A function (often called a valuation or truth assignment) that maps propositional variables or atomic formulas to truth values in a given semantic setting and extends to complex formulas according to the logic's semantic clauses.

Principle

A valuation specifies the semantic base: it gives primitive truth values to atoms and, via compositional rules (truth tables or clauses), determines the truth value of every well-formed formula in the language under a chosen interpretation.

Demonstration

In propositional logic define v such that v(P)=True and v(Q)=False; then v(P ∨ Q)=True and v(P → Q)=False by the usual truth-functional extension. In first-order contexts a valuation must be supplemented by an interpretation of predicates and a variable assignment.

Misapplication

Confusing a valuation with a proof, treating it as evidence that a formula is derivable from axioms, or ignoring the need to specify the underlying interpretation (domain, predicate denotations) in first-order settings are common mistakes.

Consequence

Valuations are the tool for model-theoretic analysis: they witness satisfiability, provide countermodels to non-entailment, and enable algorithmic checks of tautologies, satisfiable formulas, and logical consequence.

Reversal

Dropping totality or truth-functionality leads to partial valuations (truth-value gaps) or nonstandard valuations in many-valued or paraconsistent logics; these reversals change how connectives are extended and which sentences count as satisfied.

Boundary

This notion presumes a specified logic and its semantic clauses; in propositional logic valuations map variables to truth values, while first-order model-theoretic assignments additionally fix domains and predicate interpretations—valuation alone is not the whole model in richer languages.

Semantic Tension

There is tension between seeing a valuation as a purely syntactic assignment of values to symbols and seeing it as part of a semantic model that includes domain, functions and relations; conflating these roles obscures what a valuation alone accomplishes.

Synthesis

A valuation is the formal device that assigns base truth values to atomic items and, through the logic's compositional rules, yields truth values for complex formulas; it underpins satisfiability, model construction and semantic evaluation across logical systems.