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There are three important themes in the categorical approach to logic:
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's modeling of various impredicative theories, such as system F is an example of the usefulness of categorical semantics.
It was found that the connectives of pre-categorical logic were more clearly understood using the concept of adjoint functor, and that the quantifiers were also best understood using adjoint functors.
This can be seen as a formalization and generalization of proof by diagram chasing. One defines a suitable internal language naming relevant constituents of a category, and then applies categorical semantics to turn assertions in a logic over the internal language into corresponding categorical statements. This has been most successful in the theory of toposes, where the internal language of a topos together with the semantics of intuitionistic higher-order logic in a topos enables one to reason about the objects and morphisms of a topos "as if they were sets and functions". This has been successful in dealing with toposes that have "sets" with properties incompatible with classical logic. A prime example is Dana Scott's model of untyped lambda calculus in terms of objects that retract onto their own function space. Another is the Moggi-Hyland model of system F by an internal full subcategory of the effective topos of Martin Hyland.
Michael Makkai and Gonzalo E. Reyes, 1977, First order categorical logic, Springer-Verlag.
Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic. Fairly accessible introduction, but somewhat dated. The categorical approach to higher-order logics over polymorphic and dependent types was developed largely after this book was published.
Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, Elsevier. ISBN0-444-50170-3. A comprehensive monograph written by a computer scientist; it covers both first-order and higher-order logics, and also polymorphic and dependent types. The focus is on fibred category as universal tool in categorical logic, which is necessary in dealing with polymorphic and dependent types.
Jean-Pierre Marquis and Gonzalo E. Reyes (2012). The History of Categorical Logic 1963-1977. Handbook of the History of Logic: Sets and Extensions in the Twentieth Century, Volume 6, D. M. Gabbay, A. Kanamori & J. Woods, eds., North-Holland, pp. 689-800. A preliminary version is available at .