On The Law Of Excluded Middle

872 words - 4 pages

The law of excluded middle is the third of the three laws of thought first written down by Aristotle. The first law is the law of identity which states that everything is identical to itself and different from anything else. The second law is the law of non-contradiction which states that contradictory statements cannot be true and not true at the same time. The third and most controversial law is the law of excluded middle which states that every contradictory statement must either be true or false. This principle is widely used in exact sciences. There is, for example, the proof by contradiction. In a proof by contradiction one assumes that some proposition is false and shows that this leads to a contradiction. From this one concludes that the proposition must be true. The law of excluded middle is used in the following way. Because the proposition cannot be false it must be that the proposition is true. An example of a proof by contradiction is the following theorem due to Euclides [3].
Theorem. 1. There are infinitely many prime1 numbers
Proof. Suppose there are only finitely many primes p1, p2, . . . , pn. Consider the integer
P = p1p2 ···pn+1. Let p be prime dividing P. If p is equal to pi for some i, then p divides
both P and p1p2 · · · pn. This implies that p also divides their difference P − p1p2 · · · pn = 1A prime number p is an integer greater than one which is only divisible by one and itself

1, which is absurd. This contradicts our assumption that there are no other primes then p1,p2,...,pn.
Generally the laws of thought are considered the basis for any thought, discourse or discussion. They cannot be proved or disproved and to deny them is self-contradictory. It is widely known that there are mathematicians who do not accept the law of ex- cluded middle. These mathematicians are the so-called constructivists. Constructivism asserts that in order to prove a mathematical object exists one needs to give a construc- tion. According to this viewpoint one does not accept the proof by contradiction, since one only proves that some object exists because assuming the contrary leads to a con- tradiction. Thus one has not given a construction and therefore not proved its existence
according to constructivists.
The law of excluded middle is a principle which is accepted by nearly all mathemati-
cians.
Intuitionism is probably the most well know variety of constructivism. Intuitionism
was founded by the dutch...

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