Semantics for first-order superposition logic

We investigate how the sentence choice semantics (SCS) for propositional superposition logic (PLS) developed in Tzouvaras (2018, Logic Journal of the IGPL, 26, 149-190) could be extended so as to successfully apply to first-order superposition logic (FOLS). There are two options for such an extension. The apparently more natural one is the formula choice semantics (FCS) based on choice functions for pairs of arbitrary formulas of the basis language. It is proved however that the universal instantiation scheme of first-order logic, (for all nu)phi(nu)->phi(t), is false, as a scheme of tautologies, with respect to FCS. This causes the total failure of FCS as a candidate semantics. Then we turn to the other option, which is a variant of SCS, since it uses again choice functions for pairs of sentences only. This semantics however presupposes that the applicability of the connective vertical bar is restricted to quantifier-free sentences, and thus the class of well-formed formulas and sentences of the language is restricted too. Granted these syntactic restrictions, the usual axiomatizations of FOLS turn out to be sound and conditionally complete with respect to this second semantics, just like the corresponding systems of PLS.