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Finite Sum  Product Logic
 Theory Appl. Categ
, 2001
"... . In this paper we describe a deductive system for categories with finite products and coproducts, prove decidability of equality of morphisms via cut elimination, and prove a "Whitman theorem" for the free such categories over arbitrary base categories. This result provides a nice illu ..."
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Cited by 11 (2 self)
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. In this paper we describe a deductive system for categories with finite products and coproducts, prove decidability of equality of morphisms via cut elimination, and prove a "Whitman theorem" for the free such categories over arbitrary base categories. This result provides a nice illustration of some basic techniques in categorical proof theory, and also seems to have slipped past unproved in previous work in this field. Furthermore, it suggests a typetheoretic approach to 2player inputoutput games. Introduction In the late 1960's Lambek introduced the notion of a "deductive system", by which he meant the presentation of a sequent calculus for a logic as a category, whose objects were formulas of the logic, and whose arrows were (equivalence classes of) sequent derivations. He noticed that "doctrines" of categories corresponded under this construction to certain logics. The classic example of this was cartesian closed categories, which could then be regarded as the "proof...
On categorical models of classical logic and the geometry of interaction
, 2005
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this co ..."
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Cited by 4 (0 self)
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this collapse by interpreting cutreduction by a posetenrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the settheoretic product. In this article, which is selfcontained, we present an improved axiomatization of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negationfree models called Dummett categories. Examples include, besides the classical categories above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from homsemilattices which have a straightforward prooftheoretic definition. Moreover, we show that the GeometryofInteraction construction can be extended from multiplicative linear logic to classical logic, by applying it to obtain a classical
An institutional view on categorical logic and the CurryHowardTaitisomorphism
"... We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number ..."
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Cited by 1 (1 self)
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We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.
Theory and Applications of Categories, Vol. 8, No. 5, pp. 63–99. FINITE SUM – PRODUCT LOGIC
"... ABSTRACT. In this paper we describe a deductive system for categories with finite products and coproducts, prove decidability of equality of morphisms via cut elimination, and prove a “Whitman theorem ” for the free such categories over arbitrary base categories. This result provides a nice illustra ..."
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ABSTRACT. In this paper we describe a deductive system for categories with finite products and coproducts, prove decidability of equality of morphisms via cut elimination, and prove a “Whitman theorem ” for the free such categories over arbitrary base categories. This result provides a nice illustration of some basic techniques in categorical proof theory, and also seems to have slipped past unproved in previous work in this field. Furthermore, it suggests a typetheoretic approach to 2–player input–output games.
Abstract
, 2009
"... We use Bell states to provide compositional distributed meaning for negative sentences of English. The lexical meaning of each word of the sentence is a context vector obtained within the distributed model of meaning. The meaning of the sentence lives within the tensor space of the vector spaces of ..."
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We use Bell states to provide compositional distributed meaning for negative sentences of English. The lexical meaning of each word of the sentence is a context vector obtained within the distributed model of meaning. The meaning of the sentence lives within the tensor space of the vector spaces of the words. Mathematically speaking, the meaning of a sentence is the image of a quantizing functor from the compact closed category that models the grammatical structure of the sentence (using Lambek Pregroups) to the compact closed category of finite dimensional vector spaces where the lexical meaning of the words are modeled. The meaning is computed via composing eta and epsilon maps that create Bell states and do substitution and as such allow the information to flow among the words within the sentence.
unknown title
, 2005
"... Before one can attach a meaning to a sentence, one must distinguish different ways of parsing it. When analyzing a language with pregroup grammars, we are thus led to replace the free pregroup by a free compact strict monoidal category. Since a strict monoidal category is a 2category with one 0cel ..."
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Before one can attach a meaning to a sentence, one must distinguish different ways of parsing it. When analyzing a language with pregroup grammars, we are thus led to replace the free pregroup by a free compact strict monoidal category. Since a strict monoidal category is a 2category with one 0cell, we investigate the free compact 2category generated by a given category, and we describe its 2cells as labeled transition systems. In particular, we obtain a decision procedure for the equality of 2cells in the free compact 2category. 1.
An Institutional View on Categorical Logic
"... We introduce a generic notion of categorical propositional logic and provide a construction of a preorderenriched institution out of such a logic, following the CurryHowardTait paradigm. The logics are specified as theories of a metalogic within the logical framework LF such that institution com ..."
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We introduce a generic notion of categorical propositional logic and provide a construction of a preorderenriched institution out of such a logic, following the CurryHowardTait paradigm. The logics are specified as theories of a metalogic within the logical framework LF such that institution comorphisms are obtained from theory morphisms of the metalogic. We prove several logicindependent results including soundness and completeness theorems and instantiate our framework with a number of examples: classical, intuitionistic, linear and modal propositional logic. We dedicate this work to the memory of our dear friend and colleague Joseph Goguen who passed away during its preparation. 1