Totuusjoukko

Totuusjoukko, translated as “truth set”, is a term used in mathematical logic to describe the collection of interpretations or assignments that make a given logical formula true in a fixed structure or model. In this sense, it captures the semantic meaning of a formula as a relation on the domain of discourse.

In propositional logic, the truth set of a formula φ with respect to a valuation is the set of truth assignments to the propositional variables that satisfy φ. For example, the formula p ∨ q is true for all valuations except the one where p and q are both false; thus its truth set in the space of all valuations {true, false}^2 is { (true, true), (true, false), (false, true) }.

In first-order logic, a formula φ(x1, ..., xn) with free variables defines a relation on the domain. Given a structure M with domain D, the truth set of φ in M is { (a1, ..., an) in D^n | M ⊨ φ[a1, ..., an] }, the interpretation of φ at those elements. This set is also called the extension or interpretation of φ in M and is a definable subset of D^n.

The concept is sensitive to the chosen model: the same syntactic formula can have different truth sets in different structures. Logical connectives correspond to set operations on truth sets: conjunction to intersection, disjunction to union, negation to complement (within the appropriate coordinates), and quantifiers to projection operations.

Applications of truth sets appear across model theory, definability theory, and database semantics, where a formula’s truth set describes precisely which tuples of elements satisfy the stated conditions. The Finnish term totuusjoukko underscores the link between syntax (the formula) and semantics (the set of its truths) within a given model.