Online research in philosophy

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 $\mapsto \exists$ p (q (...) $\leftrightarrow$ 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. (shrink)

We consider propositional operators defined by propositional quantification in intuitionistic logic. More specifically, we investigate the propositional operators of the formA* :p q(p A(q)) whereA(q) is one of the following formulae: (¬¬q q) V ¬¬q, (¬¬q q) (¬¬q V ¬q), ((¬¬q q) (¬¬q V ¬q)) ((¬¬q q) V ¬¬q). The equivalence ofA*(p) to ¬¬p is proved over the standard topological interpretation of intuitionistic second order propositional logic over Cantor space.We relate topological interpretations of second order intuitionistic propositional logic over Cantor (...) space with the interpretation of propositional quantifiers (as the strongest and weakest interpolant in Heyting calculus) suggested by A. Pitts. One of the merits of Pitts' interpretation is shown to be valid for the interpretation over Cantor space. (shrink)