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Proof by Absurdity
alias proof by contradiction

Example One: A detective in  solving a crime may have a suspect. Then he may found the suspect has an alibi which directly or indirectly implies she did not committed the crime. So the alibi and suspicion are inconsistent - that is incompatible. The detective may drop the suspicion or challenge the alibi. Lawyers for the prosecution and defense may erect competing chains of reason, and leave it to a jury or judge to decide which one, if any, appears to be true. A conclusion may follow or not.

Example Two: In developing a story or theory, we may require its elements to be consistent.  We writing or developing a story, we note that a situation A is inconsistent with what has so far been written or determined, we will not add situation A to the plot or theory.  We will instead accept the situation did not occur and may employ that in the further composition of the plot, story or theory.  Likewise, if that the non-occurrence of a situation A is inconsistent with what has so far been written or determined, we will be forced to add the occurrence of situation A to the plot or theory. Finally, if the occurrence of A or its non-occurrence has is independent of the plot, story or theory,  we extend the story or theory to include A or not as we like, if we are writing fiction or describing what may be possible.  On the other hand, if we are trying to write non-fiction then the addition of statements of patterns A to plot depends on their correspondence with reality. 

In telling a story or developing a theory, we may look at the consequences of our assumptions - the situations we tend to assume as holding or being true.  If a chain of reason implies that a situation C occurs and does not occur, then the story or theory is inconsistent - becomes absurd. For the sake of consistency, the story or theory needs to be revised or abandoned.

In telling a story or developing a theory, there is an inconsistency if a situation A and its negation Not A both occur.  While we are developing a theory from assumptions, we cannot be certain that an inconsistency   A and Not A will not occur.  However, the assuming the law of excluded middle in the development of a theory or is equivalent to the statement that the situation A and Not A does not happen.  It equivalent to the assumption that the theory under development is consistent. That be said, when we are developing a theory from logic and underlying assumptions, we may not be able to prove that the theory is consistent. If the theory is consistent, the law of excluded middle holds and so we may used in our logical development of theory without loss of consistency. But if the theory under development is inconsistent,  assuming or using the law of excluded middle in its development may lead to the discovery of the inconsistency, sooner rather than later, if at all. Assuming the law of excluded middle, simply adds another inconsistency. . 

Example Three: Assume any infinite decimal expansion locates a point or distance on a real number line. Assume further that each ratio of two whole numbers can be expressed a  ratio of two whole numbers with no common divisors?  The Pythagorean theorem then suggests in an isosceles right triangle, the ratio of the hypotenuse to each of the others sides, the legs, by length given by the square root of 2.  Is that square root equal to a rational number? The suspicion or assumption that YES, the square root of 2 equals a rational number  implies an inconsistency.  Namely, that in any ratio or fraction that represents the square root of 2, the denominator and numerator will both be multiples of 2. So the square root of two cannot be rational. 

The Pythagoreans in finding the inconsistency in example 2 had a problem. They assume  lined segments in the plane represented numbers  and they assumed all such lengths were rational multiples of each other.  When these assumptions or their consequences clashed, reconciliation was not obvious.  Their view of numbers collapsed and a replacement was not available. That was a serious problem for the Pythagorean school in their theory of knowledge was based on and assumed rational numbers and only rational numbers.  

Today, however, we have an advantage or two. One advantage is a our assumption that infinite decimals expansions represent rational and irrational numbers.  Physically, if you imagine ruler with a unit length and its division into tenths, hundredths, thousandths and so on, then you can count the maximum number of  units, tenths, hundredths, thousandths and that may fit in a line segment.  The result is a sequence of numbers which provide a better and better approximation to the line segment length.  You can further imagine that sequence of approximations given by two, three, four, five and more decimal places in a decimal expansion locate the end of a line segment.  The decimal representation of numbers does not depend on nor require the "numbers" to be rational. 

Aside: The number 1.0000 represent a single unit of length. It also represents the limit of the sequence 0.9, 0.99, 0.999, 0.9999 and so on. The sequence of decimal approximations 0.99999 (9 recurring) is shorthand for  a sequence of lengths L(j) =1 - 10^-j < 1 where the lengths are increasing, and where the differences  d(j) = 1 - L(j) = 10^-j  is getting smaller and smaller as j increases. So the sequence approaches 1.  Because the difference tends to zero, the number 1 has several decimal expansions

• 1,  1.0, 1.00 (0 recurring finitely many times)
• 1.000 (0 repeating indefinitely)
• 0.99999 (9 recurring)

where the last one represents  1 as the limit of a sequence of approximations 0.9, 0.99, 0.999, 0.9999, 0.99999, ...  . 

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