Antinomy

Antinomy (Greek αντι-, against, plus νομος, law) literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology.

The term acquired a special significance in the philosophy of Immanuel Kant, who used it to describe the equally rational but contradictory results of applying to the universe of pure thought the categories or criteria of reason proper to the universe of sensible perception or experience (phenomena). Reason cannot here play the role of establishing rational truths because it goes beyond possible experience and is applied to the sphere of that which transcends it.

For Kant there are four antinomies connected with about each of which pure reason contradicts the empirical, as thesis and antithesis. This was part of Kant's critical program of determining limits to science and philosophical inquiry. Kant claimed to solve these contradictions by saying, that in no case is the contradiction real, however really it has been intended by the opposing partisans, or must appear to the mind without critical enlightenment. It is wrong, therefore, to impute to Kant, as is often done, the view that human reason is, on ultimate subjects, at war with itself, in the sense of being impelled by equally strong arguments towards alternatives contradictory of each other. The difficulty arises from a confusion between the spheres of phenomena and noumena. In fact no rational cosmology is possible.
 * 1) the limitation of the universe in respect of space and time,
 * 2) the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist),
 * 3) the problem of freedom in relation to universal causality,
 * 4) the existence of a necessary being

It can also be argued that antinomies do not highlight limitations in the power of logical reasoning. This is because the conclusion that there is a limitation is (supposedly) derived from the antinomy by logical reasoning; therefore any limitation in the validity of logical reasoning imposes a limitation on the conclusion that there is a limitation on logical reasoning. (This is an argument by self-reference.) In short, in terms of the validity of logical reasoning as a whole, antinomies are self-isolating: they are like scattered discontinuities within the field of logic, incapable of casting doubt on anything else but themselves.

This carefree position is incompatible with the principle of explosion. In mathematical logic, antinomies are patently not self-isolating, and are usually seen as disasters for the formal system in which they arise (as Russell's paradox in Frege's work).