Difference between revisions of "Fallacy"
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[[Fallacious proof/All numbers are equal|Explanation]] | [[Fallacious proof/All numbers are equal|Explanation]] | ||
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*Nothing is better than fame, happiness and success. | *Nothing is better than fame, happiness and success. | ||
*A few small crumbs of bread are better than nothing. | *A few small crumbs of bread are better than nothing. | ||
Revision as of 19:22, 2 November 2007
A fallacy is a step an attempted proof that is logically flawed in some way. The fact that a proof is fallacious says nothing about the validity of the original proposition.
Contents
Common false proofs
The fallacious proofs are stated first and then links to the explanations of their fallacies follow.
2 = 1
Let 
.
Then we have
(since )
(adding 
to both sides)
(factoring out a 2 on the LHS)
(dividing by 
)All horses are the same color
We shall prove that all horses are the same color by induction on the number of horses.
First we shall show our base case, that all horses in a group of 1 horse have the same color to be true. Of course, there's only 1 horse in the group so certainly our base case holds.
Now assume that all the horses in any group of 
horses are the same color. This is our inductive assumption.
Using our inductive assumption, we will now show that all horses in a group of 
horses have the same color. Number the horses 1 through . Horses 1 through must be the same color as must horses 
through . It follows that all of the horses are the same color.
All numbers are equal
Consider arbitrary reals 
and 
, and let 
= 
. Then
![\[a + b = t\]](http://latex.artofproblemsolving.com/f/8/5/f85958a404faf3e791513aaa5f27a07196e73200.png)
![\[(a + b)(a - b) = t(a - b)\]](http://latex.artofproblemsolving.com/d/3/b/d3b84cfd26fd8716977eaa15b0aa994d0039996f.png)
![\[a^2 - b^2 = ta - tb\]](http://latex.artofproblemsolving.com/f/e/b/feba7daaad737654702e5f63eb8b1a1a1bfa52cc.png)
![\[a^2 - ta = b^2 - tb\]](http://latex.artofproblemsolving.com/3/8/b/38b3b9f5b52471720de05b17a675747cf6d51c4f.png)
![\[a^2 - ta + \dfrac{t^2}{4} = b^2 - tb + \dfrac{t^2}{4}\]](http://latex.artofproblemsolving.com/d/c/1/dc1ebebc4f496412492197eccb4748b81b2beae4.png)
![\[\left(a - \dfrac{t}{2}\right)^2 = \left(b - \dfrac{t}{2}\right)^2\]](http://latex.artofproblemsolving.com/0/6/7/0675c23778651c07d2a0c909e8b10d4185d207cb.png)
![\[a - \dfrac{t}{2} = b - \dfrac{t}{2}\]](http://latex.artofproblemsolving.com/b/2/6/b26a423e88c1a92e8aed088f3bc1906584b57564.png)
![\[a = b\]](http://latex.artofproblemsolving.com/e/f/2/ef220c9aec64f58c725ba3f77e69829b424cd7a1.png)
Bread
- Nothing is better than fame, happiness and success.
- A few small crumbs of bread are better than nothing.
- Thus, a few small crumbs of bread are better than fame, happiness, and success.
