Extendable formulas in two variables in intuitionistic logic

In this paper we discuss projective, exact and extendable formulas of the intuitionistic propositional calculus IPC. Exact formulas were introduced in [6] as the formulas that axiomatize the theories of substitutions. In an attempt to find a semantic characterization of exact formulas, de Jongh and Visser [7] defined the notion of extendable formulas and proved that every exact formula is extendable. Later Ghilardi [4], motivated by finding the most general unifiers for intuitionistic formulas, introduced projective formulas and proved that every extendable formula is projective and that every projective formula is exact. Thus all these three notions are equivalent. In this paper we give an alternative proof of Ghilardi’s theorem for formulas in two variables. We also give a full description of projective, exact and extendable formulas in two variables. Our main tools are the n-universal models — the general frames that are dual to Lindenbaum-Tarski algebras of IPC in n-variables.