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/ , and â€»; in any symbolism, a contradiction may be substituted for the truth value "false", as symbolized, for instance, by "0" (as is common in boolean algebra). It is not uncommon to see Q.E.D., or some of its variants, immediately after a contradiction symbol. In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.
When Emil Post, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of the propositional calculus (i.e. the logic) beyond that of Principia Mathematica (PM), he observed that with respect to a generalized set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"—such a notion might not be contained in the postulates.In dialectical materialism: Contradiction—as derived from Hegelianism—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to Marxist theory, such a contradiction can be found, for example, in the fact that.
This entry outlines the role of the law of non-contradiction (LNC) as the foremost among the first (indemonstrable) principles of Aristotelian philosophy and its heirs, and depicts the relation between LNC and LEM (the law of excluded middle) in establishing the nature of contradictory and contrary opposition. §1 presents the classical treatment of LNC as an axiom in Aristotle's “First Philosophy” and reviews the status of contradictory and contrary opposition as schematized on the Square of Opposition. §2 explores in further detail the possible characterizations of LNC and LEM, including the relevance of future contingent statements in which LEM (but not LNC) is sometimes held to fail. §3 addresses the mismatch between the logical status of contradictory negation as a propositional operator and the diverse realizations of contradictory negation within natural language. §4 deals with several challenges to LNC within Western philosophy, including the paradoxes, and the relation between systems with truth-value gaps (violating LEM) and those with truth-value gluts (violating LNC). In §5, the tetralemma of Buddhist logic is discussed within the context of gaps and gluts; it is suggested that apparent violations of LNC in this tradition (and others) can be attributed to either differing viewpoints of evaluation, as foreseen by Aristotle, or to intervening modal and epistemic operators. §6 focuses on the problem of “borderline contradictions”: the range of acceptability judgments for apparently contradictory sentences with vague predicates as surveyed in empirical studies, and the theoretical implications of these studies. Finally, §7 surveys the ways of contradiction and its exploitation in literature and popular culture from Shakespeare to social media. (Source: plato.stanford.edu)