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So Plato's law of non-contradiction is the empirically derived necessary starting point for all else he has to say. [13] In contrast, Aristotle reverses Plato's order of derivation. Rather than starting with experience, Aristotle begins a priori with the law of non-contradiction as the fundamental axiom of an analytic philosophical system. [14]
The expressions "law of non-contradiction" and "law of excluded middle" are also used for semantic principles of model theory concerning sentences and interpretations: (NC) under no interpretation is a given sentence both true and false, (EM) under any interpretation, a given sentence is either true or false.
Formally the law of non-contradiction is written as ¬(P ∧ ¬P) and read as "it is not the case that a proposition is both true and false". The law of non-contradiction neither follows nor is implied by the principle of Proof by contradiction. The laws of excluded middle and non-contradiction together mean that exactly one of P and ¬P is true.
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In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion [a] [b] is the law according to which any statement can be proven from a contradiction. [1] [2] [3] That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion. [4] [5]
Necessary truths can be derived from the law of identity (and the principle of non-contradiction): "Necessary truths are those that can be demonstrated through an analysis of terms, so that in the end they become identities, just as in Algebra an equation expressing an identity ultimately results from the substitution of values [for variables ...
The difference between the principle of bivalence and the law of excluded middle is important because there are logics that validate the law but not the principle. [2] For example, the three-valued Logic of Paradox (LP) validates the law of excluded middle, and yet also validates the law of non-contradiction , ¬(P ∧ ¬P), and its intended ...
law of non-contradiction A fundamental principle of classical logic stating that contradictory statements cannot both be true in the same sense at the same time. left field See domain. Leibniz's Law The principle of the identity of indiscernibles, stating that if two entities share all the same properties, then they are identical. lemma