Law of excluded middle

Not to be confused with fallacy of the excluded middle.

This article uses forms of logical notation. For a concise description of the symbols used in this notation, see List of logic symbols.

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.

The law is also known as the law (or principle) of the excluded third, in Latin principium tertii exclusi. Yet another Latin designation for this law is tertium non datur: "no third (possibility) is given".

The earliest known formulation is Aristotle's principle of non-contradiction, first proposed in On Interpretation, 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. He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny, and that it is impossible that there should be anything between the two parts of a contradiction. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

$\mathbf{*2\cdot11}. \ \ \vdash . \ p \ \vee \thicksim p$.

The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false.

Classic laws of thought

The principle of excluded middle, along with its complement, the law of contradiction (the second of the three classic laws of thought), are correlates of the law of identity (the first of these laws).

Analogous laws

Some systems of logic have different but analogous laws. For some finite n-valued logics, there is an analogous law called the law of excluded n+1th. If negation is cyclic and "∨" is a "max operator", then the law can be expressed in the object language by (P ∨ ~P ∨ ~~P ∨ ... ∨ ~...~P), where "~...~" represents n−1 negation signs and "∨ ... ∨" n−1 disjunction signs. It is easy to check that the sentence must receive at least one of the n truth values (and not a value that is not one of the n).

Other systems reject the law entirely.

Examples

For example, if P is the proposition:

Socrates is mortal.

then the law of excluded middle holds that the logical disjunction:

Either Socrates is mortal, or it is not the case that Socrates is mortal.

is true by virtue of its form alone. That is, the "middle" position, that Socrates is neither mortal nor not-mortal, is excluded by logic, and therefore either the first possibility (Socrates is mortal) or its negation (it is not the case that Socrates is mortal) must be true.

An example of an argument that depends on the law of excluded middle follows. We seek to prove that there exist two irrational numbers $a$ and $b$ such that

$a^b$ is rational.

It is known that $\sqrt{2}$ is irrational (see proof). Consider the number

$\sqrt{2}^{\sqrt{2}}$.

Clearly (excluded middle) this number is either rational or irrational. If it is rational, the proof is complete, and

$a=\sqrt{2}$ and $b=\sqrt{2}$.

But if $\sqrt{2}^{\sqrt{2}}$ is irrational, then let

$a=\sqrt{2}^{\sqrt{2}}$ and $b=\sqrt{2}$.

Then

$a^b = \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = \sqrt{2}^{\left(\sqrt{2}\cdot\sqrt{2}\right)} = \sqrt{2}^2 = 2$,

and 2 is certainly rational. This concludes the proof.

In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle. An intuitionist, for example, would not accept this argument without further support for that statement. This might come in the form of a proof that the number in question is in fact irrational (or rational, as the case may be); or a finite algorithm that could determine whether the number is rational.

The Law in non-constructive proofs over the infinite

The above proof is an example of a non-constructive proof disallowed by intuitionists:

The proof is non-constructive because it doesn't give specific numbers $a$ and $b$ that satisfy the theorem but only two separate possibilities, one of which must work. (Actually $a=\sqrt{2}^{\sqrt{2}}$ is irrational but there is no known easy proof of that fact.) (Davis 2000:220)

By non-constructive Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would have to provide a method to exhibit explicitly the entities in question." (p. 85). 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. For example, to prove there exists an n such that P(n), the classical mathematician may deduce a contradiction from the assumption for all n, not P(n). Under both the classical and the intuitionistic logic, by reductio ad absurdum this gives not for all n, not P(n). 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. (Kleene 1952:49–50)

Indeed, Hilbert and Brouwer both give examples of the law of excluded middle extended to the infinite. Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and Brouwer's: "Every mathematical species is either finite or infinite." (Brouwer 1923 in van Heijenoort 1967:336).

In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g. the natural numbers). Thus intuitionists absolutely disallow the blanket assertion: "For all propositions P concerning infinite sets D: P or ~P" (Kleene 1952:48).

For more about the conflict between the intuitionists (e.g. Brouwer) and the formalists (Hilbert) see Foundations of mathematics and Intuitionism.

Putative counterexamples to the law of excluded middle include the liar paradox or Quine's Paradox. 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. In this way, the law of excluded middle is true, but because truth itself, and therefore disjunction, is not exclusive, it says next to nothing if one of the disjuncts is paradoxical, or both true and false.

History

Aristotle

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

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", which would be written in propositional logic as ¬ (P ∧ ¬P), is a statement modern logicians could call the law of excluded middle (P ∨ ¬P), as distribution of the negation of Aristotle's assertion makes them equivalent, regardless that the former claims that no statement is both true and false, while the latter requires that any statement is either true or false.

However, 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. 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. 531). In the context of Aristotle's traditional logic, this is a remarkably precise statement of the law of excluded middle, P ∨ ¬P.

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)

Bertrand Russell and Principia Mathematica

Bertrand Russell asserts a distinction between the "law of excluded middle" and the "law of noncontradiction". In The Problems of Philosophy, he cites three "Laws of Thought" as more or less "self-evident" or "a priori" in the sense of Aristotle:

1. Law of identity: "Whatever is, is."

2. Law of noncontradiction: "Nothing can both be and not be."

3. Law of excluded middle: "Everything must either be or not be."

These three laws are samples of self-evident logical principles... (p. 72)

It is correct, at least for bivalent logic—i.e. it can be seen with a Karnaugh map—that Russell's Law (2) removes "the middle" of the inclusive-or used in his law (3). And this is the point of Reichenbach's demonstration that some believe the exclusive-or should take the place of the inclusive-or.

About this issue (in admittedly very technical terms) Reichenbach observes:

The tertium non datur

29. (x)[f(x) ∨ ~f(x)]

is not exhaustive in its major terms and is therefore an inflated formula. This fact may perhaps explain why some people consider it unreasonable to write (29) with the inclusive-'or', and want to have it written with the sign of the exclusive-'or'

30. (x)[f(x) ⊕ ~f(x)], where the symbol "⊕" signifies exclusive-or

in which form it would be fully exhaustive and therefore nomological in the narrower sense. (Reichenbach, p. 376)

In line (30) the "(x)" means "for all" or "for every", a form used by Russell and Reichenbach; today the symbolism is usually $\forall$ x. Thus an example of the expression would look like this:

• (pig): (Flies(pig) ⊕ ~Flies(pig))
• (For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously)

A formal definition from Principia Mathematica

Principia Mathematica (PM) defines the law of excluded middle formally:

*2.1 : ~p ∨ p (PM p. 101) Example: Either it is true that "this is red", or it is true that "this is not red". Hence it is true that "this is red or this is not red". (See below for more about how this is derived from the primitive axioms).

So just what is "truth" and "falsehood"? At the opening PM quickly announces some definitions:

Truth-values. The "truth-values" of a proposition is truth if it is true and falsehood if it is false* [*This phrase is due to Frege]...the truth-value of "p ∨ q" is truth if the truth-value of either p or q is truth, and is falsehood otherwise ... that of "~ p" is the opposite of that of p..." (p. 7-8)

This is not much help. But later, in a much deeper discussion, ("Definition and systematic ambiguity of Truth and Falsehood" Chapter II part III, p. 41 ff ) PM defines truth and falsehood in terms of a relationship between the "a" and the "b" and the "percipient". For example "This 'a' is 'b'" (e.g. "This 'object a' is 'red'") really means "'object a' is a sense-datum" and "'red' is a sense-datum", and they "stand in relation" to one another and in relation to "I". Thus what we really mean is: "I perceive that 'This object a is red'" and this is an undeniable-by-3rd-party "truth".

PM further defines a distinction between a "sense-datum" and a "sensation":

That is, when we judge (say) "this is red", what occurs is a relation of three terms, the mind, and "this", and "red". On the other hand, when we perceive "the redness of this", there is a relation of two terms, namely the mind and the complex object "the redness of this" (pp. 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. We shall give the name "sensation" to the experience of being immediately aware of these things... The colour itself is a sense-datum, not a sensation. (p. 12)

Russell further described his reasoning behind his definitions of "truth" and "falsehood" in the same book (Chapter XII Truth and Falsehood).

Consequences of the law of excluded middle in Principia Mathematica

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".)

✸2.1 ~p ∨ p "This is the Law of excluded middle" (PM, p. 101).

The proof of ✸2.1 is roughly as follows: "primitive idea" 1.08 defines p → q = ~p ∨ q. Substituting p for q in this rule yields p → p = ~p ∨ p. Since p → p is true (this is Theorem 2.08, which is proved separately), then ~p ∨ p must be true.

✸2.11 p ∨ ~p (Permutation of the assertions is allowed by axiom 1.4)
✸2.12 p → ~(~p) (Principle of double negation, part 1: if "this rose is red" is true then it's not true that "'this rose is not-red' is true".)
✸2.13 p ∨ ~{~(~p)} (Lemma together with 2.12 used to derive 2.14)
✸2.14 ~(~p) → p (Principle of double negation, part 2)
✸2.15 (~p → q) → (~q → p) (One of the four "Principles of transposition". Similar to 1.03, 1.16 and 1.17. A very long demonstration was required here.)
✸2.16 (p → q) → (~q → ~p) (If it's true that "If this rose is red then this pig flies" then it's true that "If this pig doesn't fly then this rose isn't red.")
✸2.17 ( ~p → ~q ) → (q → p) (Another of the "Principles of transposition".)
✸2.18 (~p → p) → p (Called "The complement of reductio ad absurdum. It states that a proposition which follows from the hypothesis of its own falsehood is true" (PM, pp. 103–104).)

Most of these theorems—in particular ✸2.1, ✸2.11, and ✸2.14—are rejected by intuitionism. These tools are recast into another form that Kolmogorov cites as "Hilbert's four axioms of implication" and "Hilbert's two axioms of negation" (Kolmogorov in van Heijenoort, p. 335).

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).

This principle is commonly called "the principle of double negation" (PM, pp. 101–102). From the law of excluded middle (✸2.1 and ✸2.11), PM derives principle ✸2.12 immediately. We substitute ~p for p in 2.11 to yield ~p ∨ ~(~p), and by the definition of implication (i.e. 1.01 p → q = ~p ∨ q) then ~p ∨ ~(~p)= p → ~(~p). QED (The derivation of 2.14 is a bit more involved.)

Use in computer science proofs

The law of excluded middle can be used to prove the decidability of certain computational problems. Usually, decidability is proved by showing an algorithm that solves the problem (i.e. a constructive proof). However, in some cases it is possible to prove that a problem is decidable without showing an algorithm that solves it.

For example, consider the following constant function f:

$\qquad \displaystyle f = \begin{cases} 1 & \text{if Goldbach's conjecture is true} \\ 0 & \text{else}\end{cases}$

By the law of excluded middle, Goldbach's conjecture is either true or false. If it is true then f is 1, and the required algorithm is just "print 1". If it is false then the required algorithm is just "print 0". In either case, there is a simple, one-line algorithm that prints f, so by definition, f is computationally decidable. It is true that we don't know which algorithm to use, but we do know that an algorithm exists.

A slightly more complicated example is:

$\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$

The function f is computable because, by the Law of Excluded Middle, there are only two possibilities to consider:

• For every positive integer n, the string $0^n$ appears in the decimal representation of $\pi$. In this case, the algorithm that always returns 1 is always correct.
• There is a largest integer N such that $0^N$ appears in the decimal representation of $\pi$. In this case the following algorithm (with the value $N$ hard-coded) is always correct:

Zeros-in-pi(n):

if (n > N) then return 0 else return 1

We have no idea which of these possibilities is correct, or what value of N is the right one in the second case. Nevertheless, one of these algorithms is guaranteed to be correct. Thus, there is an algorithm to decide whether a string of n zeros appears in $\pi$; the problem is decidable.

Criticisms

Many modern logic systems replace the law of excluded middle with the concept of negation as failure. Instead of a proposition either being true or false, a proposition is either true or not able to be proven true. These two dichotomies only differ in logical systems that are not complete. The principle of negation as failure is used as a foundation for autoepistemic logic, and is widely used in logic programming. In these systems, the programmer is free to assert the law of excluded middle as a true fact, but it is not built-in a priori into these systems.

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.

See also

• Consequentia mirabilis
• Diaconescu's theorem
• Intuitionistic logic
• Law of bivalence
• Law of excluded fourth
• Law of excluded middle is untrue in many-valued logics such as ternary logic and fuzzy logic
• Laws of thought
• Liar paradox
• Logical graphs: a graphical syntax for propositional logic
• Peirce's law: another way of turning intuition classical
• Logical determinism - the application excluded middle to modal propositions

References

• Aquinas, Thomas, "Summa Theologica", Fathers of the English Dominican Province (trans.), Daniel J. Sullivan (ed.), vols. 19–20 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, IL, 1952. Cited as GB 19–20.
• Aristotle, "Metaphysics", W.D. Ross (trans.), vol. 8 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, IL, 1952. Cited as GB 8. 1st published, W.D. Ross (trans.), The Works of Aristotle, Oxford University Press, Oxford, UK.
• Martin Davis 2000, Engines of Logic: Mathematicians and the Origin of the Computer", W. W. Norton & Company, NY, ISBN 0-393-32229-7 pbk.
• Dawson, J., Logical Dilemmas, The Life and Work of Kurt Gödel, A.K. Peters, Wellesley, MA, 1997.
• van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.
• Luitzen Egbertus Jan Brouwer, 1923, On the significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort]
• Andrei Nikolaevich Kolmogorov, 1925, On the principle of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]
• Luitzen Egbertus Jan Brouwer, 1927, On the domains of definitions of functions,[reprinted with commentary, p. 446, van Heijenoort] Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
• Luitzen Egbertus Jan Brouwer, 1927(2), Intuitionistic reflections on formalism,[reprinted with commentary, p. 490, van Heijenoort]
• Stephen C. Kleene 1952 original printing, 1971 6th printing with corrections, 10th printing 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9.
• Kneale, W. and Kneale, M., The Development of Logic, Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted). Extremely difficult because of arcane symbolism, but a must-have for serious logicians.
• Bertrand Russell, An Inquiry Into Meaning and Truth. The William James Lectures for 1940 Delivered at Harvard University.
• Bertrand Russell, The Problems of Philosophy, With a New Introduction by John Perry, Oxford University Press, New York, 1997 edition (first published 1912). Very easy to read: Russell was a wonderful writer.
• Bertrand Russell, The Art of Philosophizing and Other Essays, Littlefield, Adams & Co., Totowa, NJ, 1974 edition (first published 1968). Includes a wonderful essay on "The Art of drawing Inferences".
• Hans Reichenbach, Elements of Symbolic Logic, Dover, New York, 1947, 1975.
• Tom Mitchell, Machine Learning, WCB McGraw-Hill, 1997.
• Constance Reid, Hilbert, Copernicus: Springer-Verlag New York, Inc. 1996, first published 1969. Contains a wealth of biographical information, much derived from interviews.
• Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993. Fuzzy thinking at its finest. But a good introduction to the concepts.
• David Hume, An Inquiry Concerning Human Understanding, reprinted in Great Books of the Western World Encyclopædia Britannica, Volume 35, 1952, p. 449 ff. This work was published by Hume in 1758 as his rewrite of his "juvenile" Treatise of Human Nature: Being An attempt to introduce the experimental method of Reasoning into Moral Subjects Vol. I, Of The Understanding first published 1739, reprinted as: David Hume, A Treatise of Human Nature, Penguin Classics, 1985. Also see: David Applebaum, The Vision of Hume, Vega, London, 2001: a reprint of a portion of An Inquiry starts on p. 94 ff

External links

• "Contradiction" entry in the Stanford Encyclopedia of Philosophy