##  [Finite Model Property](/finite-model-property-0) 

 Definition

A property of a formal logic or theory stating that every sentence or formula that is satisfiable (has a model) has a finite model.

 

 

 

 

 

 





## Principle

Principle

Restrict satisfiability to finite structures: if a formula can be realized at all, it can be realized in some finite structure; often used to link semantics to algorithmic checks by searching finite domains.

 

 

 

 

 





## Demonstration

Demonstration

Many modal logics enjoy the property because filtration constructs a finite Kripke model from any satisfiable formula; concretely, a satisfiable modal formula can be shown to have a finite frame validating it.

 

 

 

 

## Misapplication

Misapplication

Assuming that the finite model property implies completeness or decidability for arbitrary theories — FMP helps decidability in many cases but does not guarantee it without effective axiomatization or enumerability.

 

 

 

 

 





## Consequence

Consequence

When present and the language/axioms are effectively enumerable, one typically obtains decidability by searching finite models; FMP also restricts possible countermodels to a finite search space.

 

 

 

 

## Reversal

Reversal

The opposite situation is the infinite model property: there exist satisfiable formulas that admit only infinite models, so no finite structure realizes them.

 

 

 

 

 





## Boundary

Boundary

Applies to logics or theories and their satisfiable sentences; it depends on the signature and allowed models (e.g., relational vs. functional signatures) and is not a statement about particular finite structures or counting models of bounded size.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between finite satisfiability (existence of finite models) and general satisfiability (existence of arbitrary models); a logic can be satisfiable without FMP if realizations require infinite constructions.

 

 

 

 

 





## Synthesis

Synthesis

Finite Model Property is the specification that satisfiability can always be witnessed by finite structures; it organizes semantic search procedures and sits between abstract satisfiability and effective decidability by constraining countermodels to finite domains.