Free-quantifier composition nominative logics of partial quasiary predicates are considered. We specify the following levels of these logics: renominative, renominative with predicates of weak equality, renominative with predicates of strong equality, free-quantifier, free-quantifier with composition of weak equality, free-quantifier with composition of strong equality. The paper is mainly dedicated to investigation ofNlogics of free-quantifier levels with equality. Languages and semantic models of such logics are described, their semantic properties are studied, in particular the properties of relations of logical consequence.
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