Difference between revisions of "Fallacy"
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− | A ''' | + | A '''fallacy''' is a step an attempted [[proof]] that is [[logic]]ally flawed in some way. The fact that a proof is fallacious says nothing about the validity of the original proposition. |
== Common false proofs == | == Common false proofs == | ||
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[[Fallacious proof/2equals1 | Explanation]] | [[Fallacious proof/2equals1 | Explanation]] | ||
− | === All horses | + | === Polya's Proof That All horses Are the Same Color === |
We shall prove that all horses are the same color by [[induction]] on the number of horses. | We shall prove that all horses are the same color by [[induction]] on the number of horses. | ||
Line 30: | Line 30: | ||
[[Fallacious proof/all horses are the same color | Explanation]] | [[Fallacious proof/all horses are the same color | Explanation]] | ||
− | === All | + | === All numbers are equal === |
+ | Consider arbitrary reals <math>a</math> and <math>b</math>, and let <math>t</math> = <math>a + b</math>. Then | ||
+ | <cmath>a + b = t</cmath> | ||
− | + | <cmath>(a + b)(a - b) = t(a - b)</cmath> | |
− | |||
− | |||
− | < | + | <cmath>a^2 - b^2 = ta - tb</cmath> |
− | |||
− | |||
− | < | + | <cmath>a^2 - ta = b^2 - tb</cmath> |
− | |||
− | < | + | <cmath>a^2 - ta + \dfrac{t^2}{4} = b^2 - tb + \dfrac{t^2}{4}</cmath> |
− | + | ||
+ | |||
+ | <cmath>\left(a - \dfrac{t}{2}\right)^2 = \left(b - \dfrac{t}{2}\right)^2</cmath> | ||
+ | |||
+ | |||
+ | <cmath>a - \dfrac{t}{2} = b - \dfrac{t}{2}</cmath> | ||
+ | |||
+ | |||
+ | <cmath>a = b</cmath> | ||
+ | |||
+ | [[Fallacious proof/All numbers are equal|Explanation]] | ||
+ | === 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. | ||
+ | |||
+ | |||
+ | [[Fallacious proof/Bread|Explanation]] | ||
== See also == | == See also == | ||
* [[Proof writing]] | * [[Proof writing]] |
Latest revision as of 08:27, 5 June 2013
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
![$a^2 = ab$](http://latex.artofproblemsolving.com/d/f/6/df616fb8336779daf527060288b49d13a3c005e7.png)
![$a=b$](http://latex.artofproblemsolving.com/5/a/b/5abd2c942ae299fd26e18d0ca5d390a7ca4a8c34.png)
![$2a^2 - 2ab = a^2 - ab$](http://latex.artofproblemsolving.com/e/d/e/ede398bee73258c4e38b6b3344f5212a16930e05.png)
![$a^2-2ab$](http://latex.artofproblemsolving.com/c/3/8/c384fa448b634e71ce150ce6dc7165e016693f23.png)
![$2(a^2 - ab) = a^2 - ab$](http://latex.artofproblemsolving.com/9/d/5/9d5c2bbac98e2986ce7a579ef55be936d92e97b9.png)
![$2 = 1$](http://latex.artofproblemsolving.com/0/f/f/0ffac44b2fc291131a3d8a59e985754d905bdb26.png)
![$a^2-ab$](http://latex.artofproblemsolving.com/c/a/2/ca2d74b575919eb08e10fb014c60053f2f4c0011.png)
Polya's Proof That 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
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.