Editing Law of excluded middle (section)

=== Aristotle ===
The earliest known formulation is in Aristotle's discussion of the [[principle of non-contradiction]], first proposed in ''[[On Interpretation]],''<ref>P. T. Geach, The Law of Excluded Middle in ''Logic Matters,'' p. 74</ref> where he says that of two [[contradictory]] propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false.<ref>''On Interpretation'', c. 9</ref> He also states it as a principle in the ''[[Metaphysics (Aristotle)|Metaphysics]]'' book 4, saying that it is necessary in every case to affirm or deny,<ref>''Metaphysics'' ''B'' 2, 996b 26–30</ref> and that it is impossible that there should be anything between the two parts of a contradiction.<ref>''Metaphysics'' Γ 7, 1011b 26–27</ref>

[[Aristotle]] wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the facts themselves:

{{Blockquote|It is impossible, then, that "being a man" should mean precisely "not being a man", if "man" not only signifies something about one subject but also has one significance. … And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call "man", and others were to call "not-man"; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact. (''Metaphysics'' 4.4, W. D. Ross (trans.), GBWW 8, 525–526).}}

Aristotle's assertion that "it will not be possible to be and not to be the same thing" would be written in propositional logic as ~(''P'' ∧ ~''P''). In modern so called classical logic, this statement is equivalent to the law of excluded middle (''P'' ∨ ~''P''), through distribution of the negation in Aristotle's assertion. The former claims that no statement is ''both'' true and false, while the latter requires that any statement is ''either'' true or false.

But Aristotle also writes, "since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing" (Book IV, CH 6, p.&nbsp;531). He then proposes that "there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate" (Book IV, CH 7, p.&nbsp;531). In the context of Aristotle's [[traditional logic]], this is a remarkably precise statement of the law of excluded middle, ''P'' ∨ ~''P''.

Yet in ''On Interpretation'' Aristotle seems to deny the law of excluded middle in the case of [[Problem of future contingents|future contingents]], in his discussion on the sea battle.