[In] the
Metaphysics, Aristotle introduced (what was later to be called) the Law of Non-Contradiction (LNC) as “the most certain of all principles” (1005b24)—
firmissimum omnium principiorum, as the Medieval theologians said. Since Aristotle, there have been few sustained attempts to
defend the law. The LNC has been an (often unstated) assumption, felt to be so fundamental to rationality that some claim it
cannot be defended . . . As a challenge to the LNC, therefore, dialetheism assails what most philosophers take to be unassailable common sense, calling into question the rules for what can be called into question . . . Since the advent of
paraconsistent logic in the second half of the twentieth century, dialetheism has been developed as a view in philosophical logic, with precise formal language. Dialetheism has been most famously advanced as a response to
logical paradoxes, in tandem with a paraconsistent logic. The view has been gaining, if not acceptance, the respect of other parties in the debate . . .
2.1 Dialetheism in Western Philosophy
Aristotle takes a number of the Presocratics to endorse dialetheism, and with apparent justification. For example, in Fragment 49a, Heraclitus says: “We step and do not step into the same rivers; we are and we are not” (Robinson, 1987, p. 35). Protagorean relativism may be expressed by the view that man is the measure of all things; but according to Aristotle, since “Many men hold beliefs in which they conflict with one another”, it follows that “the same thing must be and not be” (1009a10–12). The Presocratic views triggered Aristotle’s attack in
Metaphysics. Chapter 4 of this Book contains Aristotle’s defence of the LNC. As we said above, historically Aristotle was almost completely successful: the LNC has been orthodoxy in Western Philosophy ever since.