Modal logics recently have found many applications in various fields, including theoretical and applied computer science, philosophy and linguistics ([1,2]). Traditional modal logics are usually based on classical predicate logic. However, classical logic has some fundamental restrictions which don’t allow taking into account sufficiently incompleteness, partiality and uncertainty of information. Composition-nominative logics of partial quasiary predicates (see [3,4]) is a program-oriented logical formalism based on wide classes of partial mappings over nominative data. Composition-nominative modal logics (CNML) combine facilities of traditional modal logics and composition-nominative logics. Their important variant, modal transitional logics (MTL), can adequately represent the fact of changing and evolution in subject domains. Traditional modal logics (alethic, temporal, deontic etc) can be easily considered within a scope of MTL. In this paper we introduce pure first-order MTL of partial quasiary predicates without monotonicity restriction. We define languages and semantic models of pure first-order MTL of non-monotone predicates and investigate their semantic properties, including properties of logical consequence relations for sets of formulas, specified with states. We distinguish various classes of MTL: multimodal, temporal, epistemic and general MTL. Significant difference between MTL of monotone and non-monotone predicates is demonstrated. Properties of logical consequence relations for sets of formulas, specified with states, are considered. Basing on these properties, corresponding sequent calculi can be constructed.