Łukasiewicz on the Principle of Contradiction

Abstract
Łukasiewicz distinguishes three formulations of the principle of contradiction in Aristotle’s works: ontological, logical, and psychological. The first two formulations are equivalent though not synonymous, but neither of them is equivalent to the psychological one, which expresses not a principle but only an empirical law. Furthermore, the principle of contradiction is neither a simple and ultimate law nor is it necessary for conducting an inference, because the syllogism is independent of it. The further explanation of this concept leads Łukasiewicz to formulate the idea of a non-Aristotelian logic, that is, a logic operating without the principle of contradiction. If the principle of contradiction shall be valid, it must be proved. A proof can be supplied only on the basis of a definition of object, as something that cannot have and not have the same property at the same time. However, this definition does not hold for all objects, i.e., for contradictory objects. In virtue of its ontological character the Aristotelian principle of contradiction is then different from that of symbolic logic
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