Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension 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. Year 1970 ( MCMLXX) was a Common year starting on Thursday (link shows full calendar of the Gregorian calendar.
Categorical logic originated with Bill Lawvere's Functorial Semantics of Algebraic Theories (1963), and Elementary Theory of the Category of Sets (1964). Francis William Lawvere (b February 9 1937 in Muncie Indiana is a Mathematician known for his work in Category theory, topos theory and the Philosophy Lawvere recognised the Grothendieck topos, introduced in algebraic topology as a generalised space, as a generalisation of the category of sets (Quantifiers and Sheaves (1970)). In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets With Myles Tierney, Lawvere then developed the notion of elementary topos, thus establishing the fruitful field of topos theory, which provides a unified categorical treatment of the syntax and semantics of higher-order predicate logic. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets The resulting logic is formally intuitionistic. Andre Joyal is credited, in the term Kripke–Joyal semantics, with the observation that the sheaf models for predicate logic, provided by topos theory, generalise Kripke semantics. Kripke semantics (also known as relational semantics or frame semantics, and often confused with Possible world semantics) is a formal Semantics Joyal and others applied these models to study higher-order concepts such as the real numbers in the intuitionistic setting. In Mathematics, the real numbers may be described informally in several different ways
An analogous development was the link between the typed lambda calculus and cartesian-closed categories (Lawvere, Lambek, Scott), which provided a setting for the development of domain theory. A In Category theory, a category is cartesian closed if roughly speaking any Morphism defined on a product of two objects can be naturally identified with a morphism Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains. Less expressive theories, from the mathematical logic viewpoint, have their own category theory counterparts. For example the concept of an algebraic theory leads to Gabriel–Ulmer duality. The view of categories as a generalisation unifying syntax and semantics has been productive in the study of logics and type theories for applications in computer science.
The founders of elementary topos theory were Lawvere and Tierney. Francis William Lawvere (b February 9 1937 in Muncie Indiana is a Mathematician known for his work in Category theory, topos theory and the Philosophy Lawvere's writings, sometimes couched in a philosophical jargon, isolated some of the basic concepts as adjoint functors (which he explained as 'objective' in a Hegelian sense, not without some justification). A subobject classifier is a strong property to ask of a category, since with cartesian closure and finite limits it gives a topos (axiom bashing shows how strong the assumption is). In Category theory, a subobject classifier is a special object &Omega of a category intuitively the Subobjects of an object X correspond to the morphisms In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts Lawvere's further work in the 1960s gave a theory of attributes, which in a sense is a subobject theory more in sympathy with type theory. In Category theory, there is a general definition of subobject extending the idea of Subset and Subgroup. Major influences subsequently have been Martin-Löf type theory from the direction of logic, type polymorphism and the calculus of constructions from functional programming, linear logic from proof theory, game semantics and the projected synthetic domain theory. Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a Logical system and a Set theory In Computer science, polymorphism is a Programming language feature that allows values of different Data types to be handled using a The calculus of constructions (CoC is a higher-order Typed lambda calculus, initially developed by Thierry Coquand, where types are First-class values In Mathematical logic, linear logic is a type of Substructural logic that denies the Structural rules of weakening and contraction. Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Game semantics ( German: dialogische Logik) is an approach to Formal semantics that grounds the concepts of Truth or Validity on The abstract categorical idea of fibration has been much applied. In Mathematics, especially Algebraic topology, a fibration is a continuous mapping pE\to B\ satisfying the
To go back historically, the major irony here is that in large-scale terms, intuitionistic logic had reappeared in mathematics, in a central place in the Bourbaki–Grothendieck program, a generation after the messy Hilbert–Brouwer controversy had ended, with Hilbert the apparent winner. Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Luitzen Egbertus Jan Brouwer ɛxˈbɛʁtəs jɑn ˈbʁʌuəʁ ( February 27 1881, Overschie – December 2 1966, Blaricum Bourbaki, or more accurately Jean Dieudonné, having laid claim to the legacy of Hilbert and the Göttingen school including Emmy Noether, had revived intuitionistic logic's credibility (although Dieudonné himself found Intuitionistic Logic ludicrous), as the logic of an arbitrary topos, where classical logic was that of 'the' topos of sets. Jean Alexandre Eugène Dieudonné ( July 1 1906, Lille - November 29 1992, Nice) was a French mathematician Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and This was one consequence, certainly unanticipated, of Grothendieck's relative point of view; and not lost on Pierre Cartier, one of the broadest of the core group of French mathematicians around Bourbaki and IHES. Pierre Cartier (born in Sedan France in 1932 is a Mathematician. The Institut des Hautes Études Scientifiques ( IHÉS, en Institute of Advanced Scientific Studies is a French institute supporting advanced research in Cartier was to give a Séminaire Bourbaki exposition of intuitionistic logic. The Séminaire Nicolas Bourbaki ( Bourbaki Seminar) is a series of Seminars (in fact public lectures with printed notes distributed that has been held in Paris since
In an even broader perspective, one might take category theory to be to the mathematics of the second half of the twentieth century, what measure theory was to the first half. The twentieth century of the Common Era began on In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with It was Kolmogorov who applied measure theory to probability theory, the first convincing (if not the only) axiomatic approach. Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet Probability theory is the branch of Mathematics concerned with analysis of random phenomena Kolmogorov was also a pioneer writer in the early 1920s on the formulation of intuitionistic logic, in a style entirely supported by the later categorical logic approach (again, one of the formulations, not the only one; the realizability concept of Stephen Kleene is also a serious contender here). The 1920s is sometimes referred to as the " Jazz Age " or the " Roaring Twenties " when speaking about the United States and Canada Realizability is a part of Proof theory which can be used to handle information about formulas instead of about the proofs of formulas Stephen Cole Kleene ( January 5, 1909, Hartford Connecticut, USA &ndash January 25, 1994, Madison Wisconsin Another route to categorical logic would therefore have been through Kolmogorov, and this is one way to explain the protean Curry–Howard isomorphism. The Curry-Howard correspondence is the direct relationship between computer programs and mathematical proofs