In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. Symbolically expressed, the law is p ∨ ¬ p {\displaystyle p\lor \neg p} . The law of the excluded middle is also known as the law/principle of the excluded third, in Latin principium tertii exclusi. Another Latin designation for the law is tertium non datur or "no third [possibility] is given". In classical logic, the law of the excluded middle is taken as a tautology. Intuitionistic logic, by contrast, does not affirm the law. To prove p ∨ ¬ p {\displaystyle p\lor \neg p} in intuitionist logic, it is necessary to prove either p {\displaystyle p} or ¬ p {\displaystyle \neg p} .
History
Aristotle William Hamilton writes in a history of the so-called laws of thought:
The law of Excluded Middle between two contradictories remounts, as I have said, also to Plato, though the Second Alcibiades, the dialogue in which it is most clearly expressed, must be admitted to be spurious. It is also in the fragments of Pseudo-Archytas, to be found in Stobæus. [Hamilton LECT. V. LOGIC. 65] Hamilton further observes that "It is explicitly and emphatically enounced by Aristotle in many passages both of his Metaphysics (l. iii. (iv.) c.7.) and of his Analytics, both Prior (l. i. c. 2) and Posterior (1. i. c. 4). In the first of these, he says: "It is impossible that there should exist any medium between contradictory opposites, but it is necessary either to affirm or to deny everything of everything." [Hamilton LECT. V. LOGIC. 65]
The Sea Battle Yet in On Interpretation, Book 9, Aristotle seems to deny the law of excluded middle in the case of future contingents, in his discussion on the sea battle. It would seem to entail fatalism or logical determinism; and for this reason, the Stoics like Chrysippus affirmed it and embraced fatalism. Epicureans denied the law of excluded middle for this reason. Some take Aristotle to be more strictly denying the principle of bivalence, which states that every proposition is either true or false. However, the principle of bivalence always implies the law of excluded middle.
Leibniz Its usual form, "Every judgment is either true or false" [footnote 9] …"(from Kolmogorov in van Heijenoort, p. 421) footnote 9: "This is Leibniz's very simple formulation (see Nouveaux Essais, IV,2)" (ibid p 421)
Russell and Whitehead The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: ✸2.1 ~p ∨ p
Formalists versus Intuitionists From the late 1800s through the 1930s, Hilbert and his followers had a bitter, persistent debate with Hermann Weyl and L. E. J. Brouwer. Brouwer's philosophy, called intuitionism, started in earnest with Leopold Kronecker in the late 1800s. Hilbert intensely disliked Kronecker's ideas:
Kronecker insisted that there could be no existence without construction. For him, as for Paul Gordan [another elderly mathematician], Hilbert's proof of the finiteness of the basis of the invariant system was simply not mathematics. Hilbert, on the other hand, throughout his life was to insist that if one can prove that the attributes assigned to a concept will never lead to a contradiction, the mathematical existence of the concept is thereby established (Reid p. 34) It was his [Kronecker's] contention that nothing could be said to have mathematical existence unless it could actually be constructed with a finite number of positive integers (Reid p. 26) The debate had a profound effect on Hilbert. Reid indicates that Hilbert's second problem (one of Hilbert's problems from the Second International Conference in Paris in 1900) evolved from this debate (italics in the original):
In his second problem, [Hilbert] had asked for a mathematical proof of the consistency of the axioms of the arithmetic of real numbers. To show the significance of this problem, he added the following observation: "If contradictory attributes be assigned to a concept, I say that mathematically the concept does not exist" (Reid p. 71) Thus, Hilbert was saying: "If p and ~p are both shown to be true, then p does not exist", invoking the law of excluded middle in the form of the law of contradiction.
And finally constructivists … restricted mathematics to the study of concrete operations on finite or potentially (but not actually) infinite structures; completed infinite totalities … were rejected, as were indirect proof based on the Law of Excluded Middle. Most radical among the constructivists were the intuitionists, led by the erstwhile topologist L. E. J. Brouwer (Dawson p. 49) The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about "polemicizing against it [intuitionism] in sneering tones" (Brouwer in van Heijenoort, p. 492). But the debate was fertile: it resulted in Principia Mathematica (1910–1913), which precisely defined the law of excluded middle, and all this provided an intellectual setting and the tools necessary for the mathematicians of the early 20th century:
Out of the rancor, and spawned in part by it, there arose several important logical developments; Zermelo's axiomatization of set theory (1908a), that was followed two years later by the first volume of Principia Mathematica, in which Russell and Whitehead showed how, via the theory of types: much of arithmetic could be developed by logicist means (Dawson p. 49) Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof:
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