Categorical Logic Lambek, J. and Scott, P. Introduction to HigherOrder Categorical Logic. Borceux, F. Handbook of Categorical Algebra (Encyclopedia of Mathematics and http://www.andrew.cmu.edu/user/awodey/catlog/
Extractions: Secretary: Baker 135 This course focuses on applications of category theory in logic and computer science. A leading idea is functorial semantics, according to which a model of a logical theory is a set-valued functor on a structured category determined by the theory. This gives rise to a syntax-invariant notion of a theory and introduces many algebraic methods into logic, leading naturally to the universal and other general models that distinguish functorial from classical semantics. Such categorical models occur, for example, in denotational semantics. In this connection the lambda-calculus is treated via the theory of cartesian closed categories. Similarly higher-order logic is modelled by the categorical notion of a topos. Using sheaves, topos theory also subsumes Kripke semantics for intuitionistic logic. 80-413/713 Category Theory, or equivalent.
Category Theory To be followed by a Fall course on categorical logic. Borceux Handbook http://www.andrew.cmu.edu/course/80-413-713/
Extractions: Webpage: www.andrew.cmu.edu/course/80-413-713 Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science. Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations. This course is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines.
Categorical Logic - Wikipedia, The Free Encyclopedia This has enabled proofs of metatheoretical properties of some logics by means of an appropriate categorical algebra. For instance, Freyd gave a proof of http://en.wikipedia.org/wiki/Categorical_logic
Extractions: From Wikipedia, the free encyclopedia Jump to: navigation search This article is about mathematical logic in the context of category theory. For Aristotle's system of logic, see Term logic Categorical logic is a branch of category theory within mathematics , adjacent to mathematical logic but more notable for its connections to theoretical computer science . In broad terms, categorical logic represents both syntax and semantics by a category , and an interpretation by a functor . The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970. There are three important themes in the categorical approach to logic: Categorical semantics . Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions . This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G. Seely
Extractions: From Wikipedia, the free encyclopedia Jump to: navigation search A category with objects X, Y, Z and morphisms f g This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations Please improve this article by introducing more precise citations where appropriate (November 2009) Category theory is an area of study in mathematics that deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows One of the simplest examples of a category (which is a very important concept in topology ) is that of groupoid , defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types , and mathematical physics where they can be used to describe vector spaces . Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942â€“45, in connection with
Algebra And Logic File Format PDF/Adobe Acrobat Quick View http://www.dpmms.cam.ac.uk/~martin/Research/Slides/algebralogic.pdf
Springer Online Reference Works A branch of mathematics dealing with the interaction between logic (cf. also Mathematical .. Appl. Categorical Algebra , Amer. Math. Soc. (1970) pp. 1–14 http://eom.springer.de/c/c120060.htm
Axiomatic And Categorical Foundations Of Mathematics I I 30 Mac Lane, S., 1971, Categorical algebra and SetTheoretic Sets, Topoi, and Internal Logic in Categories, Studies in Logic and the Foundations of http://myyn.org/m/article/axiomatic-and-categorical-foundations-of-mathematics-i
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Extractions: Please Read How You Can Help Keep the Encyclopedia Free First published Fri Dec 6, 1996; substantive revision Thu Feb 25, 2010 et al . 2000, 2001, 2002). The very definition of a category is not without philosophical importance, since one of the objections to category theory as a foundational framework is the claim that since categories are defined Definition : A mapping e will be called an identity if and only if the existence of any product e e implies that e e Definition C is an aggregate Ob of abstract elements, called the objects of C , and abstract elements Map , called mappings of the category. The mappings are subject to the following five axioms: is defined. When either is defined, the associative law are defined. e e is defined, and at least one identity e such that e (C4) The mapping e X corresponding to each object X is an identity.
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Notes On: The Logical Foundations Of Mathematics Mar 16, 1995 Chapter 1 First Order Logic A presentation of first order logic including a general treatment of Chapter 8 - Categorial Algebra http://www.rbjones.com/rbjpub/philos/bibliog/hatch82.htm