Happiness or a ham sandwich?
is better, eternal happiness or a ham sandwich? It would appear that
eternal happiness is better, but this is really not so! After all,
nothing is better than eternal happiness, and a ham sandwich is
certainly better than nothing. Therefore a ham sandwich is better than
Smullyan (1), p. 219
wish to prove to you that there exists a unicorn. To do this it
obviously suffices to prove the (possibly) stronger statement that
there exists an existing unicorn. (By an existing unicorn I of course
mean one that exists.) Surely if there exists an existing unicorn, then
there must exist a unicorn. So all I have to do is prove that an
existing unicorn exists. Well, there are exactly two possibilities:
(1) An existing unicorn exists.
(2) An existing unicorn does not exist.
(2) is clearly contradictory: How could an existing unicorn not exist?
Just as it is true that a blue unicorn is necessarily blue, an existing
unicorn must necessarily be existing.
"Interesting" and "uninteresting" numbers
question arises: Are there any uninteresting numbers? We can prove that
there are none by the following simple steps. If there are dull
numbers, then we can divide all numbers into two sets – interesting and
dull. In the set of dull numbers there will be only one number that is
the smallest. Since it is the smallest uninteresting number it becomes,
ipso facto , an interesting number. We must therefore remove
it from the dull set and place it in the other. But now there will be
another smallest uninteresting number. Repeating this process will make
any dull number interesting.
The ship of Theseus
The ship wherein Theseus and the youth of Athens
returned had thirty oars, and was preserved by the Athenians down even
to the time of Demetrius Phalereus, for they took away the old planks
as they decayed, putting in new and stronger timber in their place,
insomuch that this ship became a standing example among the
philosophers, for the logical question of things that grow; one side
holding that the ship remained the same, and the other contending that
it was not the same.
Plutarch, Vita Thesei, 22-23
2 = 1
is the version offered by Augustus De Morgan: Let x = 1.
Then x² = x. So x² – 1 = x -1. Dividing
both sides by x -1, we conclude that x + 1 = 1; that
is, since x = 1, 2 = 1.
= b. (1)
both sides by a,
= ab. (2)
b² from both sides,
– b² = ab – b² . (3)
+ b)(a – b) = b(a – b). (4)
both sides by (a – b),
+ b = b. (5)
now we take a = b = 1, we conclude that 2 = 1. Or
we can subtract b from both sides and conclude that a,
which can be taken as any number, must be equal to zero.
Or we can substitute b for a and conclude that any
number is double itself. Our result can thus be interpreted in a
number of ways, all equally ridiculous.
arises from a disguised breach of the arithmetical prohibition on
division by zero, occurring at (5): since a = b, dividing
both sides by (a – b) is dividing by zero, which renders
the equation meaningless. As Northrop goes on to show, the same
trick can be used to prove, e.g., that any two unequal numbers are
equal, or that all positive whole numbers are equal.
Of course paradoxes are everywhere, you just need to look carefully