Law of Excluded Third

The law of the excluded third ( lat. Tertium non datur , that is, “no third is given”) is the law of classical logic , consisting in the fact that of two statements - “A” or “not A” - one is necessarily true, that is, two judgments , one of which formulates the negation of the other, cannot be simultaneously false. The law of the excluded third is one of the fundamental principles of "classical mathematics ."

From an “ intuitionistic ” (and, in particular, “ constructivist ”) point of view, establishing the truth of a statement of the form “ A or not A ” means:

either truth $A$ ${\ displaystyle A}$ $A$ ;
either establishing the truth of his denial $\neg A$ ${\ displaystyle \ neg A}$ $\ neg A$ .

Since, generally speaking, there is no general method allowing for any statement in a finite number of steps to establish its truth or the truth of its negation, the law of the excluded third should not be applied as an axiom in the framework of intuitionistic and constructive directions in mathematics.

Wording

In mathematical logic, the law of the excluded third is expressed identically by the true formula ^[1] :

$A\vee \neg A,$ ${\ displaystyle A \ vee \ neg A,}$ $A\vee \neg A,$

Where:

" $\vee$ ${\ displaystyle \ vee}$ $\vee$ "- a sign of disjunction ;
" $\neg$ ${\ displaystyle \ neg}$ $\neg$ "- a sign of negation .

Other wording

Other logical laws have a similar meaning, many of which have developed historically.

In particular, the law of double negation $\neg (\neg A)\rightarrow A$ ${\ displaystyle \ neg (\ neg A) \ rightarrow A}$ $\neg (\neg A)\rightarrow A$ and pierce law $((P\rightarrow Q)\rightarrow P)\rightarrow P$ ${\ displaystyle ((P \ rightarrow Q) \ rightarrow P) \ rightarrow P}$ $((P\rightarrow Q)\rightarrow P)\rightarrow P$ equivalent to the law of the excluded third in intuitionistic logic . This means that the expansion of the system of axioms of intuitionistic logic by any of these three laws in any case leads to classical logic . And yet, in the general case, there are logics in which all three laws are nonequivalent ^[2] .

Examples

Suppose P is the statement Socrates is mortal . Then the law of the excluded third for P will take the form: "Socrates is mortal or Socrates is immortal," - from which it is clear that the law cuts off all other options in which Socrates is neither mortal nor immortal. The last - this is the very "third" that is excluded.

A much more subtle example of the application of the law of the excluded third, which demonstrates well why it is not acceptable from the point of view of intuitionism, is as follows. Suppose we want to prove a theorem that there are irrational numbers $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ such that $a^{b}$ ${\ displaystyle a ^ {b}}$ rationally .

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

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

It is obvious (excluding the third option) that this number is either rational or irrational. If this number is rational, then the theorem is proved. Searched numbers:

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

But if the number ${\sqrt {2}}^{\sqrt {2}}$ ${\ displaystyle {\ sqrt {2}} ^ {\ sqrt {2}}}$ is irrational then let $a={\sqrt {2}}^{\sqrt {2}}$ ${\ displaystyle a = {\ sqrt {2}} ^ {\ sqrt {2}}}$ and $b={\sqrt {2}}$ ${\ displaystyle b = {\ sqrt {2}}}$ . Consequently,

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

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

i.e $a^{b}$ ${\ displaystyle a ^ {b}}$ Is a rational number .

According to the law of the excluded third, there can be no other options. Therefore, the theorem is proved in the general case. Moreover, the proof is extremely simple and elementary. On the other hand, if we accept the intuitionistic point of view and abandon the law of the excluded third, the theorem, although it can be proved, but its proof becomes extremely complicated.

Notes

↑ Edelman, 1975 , p. 21.
↑ Zena M. Ariola and Hugo Herbelin. Minimal classical logic and control operators. In Thirtieth International Colloquium on Automata, Languages and Programming, ICALP'03, Eindhoven, The Netherlands, June 30 - July 4, 2003, volume 2719 of Lecture Notes in Computer Science, pages 871–885. Springer-Verlag, 2003. [1]

Literature

Edelman S.L. Mathematical logic. - M .: Higher school, 1975 .-- 176 p.