##  [Satisfiability](/satisfiability-0) 

 Definition

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

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

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

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

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

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

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

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

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.