proof by contradiction

In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. It is an example of the weaker logical refutation '' reductio ad absurdum''.

A mathematical proof employing proof by contradiction usually proceeds as follows:

#The proposition to be proved is ''P''.
#We assume ''P'' to be false, i.e., we assume ''¬P''.
#It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, ''Q'' and ''¬Q'', and appealing to the Law of noncontradiction.
#Since assuming ''P'' to be false leads to a contradiction, it is concluded that ''P'' is in fact true.

An important special case is the existence proof by contradiction: in order to demonstrate the existence of an object satisfying a given property, we assume that no such object exists and derive a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid.

Formalization

The principle may be formally expressed as the propositional formula ''¬¬P ⇒ P'', equivalently ''(¬P ⇒ ⊥) ⇒ P'', which reads: "If assuming ''P'' to be false implies falsehood, then ''P'' is true."

In natural deduction the principle takes the form of the rule of inference

: $\cfrac$

which reads: "If $\lnot\lnot P$ is proved, then $P$ may be concluded."

In sequent calculus the principle is expressed by the sequent

: $\Gamma, \lnot\lnot P \vdash P, \Delta$

which reads: "Hypotheses $\Gamma$ ''and'' $\lnot\lnot P$ entail the conclusion $P$ ''or'' $\Delta$."

Justification

In classical logic the principle may be justified by the examination of the truth table of the proposition ''¬¬P ⇒ P'', which demonstrates it to be a tautology:

Another way to justify the principle is to derive it from the Law of the excluded middle, as follows. We assume ''¬¬P'' and seek to prove ''P''. By the law of excluded middle ''P'' either holds or it does not:

# if ''P'' holds, then of course ''P'' holds.
# if ''¬P'' holds, then we derive falsehood by applying the law of noncontradiction to ''¬P'' and ''¬¬P'', after which the principle of explosion allows us to conclude ''P''.

In either case, we established ''P''. It turns out that, conversely, proof by contradiction can be used to derive the law of excluded middle.

In classical sequent calculus LK proof by contradiction is derivable from the inference rules for negation:

: $\cfrac \; (L)$

Relationship with other proof techniques

Refutation by contradiction

Proof by contradiction is frequently confused with the similar-looking refutation by contradiction, a.k.a. proof of negation, which states that ''¬P'' is proved as follows:

# The proposition to be proved is ''¬P''.
# Assume ''P''.
# Derive falsehood.
# Conclude ''¬P''.

In contrast, proof by contradiction proceeds as follows:

# The proposition to be proved is ''P''.
# Assume ''¬P''.
# Derive falsehood.
# Conclude ''P''.

Refutation by contradiction applies only when the proposition to be proved is negated, whereas proof by contradiction may be applied to any proposition whatsoever.

Confusingly, in mathematical practice, both proof by contradiction and refutation by contradiction are referred to as "proof by contradiction", as they proceed in a similar manner. However, they are not equivalent—unless we assume the law of excluded middle, in which case they become equivalent.

Law of non-contradiction

The law of noncontradiction was first formalized as a metaphysical principle by Aristotle. It states that a proposition and its negation cannot both be true. Equivalently, it states that a proposition cannot be both true and false. Formally the law of non-contradiction is written as ''¬(Q ∧ ¬Q)'' 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.

Law of the excluded middle

Proof by contradiction is equivalent to the law of the excluded middle, also first formulated by Aristotle, which states that either an assertion or its negation is true, ''P ∨ ¬P''.
Combined with the principle of noncontradiction, this means that exactly one of ''P'' and ''¬P'' is true.

Proof by contradiction in intuitionistic logic

In intuitionistic logic proof by contradiction is not generally valid, although some particular instances can be derived. In contrast, proof of negation and principle of noncontradiction are both intuitionistically valid.

Brouwer–Heyting–Kolmogorov interpretation of proof by contradiction gives the following intuitionistic validity condition:

:"If there is no method for establishing that a proposition is false, then there is a method for establishing that the proposition is true."

If we take "method" to mean algorithm, then the condition is not acceptable, as it would allow us to solve the Halting problem. To see how, consider the statement ''H(M)'' stating "Turing machine ''M'' halts or does not halt". Its negation ''¬H(M)'' states that "''M'' neither halts nor does not halt", which is false by the law of noncontradiction (which is intuitionistically valid). If proof by contradiction were intuitionistically valid, we would obtain an algorithm for deciding whether an arbitrary Turing machine ''M'' halts, thereby violating the (intuitionistically valid) proof of non-solvability of the Halting problem.

A proposition ''P'' which satisfies $\lnot\lnot P \Rightarrow P$ is known as a ''¬¬-stable proposition''. Thus in intuitionistic logic proof by contradiction is not universally valid, but can only be applied to the ¬¬-stable propositions. An instance of such a proposition is a decidable one, i.e., satisfying $P \lor \lnot P$. Indeed, the above proof that the law of excluded middle implies proof by contradiction can be repurposed to show that a decidable proposition is ¬¬-stable. A typical example of a decidable proposition is a statement that can be checked by direct computation, such as "$n$ is prime" or "$a$ divides $b$".

Examples of proofs by contradiction

Euclid's Elements

An early occurrence of proof by contradiction can be found in Euclid's Elements, Book 1, Proposition 6:

: If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

The proof proceeds by assuming that the opposite angles are not equal, and derives a contradiction.

Hilbert's Nullstellensatz

An influential proof by contradiction was given by David Hilbert. His
Nullstellensatz states:

: If $f_1,\ldots,f_k$ are polynomials in indeterminates with complex coefficients, which have no common complex zeros, then there are polynomials $g_1,\ldots, g_k$ such that $f_1g_1+\ldots +f_kg_k=1.$

Hilbert proved the statement by assuming that there are no such polynomials $g_1, \ldots, g_k$ and derived a contradiction.

Infinitude of primes

Euclid's theorem states that there are infinitely many primes. In Euclid's Elements the theorem is stated in Book IX, Proposition 20:

: Prime numbers are more than any assigned multitude of prime numbers.

Depending on how we formally write the above statement, the usual proof takes either the form of a proof by contradiction or a refutation by contradiction. We present here the former, see below how the proof is done as refutation by contradiction.

If we formally express Euclid's theorem as saying that for every natural number $n$ there is a prime bigger than it, then we employ proof by contradiction, as follows.

Given any number $n$, we seek to prove that there is a prime larger than $n$. Suppose to the contrary that no such ''p'' exists (an application of proof by contradiction). Then all primes are smaller than or equal to $n$, and we may form the list $p_1, \ldots, p_k$ of them all. Let $P = p_1 \cdot \ldots \cdot p_k$ be the product of all primes and $Q = P + 1$. Because $Q$ is larger than all prime numbers it is not prime, hence it must be divisible by one of them, say $p_i$. Now both $P$ and $Q$ are divisible by $p_i$, hence so is their difference $Q - P = 1$, but this cannot be because 1 is not divisible by any primes. Hence we have a contradiction and so there is a prime number bigger than $n$

Examples of refutations by contradiction

Because proof by contradiction is frequently conflated with refutation by contradiction, the following are often erroneously claimed to be proofs by contradiction, but they really are refutations by contradiction (and therefore intuitionistically valid).

Infinitude of primes

Let us take a second look at Euclid's theorem – Book IX, Proposition 20:

: Prime numbers are more than any assigned multitude of prime numbers.

We may read the statement as saying that for every finite list of primes, there is another prime not on that list,
which is arguably closer to and in the same spirit as Euclid's original formulation. In this case Euclid's proof applies refutation by contradiction at one step, as follows.

Consider any finite list of prime numbers $p_1, \ldots, p_n$. It will be shown that at least one additional prime number not in this list exists. Let $P = p_1 \cdot p_2 \cdots p_n$ be the product of all the prime numbers in the list and $Q = P + 1$. Then $Q$ is either prime or not:

* If $Q$ is prime, then $Q$ itself is a prime not in the list, as it is greater than all primes in the list.
* If $Q$ is not prime, then it has a prime factor $p$. We claim that $p$ is not in the given list of primes. Suppose to the contrary that it were (an application of refutation by contradiction). Then $p$ would divide $P$, and because it also divides $Q$ it divides their difference $Q - P = 1$. This gives a contradiction, since no prime number divides 1. Thus $p$ is a prime number not in the list.

Irrationality of the square root of 2

The classic proof that the square root of 2 is irrational is a refutation by contradiction.
Indeed, we set out to prove the negation ''¬ ∃ a, b ∈ $\mathbb$ . a/b = '' by assuming that there exist natural numbers ''a'' and ''b'' whose ratio is the square root of two, and derive a contradiction.

Proof by infinite descent

Proof by infinite descent is a method of proof whereby a smallest object with desired property is shown not to exist as follows:

* Assume that there is a smallest object with the desired property.
* Demonstrate that an even smaller object with the desired property exists, thereby deriving a contradiction.

Such a proof is not a proof by contradiction but once again a refutation by contradiction. A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number ''q'' and derive a contradiction by observing that is even smaller than ''q'' and still positive.

Russell's paradox

Russell's paradox, stated set-theoretically as "there is no set whose elements are precisely those sets that do not contain themselves", is a negated statement whose usual proof is a refutation by contradiction.

Notation

Proofs by contradiction sometimes end with the word "Contradiction!". Isaac Barrow and Baermann used the notation Q.E.A., for "''quod est absurdum''" ("which is absurd"), along the lines of Q.E.D., but this notation is rarely used today. A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley. Others sometimes used include a pair of opposing arrows (as $\rightarrow\!\leftarrow$ or $\Rightarrow\!\Leftarrow$), struck-out arrows ($\nleftrightarrow$), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※), or $\times\!\!\!\!\times$.The Comprehensive LaTeX Symbol List, pg. 20. http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf

Hardy's view

G. H. Hardy described proof by contradiction as "one of a mathematician's finest weapons", saying "It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." G. H. Hardy, '' A Mathematician's Apology; Cambridge University Press, 1992. .
PDF p.19

See also

* Law of excluded middle
*Law of noncontradiction
* Proof by exhaustion
* Proof by infinite descent
* Modus tollens

References

Further reading and external links

*

Proof by Contradiction
from Larry W. Cusick'


Reductio ad Absurdum
Internet Encyclopedia of Philosophy; ISSN 2161-0002

{{DEFAULTSORT:Proof By Contradiction

Mathematical proofs
Methods of proof
Theorems in propositional logic