Property of a formula or set of formulas that there exists at least one interpretation, model, or assignment of truth-values making every formula in the set true.

Principle

A formula or theory is satisfiable iff a model exists in which all its sentences evaluate to true under the semantics of the logic in question.

Demonstration

In propositional logic, the clause set {p ∨ q, ¬p} is satisfiable because the assignment p = false, q = true makes both clauses true; in first-order logic, a theory with a single existential sentence ∃x P(x) is satisfiable if there is any interpretation with at least one element satisfying P.

Misapplication

Treating syntactic provability from axioms as equivalent to satisfiability: a set can be provable-consistent but unsatisfiable in some nonstandard semantics, or conversely a satisfiable set may not be derivable using a restricted proof system.

Consequence

When a set is satisfiable one can exhibit or reason about at least one concrete model; satisfiability enables model-based methods such as counterexample search and model checking.

Reversal

The negation of satisfiability is unsatisfiability (no model exists); contrastingly, validity is not simply the negation of satisfiability but a universal truth condition across all models.

Boundary

Satisfiability is semantic and depends on the chosen logic and its admissible models; some non-classical logics change what counts as a model, and finite versus infinite model existence may differ.

Semantic Tension

Satisfiability vs provability: satisfiability is about existence of models, while provability concerns syntactic derivations—completeness theorems bridge but do not eliminate differences in practice.

Synthesis

Satisfiability identifies when at least one interpretation makes every sentence true; it is a semantic existence claim that underpins model construction, counterexample search, and many automated reasoning tasks.