Rasiowa-Sikorski proof system for the non-Fregean sentential logic SCI

The sentential logic SCI is obtained from the classical sentential calculus by adding a new identity connective ≡ (different from ↔) and axioms which say ”α ≡ β” means ”α is identical to β” . From the axioms for ≡ it follows that the range of sentences has at least two elements. No other special presupposition about the meaning of ’is identical to’ nor the range of sentences are assumed. Any additional conditions for the range or the nature of connectives lead to an extension of SCI. In other words, SCI seems to be as weak as it is possible. Most of known sentential calculus classical, modal, intuitionistic are extensions of SCI. In this section we present the basic definitions of the non-Fregean sentential logic. They can be found in [1], [7] and [8], among others.