Two classes of intermediate propositional logics without disjunction property (Q1115419)

The author's model theoretic arguments presuppose familiarity with the notation and some results of his Journal of Symbolic Logic essays: ``Finitely generated free Heyting algebras'' [J. Symb. Logic 51, 152-165 (1986; Zbl 0616.03021)] and ``Finite and finitely separable intermediate propositional logics'' [ibid. 53, 403-420 (1988)]. He shows that there are two classes of logics between intuitionistic and classical propositional logics which are ``close to'' intuitionistic logic but yet lack intuitionistic logic's disjunctive property \(\vdash A\vee B\Rightarrow \vdash A\) or \(\vdash B\). The first class agrees with intuitionistic logic on theorems with at most n variables. In the second class the disjunctive property fails for formulas with a single variable in each disjunct although each disjunct is intuitionistically equivalent to some formula with a single variable.