Program-oriented logics defined for classes of quasiary predicates are studied. Such predicates are partial predicates over partial states (partial assignments) of variables. Conventional n-ary predicates can be considered as a special case of quasiary predicates. We define extended first-order logics of quasiary nondeterministic predicates. A special consequence relation, adequate for such logics, is introduced and its semantic properties are studied. Obtained results are used to prove logic validity and completeness.