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 d...
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...roperty P, since X is infinite. There are properties P and sets X for which we can prove that all elements of X possess P or are able to point to an element that possesses P. However, this is not possible for all properties P.
Wade [2] gives the following example: let E be the proposition that there are infinitely many twin primes i.e., infinitely many integers p such that both p and p + 2 are prime. This is known as the Twin Primes Conjecture and it is at the present time unknown whether this conjecture is true or false. Hence, we cannot apply the law of excluded middle.
Works Cited
[1] S. Kleene. Introduction to Metamathematics. Van Nostrand, New York, 1952.
[2] C. Wade. Why does Intuitionistic Logic not allow the ’Law of the Excluded Middle’.
University of Southampton Journal of Philosophy, 2011.
[3] J. Williamson. The Elements of Euclid. Clarendon Press, 1781.
Despite the efforts of skeptics, there are no counter-examples that are sufficient in proving that the Closure Principle is invalid. This is Jonathan Vogel’s main argument in his paper, Are There Counterexamples to the Closure Principle? Vogel presents an interesting argument against counterexamples like Fred Dretske’s “Zebra Case”. He introduces a set of conditions required for such counterexamples to work, and in doing so, demonstrates why the Zebra Case is not even a genuine counterexample to the Closure Principle. In fact, Vogel’s own examples do a much better job of what the Zebra Case intended to accomplish, and even those fail. Interestingly, what accounts for the failure of both Vogel’s and Dretske’s counterexamples are what Vogel takes to be the main features of the very counterexamples that he presents. Those three main conditions have to do with non-arbitrariness, statistical probability, and abnormality. Vogel demonstrates why these counterexamples are insufficient against the Closure Principle through how these conditions are required to make a case against the principle, and how the skeptic line of questions are insufficient in dealing with the principle directly. What is most notable are the conditions put forth that constitute, what Vogel believes, are the best counterexamples against the Closure Principle. He brilliantly reveals how the epistemic weight of those features hold lightly against the Closure Principle, and heavily against the skeptic’s argument. It is this operation of Vogel’s three conditions that will be explored at length for the purpose of establishing several facts. The first is that Vogel’s three conditions adequately take away the plausibility of Dretske’s Zebra Case, and prove the validity of ...
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In 1950, a man, Enrico Fermi, during a lunch break conversation he causally asked his co-workers an interesting question, “where is everybody”. (Howell, 2014) By which he meant, since there are over a million planets which are proficient enough to support life and possibly some sort of intelligent species, so how come no one has visited earth? This became known as The Fermi Paradox, which came from his surname and two Greek words, para meaning contrary and Doxa meaning opinion, about a 100 years ago. (Webb 2002) A paradox arises when you set undisputable evidence and then a certain conclusion contradicts the idea. For example, Fermi realized that extra-terrestrials have had a large amount of time to appear
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Almost all epistemologists, since Edmund Gettier’s 1963 article, have agreed that he disproved the justified-true-belief conception of knowledge. He proposed two examples
Wittgenstein, Ludwig; G. E. M. Anscombe, P.M.S. Hacker and Joachim Schulte (eds. and trans.). Philosophical Investigations. 4th edition, Oxford: Wiley-Blackwell, 2009. Print.
Gettier indicates at the beginning of his selection, he is concerned with attempts to provide sufficient conditions for someone knowing that a proposition is true (Gettier, 43). He is responding to several accounts that have it that a proposition being true, a person’s believing that proposition to be true, and that persons justification in the belief of the truth of the proposition are jointly enough for the subjects of knowing the proposition. Gettier argues that these kinds, while they may initially seem plausible are in fact false. A is false in the conditions stated therein do not constitute a sufficient condition for the truth of the proposition that S knows tha...
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By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.
"Internet Encyclopedia of Philosophy." Beauvoir, Simone de []. N.p., n.d. Web. 28 Apr. 2014. .
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