Implicational Formulas in Intuitionistic Logic

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P . He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra I n on n free generators. In the present paper we give an alternative proof of the finiteness of I n , and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that I n is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1]. Let V be a finite set of propositional variables, and let F ( V ) be the set of all formulas built up from V ⋃ { t } using → alone. The algebra defined on the equivalence classes , by setting is a free Hilbert algebra I ( V ) on the free generators . A set T ⊆ F ( V ) is a theory if ⊦ p A implies A ∈ T , and T is closed under modus ponens . For T a theory, T [ A ] is the theory { B ∣ A → B ∈ T }. A theory T is p-prime , where p ∈ V , if p ∉ T and, for any A ∈ F ( V ), A ∈ T or A → p ∈ T . A theory is prime if it is p -prime for some p . P p ( V ) denotes the set of p -prime theories in F ( V ), P ( V ) the set of prime theories. T ∈ P ( V ) is minimal if there is no theory in P ( V ) strictly contained in T . Where X = { A 1 , …, A n } is a finite set of formulas, let X → B be A 1 →····→· A n → B (ϕ → B is B ). A formula A is a p-formula if p is the right-most variable occurring in A , i.e. if A is of the form X → p .