This chapter deals with attempts, contemporary with Vasil'ev's own, to develop non-Aristotelian logics that present affinities with imaginary logic. Already in Aristotle's work there are passages that press in the direction of a non-Aristotelian logic, in so far as they show that the syllogism is independent of the principle of contradiction. Some Aristotelian scholars like Heinrich Maier and Isaac Husik had drawn attention to such passages. Husik in particular proposes, on the basis of them and of Herbert Spencer's philosophy, a hypothetical logic in which the syllogism is independent of the principle of contradiction; judgments are allowed that assert contradictory predicates of the same subject; contradictory objects are subjects of true propositions; and a hypothetical world is assumed, for which such a different logic would be valid. Jan Łukasiewicz was familiar with Maier and Husik's works. He subjects to rigorous critique the Aristotelian principle of contradiction, claiming that it is uncertain, that it is not a simple, ultimate and necessary principle, and that in relation to contradictory objects it is actually false. Łukasiewicz took the notion of contradictory objects from Meinong, according to whom such objects — which are overdetermined objects of higher order in which a surplus of determinations inheres, amongst which there is a relation of incompatibility — can occur as genuine subjects in true propositions. The chapter concludes with an outline of the controversy between Meinong and Russell, with which Łukasiewicz was thoroughly acquainted, and his proposal of a non-Aristotelian logic in which the principle of contradiction is insignificant.