Knowra Structure (mathematical logic) Structure (mathematical logic) A structure consists of a domain and interpretations of the constant, function, and relation symbols in a formal language. It gives those symbols meaning while preserving the language’s specified arities.
First-order language : A formal language with symbols for variables, logical operations, quantifiers, and nonlogical vocabulary. Its nonlogical symbols are the ones a structure interprets.
Term (logic) : An expression denoting an object, built from variables, constants, and function symbols. Terms denote domain elements using the structure’s constants and functions.
Model (mathematical logic) : A structure in which a specified set of sentences is true. The term highlights a structure’s role as a satisfying interpretation of a theory.
Natural numbers structure : A structure whose domain is the natural numbers and whose symbols may interpret arithmetic operations and relations. It exemplifies a domain equipped with interpretations for arithmetic symbols.
Domain of discourse : The collection of objects over which variables range in a logical interpretation. The domain supplies the objects on which this structure’s symbols are interpreted.
Satisfaction relation : A relation stating when a formula is true in a structure under a variable assignment. It formally evaluates formulas against this structure’s interpretations.
Theory (mathematical logic) : A set of sentences in a formal language, often closed under logical consequence. A theory describes the properties its models, including structures, must satisfy.
Random graph : A countably infinite graph in which every finite adjacency pattern occurs among suitable vertices. It is a graph viewed as a structure with an adjacency relation.
Signature (logic) : A specification of the nonlogical symbols in a language and their arities or types. The signature states which constants, functions, and relations this structure must interpret.
Model theory : The study of formal languages through their interpretations and the structures that satisfy sentences. Structures are the central semantic objects studied in model theory.
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