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The categorical semantics of a logic consists in describing a category of structured categories that is related to the category of theories in that logic by an adjunction, where the two functors in the adjunction give the internal language of a structured category on the one hand, and the term model of a theory on the other.
In logic, a categorical proposition, ... Sentences in natural language may be translated into standard forms. ... English Sentence
A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley ( 1965 ) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality , then it is categorical in all uncountable ...
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categorical logic A branch of logic that studies the categorization of objects and the logical foundations of categories, often using the framework of category theory. categorical proposition A proposition that asserts or denies that all or some of the members of one category are included in another category, fundamental in syllogistic reasoning.
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
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.
Founds categorical logic, discovers internal logics of categories and recognizes its importance and introduces Lawvere theories. Essentially categorical logic is a lift of different logics to being internal logics of categories. Each kind of category with extra structure corresponds to a system of logic with its own inference rules.