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Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. [1] In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor.
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term). [1]
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). [a] Such a theory can be viewed as defining its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model.
Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus.
In traditional logic, a proposition (Latin: propositio) is a spoken assertion (oratio enunciativa), not the meaning of an assertion, as in modern philosophy of language and logic. A categorical proposition is a simple proposition containing two terms, subject (S) and predicate (P), in which the predicate is either asserted or denied of the subject.
A declarative statement that is capable of being true or false, serving as the basic unit of meaning in logic and philosophy. propositional attitude A mental state expressed by verbs such as believe, desire, hope, and know, followed by a proposition, reflecting an individual's attitude towards the truth of the proposition. propositional connective
categorical logic Categorical logic is an approach to mathematical logic that uses category theory. categorical probability categorical probability categorification categorification is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors ...
Categorical may refer to: Categorical imperative, a concept in philosophy developed by Immanuel Kant; Categorical theory, in mathematical logic; Morley's categoricity theorem, a mathematical theorem in model theory; Categorical data analysis; Categorical distribution, a probability distribution