The principle of bivalence always implies the law of excluded middle, while the converse is not always true.
A commonly cited counterexample uses statements unprovable now, but provable in the future to show that the law of excluded middle may apply when the principle of bivalence fails.
[5] He also states it as a principle in the Metaphysics book 4, saying that it is necessary in every case to affirm or deny,[6] and that it is impossible that there should be anything between the two parts of a contradiction.
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.
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. 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 future contingents, in his discussion on the sea battle.
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)The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:
43–44).Russell reiterated his distinction between "sense-datum" and "sensation" in his book The Problems of Philosophy (1912), published at the same time as PM (1910–1913): Let us give the name of "sense-data" to the things that are immediately known in sensation: such things as colours, sounds, smells, hardnesses, roughnesses, and so on.
From the law of excluded middle, formula ✸2.1 in Principia Mathematica, Whitehead and Russell derive some of the most powerful tools in the logician's argumentation toolkit.
(In Principia Mathematica, formulas and propositions are identified by a leading asterisk and two numbers, such as "✸2.1".)
Propositions ✸2.12 and ✸2.14, "double negation": The intuitionist writings of L. E. J. Brouwer refer to what he calls "the principle of the reciprocity of the multiple species, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property" (Brouwer, ibid, p. 335).
Thus an example of the expression would look like this: From the late 1800s through the 1930s, a bitter, persistent debate raged between Hilbert and his followers versus Hermann Weyl and L. E. J. Brouwer.
Brouwer's philosophy, called intuitionism, started in earnest with Leopold Kronecker in the late 1800s.
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.
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.
But the debate was fertile: it resulted in Principia Mathematica (1910–1913), and that work gave a precise definition to 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: In his lecture in 1941 at Yale and the subsequent paper, Gödel proposed a solution: "that the negation of a universal proposition was to be understood as asserting the existence … of a counterexample" (Dawson, p. 157) Gödel's approach to the law of excluded middle was to assert that objections against "the use of 'impredicative definitions'" had "carried more weight" than "the law of excluded middle and related theorems of the propositional calculus" (Dawson p. 156).
Among them were a proof of the consistency with intuitionistic logic of the principle ~ (∀A: (A ∨ ~A)) (despite the inconsistency of the assumption ∃ A: ~ (A ∨ ~A))" (Dawson, p. 157) The debate seemed to weaken: mathematicians, logicians and engineers continue to use the law of excluded middle (and double negation) in their daily work.
Their difficulties with the law emerge: that they do not want to accept as true implications drawn from that which is unverifiable (untestable, unknowable) or from the impossible or the false.
In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle.
Such proofs presume the existence of a totality that is complete, a notion disallowed by intuitionists when extended to the infinite—for them the infinite can never be completed: In classical mathematics there occur non-constructive or indirect existence proofs, which intuitionists do not accept.
The classical logic allows this result to be transformed into there exists an n such that P(n), but not in general the intuitionistic … the classical meaning, that somewhere in the completed infinite totality of the natural numbers there occurs an n such that P(n), is not available to him, since he does not conceive the natural numbers as a completed totality.
[11] (Kleene 1952:49–50)David Hilbert and Luitzen E. J. Brouwer both give examples of the law of excluded middle extended to the infinite.
Thus intuitionists absolutely disallow the blanket assertion: "For all propositions P concerning infinite sets D: P or ~P" (Kleene 1952:48).
Certain resolutions of these paradoxes, particularly Graham Priest's dialetheism as formalised in LP, have the law of excluded middle as a theorem, but resolve out the Liar as both true and false.
The Catuṣkoṭi (tetralemma) is an ancient alternative to the law of excluded middle, which examines all four possible assignments of truth values to a proposition and its negation.
Many modern logic systems replace the law of excluded middle with the concept of negation as failure.
Mathematicians such as L. E. J. Brouwer and Arend Heyting have also contested the usefulness of the law of excluded middle in the context of modern mathematics.
[14] In modern mathematical logic, the excluded middle has been argued to result in possible self-contradiction.