(I am indebted to an anonymous reader for clarifying this point; see also the papers in the "Against the LNC" section of Priest et al. 2004).
For example, "P is happy", and "P is sad" are contraries and thus mutually inconsistent and cannot both be true (LNC).
While Sophists and "even many physicists" may claim that it is possible for the same thing to be and not to be at the same time and in the same respect, such a position self-destructs "if only our opponent says something", since as soon as he opens his mouth to make an assertion, any assertion, he must accept LNC.
As such, dialetheism opposes the so-called Law of Non-Contradiction (LNC) (sometimes also called the Law of Contradiction).
Heraclitus could not have literally rejected LNC, as he is often accused of (or praised for) doing, as his writings preceded the statement of that principle in Metaphysics Γ by well over a century.
Nevertheless, as we will see below, many paraconsistent logics validate the Law of Non-Contradiciton (LNC) (⊨ ¬(A ∧ ¬A)) even though they invalidate ECQ.
That is, P and ~P are mutually inconsistent and cannot be both true (LNC), and they are mutually exhaustive (LEM).
Hence meaningful language presupposes the LNC.
This may be seen as confirming that the Law of Identity is not a basic principle; after all, LNC and LEM advocates can retort, "Petition drive?
Not everyone has found the reductio arguments in Metaphysics Γ for the non-demonstrable status of LNC as convincing as Aristotle did (cf. Barnes 1969 and Dialetheism).
Yet this proposition can only constitute the serious paradox Kierkegaard intends if LNC is generally valid.
