Semantics for Two Second-Order Logical Systems: $\equiv$RRC* and Cocchiarella's RRC*

Abstract

We develop a set-theoretic semantics for Cocchiarella's second-order logical system ${\bf RRC^\ast}$. Such a semantics is a modification of the nonstandard sort of second-order semantics described, firstly, by Simms and later extended by Cocchiarella. We formulate a new second order logical system and prove its relative consistency. We call such a system ${\bf \equiv RRC^\ast}$ and construct its set-theoretic semantics. Finally, we prove completeness theorems for proper normal extensions of the two systems with respect to certain notions of validity provided by the semantics.