Predikáty

Predikáty, in logic, are relations of a fixed arity that are denoted by predicate symbols in formal languages. They are interpreted in a structure as actual relations on the domain of discourse. A k-ary predicate symbol P represents a relation P^M ⊆ D^k in a structure M with domain D. When used in formulas, a k-ary predicate yields atomic formulas of the form P(t1, ..., tk), where the ti are terms.

Predikáty can have various arities, including unary predicates (k = 1) and binary predicates (k = 2). Zero-ary predicates, if present, are propositional constants that evaluate to true or false in a given structure. The truth of more complex formulas is determined by logical connectives (and, or, not, implies) and quantifiers (forall, exists). For a given structure M and variable assignment g, one writes M ⊨_g φ to indicate that φ is satisfied; sentences without free variables are either true in M or false in M.

In practice, predikáty enable the expression of properties and relations, such as “is prime” or “is parent of,” once appropriate arity is chosen. They are central to predicate (first-order) logic, which extends propositional logic by incorporating quantification and relations. Applications span mathematics, computer science, linguistics, and philosophy, where predicate logic underpins formal specification, database query languages, and program verification. The term predikát is used in Slovak and Czech to denote the concept, with predikáty as its plural form.