The History of Categorical Logic: 1963–1977 [chapter]

Jean-Pierre Marquis, Gonzalo E. Reyes
2012 Handbook of the History of Logic  
Categorical logic, as its name indicates, is logic in the setting of category theory. But this description does not say much. Most readers would probably find more instructive to learn that categorical logic is algebraic logic, pure and simple. It is logic in an algebraic dressing. Just as algebraic logic encodes propositional logic in its different guises (classical, intuitionistic, etc.) by their Lindenbaum-Tarski algebras (Boolean algebras, Heyting algebras and so on), categorical logic
more » ... es first-order and higher-order logics (classical, intuitionistic, etc.) by categories with additional properties and structure (Boolean categories, Heyting categories and so on). Thus, from the purely technical point of view, categorical logic constitutes a generalization of the algebraic encoding of propositional logic to firstorder, higher-order and other logics. Furthermore, we shall present and discuss arguments (given by the main actors) to show that this encoding constitutes the correct generalization of the well-known algebraic encoding of propositional logics by the Lindenbaum-Tarski algebras. The proper algebraic structures are not only categories, but also morphisms between categories, mainly functors and more specially adjoint functors. A key example is provided by the striking fact that quantifiers, which were the stumbling block to the proper algebraic generalization of propositional logic, can be seen to be adjoint functors and thus entirely within the categorical framework. As is usually the case when algebraic techniques are imported and developed within a field, e.g. geometry and topology, vast generalizations and unification become possible. Furthermore, unexpected concepts and results show up along the way, often allowing a better understanding of known concepts and results. Categorical logic is not merely a convenient tool or a powerful framework. Again, as is usually the case when algebraic techniques are imported and used in a field, the very nature of the field has to be thought over. Furthermore, various results shed a new light on what was assumed to be obvious or, what turns out to be often the same on careful analysis, totally obscure. Thus, categorical logic is philosophically relevant in more than one way. The way it encodes logical concepts and operations reveals important, even essential, aspects and properties of these concepts and operations. Again, as soon as quantifiers are seen as adjoint functors, the traditional question of the nature of variables in logic receives a satisfactory analysis. Furthermore, many results obtained via categorical techniques have clear and essential philosophical implications. The systematic development of higher-order
doi:10.1016/b978-0-444-51621-3.50010-4 fatcat:xlm77neeubhl7omhhrxhdtuo4q