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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.
Categorial grammar posits a close relationship between the syntax and semantic composition, since it typically treats syntactic categories as corresponding to semantic types. Categorial grammars were developed in the 1930s by Kazimierz Ajdukiewicz and in the 1950s by Yehoshua Bar-Hillel and Joachim Lambek .
Combinatory categorial grammar (CCG) is an efficiently parsable, yet linguistically expressive grammar formalism.It has a transparent interface between surface syntax and underlying semantic representation, including predicate–argument structure, quantification and information structure.
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 ...
Categorical (or "functorial") semantics [11] uses category theory as the core mathematical formalism. Categorical semantics is usually proven to correspond to some axiomatic semantics that gives a syntactic presentation of the categorical structures. Also, denotational semantics are often instances of a general categorical semantics; [12]
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 ...
Categorical semantics [ edit ] The simply typed lambda calculus enriched with product types, pairing and projection operators (with β η {\displaystyle \beta \eta } -equivalence) is the internal language of Cartesian closed categories (CCCs), as was first observed by Joachim Lambek . [ 11 ]
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