Predikaatiologiikka

Predikaatiologiikka, or predicate logic, is a branch of mathematical logic that extends propositional logic by introducing variables, predicates and quantifiers to describe properties of objects and relations between them. In this framework, formulas are built from predicate symbols with certain arities, function symbols, variables, and the logical connectives and quantifiers ∀ (for all) and ∃ (exists).

In syntax, atomic formulas have the form P(t1,...,tn) or R(t1,...,tm), where P and R are predicate symbols and t1,...,tn are terms. More complex formulas are obtained by applying logical connectives (and, or, not, implies) and by applying quantifiers to bind variables. The central fragment is first-order logic, which permits quantification over individuals but not over predicates or functions.

In semantics, a structure consists of a nonempty domain and interpretations of all nonlogical symbols. An assignment maps variables to elements of the domain. The truth of a sentence is defined relative to a structure and an assignment; a sentence is true in a structure if it is satisfied under every assignment. This framework supports various extensions, such as adding equality or higher-order features.

Key results include Gödel’s completeness theorem, which links semantic truth to syntactic derivability, and the compactness and Löwenheim–Skolem theorems. Full first-order logic is undecidable (Church’s theorem), though many decidable fragments exist, such as monadic first-order logic.

Applications span formalizing mathematics, computer science, databases and artificial intelligence. Examples express general claims like ∀x (Human(x) → Mortal(x)) or existence statements such as ∃x (Dog(x) ∧ Barking(x)). The field provides foundations for automated reasoning, knowledge representation and formal verification.