We study the monadic fragment of second order intuitionistic propositional logic in the language containing the standard propositional connectives and propositional quantifiers. It is proved that under the topological interpretation over any dense-in-itself metric space, the considered fragment collapses to Heyting calculus. Moreover, we prove that the topological interpretation over any dense-in-itself metric space of fragment in question coincides with the so-called Pitts' interpretation. We also prove that all the nonstandard propositional operators of the form q ↦ ∃p (q ↔ F(p)), where F is an arbitrary monadic formula of the variable p, are definable in the language of Heyting calculus under the topological interpretation of intuitionistic logic over sufficiently regular spaces.

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We show that Pitts' modeling of propositional quantification in intuitionistic logic (as the appropriate interpolants) does not coincide with the topological interpretation. This contrasts with the case of the monadic language and the interpretation over sufficiently regular topological spaces. We also point to the difference between the topological interpretation over sufficiently regular spaces and the interpretation of propositional quantifiers in Kripke models.When we consider propositional quantification and think of classical logic we easily find out that the problem is trivial: the truth functional interpretation allows us to express quantification by means of propositional connectives and constants only. However, in case of nonclassical logics the situation is different-there propositional quantification usually gives rise to interesting extensions of the logic in question. The problem of propositional quantification was investigated in case of modal logics (see e.g., Fine [2], Bull [1], Ghilardi and Zawadowski [6], Kaplan [8], Kremer [11]), relevance logic (see Kremer [12]) and intuitionistic logic (see Gabbay [4], Scedrov [17], Kremer [10], Pitts [13], Połacik [16], Ghilardi and Zawadowski [5], Visser [22], Skvortsov [19]). The study of propositional quantification in intuitionistic logic is continued in this paper.We consider the Heyting calculus which corresponds to (the fragment of) intuitionistic propositional logic in the language of the standard propositional connectives: ¬, ∨, ∧, →. In this language, the constants , ⊥, and equivalence ≡ can be defined in the usual way. In the sequel, p, q, r, s, . . . will range over the set of propositional variables and the letters F, G, H . . . will serve as the metavariables for formulas. The symbol will be used to denote provability in Heyting calculus. We extend the language by adding propositional quantifiers ∃ p , ∀ p , . . .; the notions of formula (in the extended language) and free variable are as usual.One way of introducing propositional quantification into a propositional logic is to specify the characteristic properties of quantification in the form of axioms and

A one-premiss rule is said to be archetypal for a consequence relation when not only is the conclusion of any application of the rule a consequence (according to that relation) of the premiss, but whenever one formula has another as a consequence, these formulas are respectively equivalent to a premiss and a conclusion of some application of the rule. We are concerned here with the consequence relation of classical propositional logic and with the task of extending the above notion of archetypality to rules with more than one premiss, and providing an informative characterization of the set of rules falling under the more general notion.