Certain (and Possible) Answers

Synonyms

True answer (Maybe answer); Validity (Satisfiability)

Definition

LetT be a finite theory expressed in a language L, and φ an L-sentence. Then T finitely entails φ, in notation T ⊧ φ, if all finite models of T also are models of φ. A theory T is said to be complete in the finite if for each L-sentence φ either T ⊧ φ or T ⊧ ¬φ. In particular, if T is incomplete (not complete in the finite), then there is an L-sentence φ, such that T ⊭ φ and T ⊭ ¬φ. It follows from classical logic that a first order theory is complete in the finite if and only if all its finite models are isomorphic. Consider now a theory

$$ {T}_1\,{=}\!\left\{\!\begin{array}{c}\hfill R(a,b){\wedge} R(a,c),\hfill \\ {}\hfill \forall x,y{:}R(x,y){\to} (x,y){=}(a,b){\vee} (a,c),\hfill \\ {}\hfill a\ne b,a\ne c,b\ne c.\hfill \end{array}\right. $$

where a, b, and c are constants. This theory is complete, and clearly for instance T ⊧ R(a, b), T ⊧ R(a, c), and T ⊭ R(d, c), for all constants d different from a and b....