##  [Unsatisfiability](/unsatisfiability-0) 

 Definition

Property of a formula or set of formulas for which no interpretation, model, or assignment makes every formula in the set true; equivalently, the set has an empty model class.

 

 

 

 

 

 





## Principle

Principle

A set is unsatisfiable when the semantic conditions of the logic rule out any possible world or structure that would render all members true simultaneously.

 

 

 

 

 





## Demonstration

Demonstration

In propositional logic, the set {p, ¬p} is unsatisfiable because no truth assignment can make both p and ¬p true; in first-order logic, the set {∀x P(x), ∃x ¬P(x)} is unsatisfiable in standard semantics.

 

 

 

 

## Misapplication

Misapplication

Calling a theory unsatisfiable because it lacks proofs of a particular sentence: absence of derivations does not imply absence of models, and decidability issues can mask satisfiability status.

 

 

 

 

 





## Consequence

Consequence

Unsatisfiability allows proof by contradiction: deriving an explicit contradiction shows no model can satisfy the premises; it also triggers refutation-based automated methods like SAT-unsat detection.

 

 

 

 

## Reversal

Reversal

The inverse concept is satisfiability; unsatisfiability corresponds semantically to the impossibility of a model, while syntactically it often corresponds to derivability of an explicit contradiction.

 

 

 

 

 





## Boundary

Boundary

Unsatisfiability depends on the semantics and domain assumptions (e.g., finite-domain vs arbitrary-domain models); paraconsistent logics alter the link between contradiction and unsatisfiability.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Unsatisfiability vs syntactic inconsistency: unsatisfiability is a semantic notion (no model), while inconsistency often denotes that a contradiction is derivable under a particular consequence relation; they coincide in sound and complete systems but diverge otherwise.

 

 

 

 

 





## Synthesis

Synthesis

Unsatisfiability declares that no interpretation can make all formulas true; it is the semantic signal of a contradiction in the model-theoretic sense and underlies refutation and countermodel reasoning.