical structure of categories. It is motivated by those concrete constructions and it reflects the implicit permission of sequencing of objects. We show that the Lambek's noncommutative product corresponds to a noncommutative biendofunctor into a category enclosing all categories of such hierarquical structure, presenting a soundne...