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Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 63 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
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"... The origin of this apparent ability to signal is postselection, i.e. conditioning on the outcome of the quantum measurement, which is easily seen to be a (virtual) resource that enables signaling. Hence ‘quantum processes in a causal universe ’ cannot form a dagger or a compact symmetric monoidal c ..."
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The origin of this apparent ability to signal is postselection, i.e. conditioning on the outcome of the quantum measurement, which is easily seen to be a (virtual) resource that enables signaling. Hence ‘quantum processes in a causal universe ’ cannot form a dagger or a compact symmetric monoidal category, since both the dagger and the transpose (induced by the compact structure) of a state yield a postselected effect. In order to retain compatibility of quantum mechanics with relativity one needs to exclude postselection and only consider processes with a certain ‘overall ’ probability; that is, one always needs to consider all possible measurement outcomes together. Formally, still in terms of monoidal categories, this can be achieved by using internal dagger Frobenius structures to index over all possible outcome scenarios as in [8]. In that case, an ‘indexchannel’ prevents signaling in the teleportation protocol:
Elsevier www.elsevier.com/locate/fss A Categorical Semantics for Fuzzy Predicate Logic
, 2010
"... The object of this study is to look at categorical approaches to many valued logic, both propositional and predicate, to see how different logical properties result from different parts of the situation. In particular, the relationship between the categorical fabric I introduced at Linz ..."
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The object of this study is to look at categorical approaches to many valued logic, both propositional and predicate, to see how different logical properties result from different parts of the situation. In particular, the relationship between the categorical fabric I introduced at Linz
Fuzzy Sets and Systems 161,2010, pp. 24622478 Elsevier, www.elsevier.com/locate/fss Categorical Approaches to NonCommutative Fuzzy Logic
, 2010
"... In this paper we consider what it means for a logic to be noncommutative, how to generate examples of structures with a non commutative operation * which have enough nice properties to serve as the truth values for a logic. Inference in the propositional logic is gotten from the categorical proper ..."
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In this paper we consider what it means for a logic to be noncommutative, how to generate examples of structures with a non commutative operation * which have enough nice properties to serve as the truth values for a logic. Inference in the propositional logic is gotten from the categorical properties (products, coproducts, monoidal and closed structures, adjoint functors) of the categories of truth values. We then show how to extend this view of propositional logic to a predicate logic using categories of propositions about a type A with functors giving change of type and adjoints giving quantifiers. In the case where the