Difference between revisions of "Fallacy"

(All horses are the same color)
 
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A '''fallacious proof''' is a 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.
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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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Then we have
 
Then we have
  
<math> a^2 = ab </math> (since <math>a=b</math>)
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<div style="text-align:center"><math> a^2 = ab </math> (since <math>a=b</math>)</div>
  
<math> 2a^2 - 2ab = a^2 - ab </math> (adding <math>a^2-2ab</math> to both sides)
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<div style="text-align:center"><math> 2a^2 - 2ab = a^2 - ab </math> (adding <math>a^2-2ab</math> to both sides)</div>
  
<math> 2(a^2 - ab) = a^2 - ab </math> (factoring out a 2 on the [[LHS]])
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<div style="text-align:center"><math> 2(a^2 - ab) = a^2 - ab </math> (factoring out a 2 on the [[LHS]])</div>
  
<math> 2 = 1 </math> (dividing by <math>a^2-ab</math>)
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<div style="text-align:center"><math> 2 = 1 </math> (dividing by <math>a^2-ab</math>)</div>
  
 
[[Fallacious proof/2equals1 | Explanation]]
 
[[Fallacious proof/2equals1 | Explanation]]
  
=== All horses are the same color ===
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=== 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.
  
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.
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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 <math>k</math> horses are the same color.  This is our inductive assumption.
 
Now assume that all the horses in any group of <math>k</math> horses are the same color.  This is our inductive assumption.
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[[Fallacious proof/all horses are the same color | Explanation]]
 
[[Fallacious proof/all horses are the same color | Explanation]]
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=== All numbers are equal ===
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Consider arbitrary reals <math>a</math> and <math>b</math>, and let <math>t</math> = <math>a + b</math>. Then
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<cmath>a + b = t</cmath>
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<cmath>(a + b)(a - b) = t(a - b)</cmath>
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<cmath>a^2 - b^2 = ta - tb</cmath>
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<cmath>a^2 - ta = b^2 - tb</cmath>
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<cmath>a^2 - ta + \dfrac{t^2}{4} = b^2 - tb + \dfrac{t^2}{4}</cmath>
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<cmath>\left(a - \dfrac{t}{2}\right)^2 = \left(b - \dfrac{t}{2}\right)^2</cmath>
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<cmath>a - \dfrac{t}{2} = b - \dfrac{t}{2}</cmath>
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<cmath>a = b</cmath>
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[[Fallacious proof/All numbers are equal|Explanation]]
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=== Bread ===
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*Nothing is better than fame, happiness and success.
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*A few small crumbs of bread are better than nothing.
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*Thus, a few small crumbs of bread are better than fame, happiness, and success.
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[[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.

Common false proofs

The fallacious proofs are stated first and then links to the explanations of their fallacies follow.

2 = 1

Let $a=b$.

Then we have

$a^2 = ab$ (since $a=b$)
$2a^2 - 2ab = a^2 - ab$ (adding $a^2-2ab$ to both sides)
$2(a^2 - ab) = a^2 - ab$ (factoring out a 2 on the LHS)
$2 = 1$ (dividing by $a^2-ab$)

Explanation

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 $k$ 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 $k+1$ horses have the same color. Number the horses 1 through $k+1$. Horses 1 through $k$ must be the same color as must horses $2$ through $k+1$. It follows that all of the horses are the same color.

Explanation

All numbers are equal

Consider arbitrary reals $a$ and $b$, and let $t$ = $a + b$. Then \[a + b = t\]

\[(a + b)(a - b) = t(a - b)\]


\[a^2 - b^2 = ta - tb\]


\[a^2 - ta = b^2 - tb\]


\[a^2 - ta + \dfrac{t^2}{4} = b^2 - tb + \dfrac{t^2}{4}\]


\[\left(a - \dfrac{t}{2}\right)^2 = \left(b - \dfrac{t}{2}\right)^2\]


\[a - \dfrac{t}{2} = b - \dfrac{t}{2}\]


\[a = b\]

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.


Explanation

See also

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