But you can’t be too sure. Especially not when nutcases like George Pólya are around. His argument is adduced much like this...
There are five very pretty horses. Their color(s), unknown as of yet, but we’ll arrive at that shortly through the givens.
Now the given: Of the five horses, any set of four horses contains horses of the same color. (Will be proven later).
Meaning, if I made a set of four horses out of the first four horses, they’d all be of the same color. If I made a set of the last four horses, they’d again be of the same color. Likewise, a set of first two and last two horses would also give me horses of the same color, and so with any other combination of horses in sets of four. Which color, we don’t care. But do pay heed to the fact that the statement doesn’t say all horses are of the same color. No. Let’s sketch out one set, set 1, for clarity…
Similarly, we have another set, set 2…
Math time now. Let the first horse in the entire group of five be denoted by A, the fifth horse be denoted by B, and the middle three horses be collectively called C. From set 1, it is clear that A=C. From set 2, B=C. And anybody with an understanding of even elemental math knows that when all humans are monkeys and all monkeys are apes, then all humans are apes. So, we have A=B from those two equations, leading us to the (obvious) conclusion that the first horse is of the same color as the fifth. Meaning, all five horses are of the same color. Phew.
That wasn’t so difficult a reasoning as I made it seem, but here comes the fun part. We have to prove our initial given statement. To do that, let’s shoo one horse away into the fields and be left with only four. Now let’s commence the whole logic again with the same initial givens. Color(s) of horses, unknown. A set of three horses in this group of four will always be of the same color, regardless of which three of them you choose for your set. Follow the same line of reasoning and end up arriving at the same conclusion that all four horses are of the same color. Shoo off another horse and do these calculations with three and two and one horse. Since a set of one horse has to be of the same color, the initial statement that any set of four horses in a group of five contains horses of the same color is proved by backtracking the deduction.
Okay, recess time just got over for our horses in the field and it’s time to get back to the stable. All five horses are back. Add another one. Wherever it came from it doesn’t matter, just add. Now you have six. Apply the logic all over again. Arrived at all horses of same color? Make it seven horses. Again all same color. Go on increasing the set one horse at a time to get the same result. 1000 horses, all same color. 10,000 horses, all same color. A million, still all same color.
ALL horses in the world…why would that be any different? They would all be of the same color, wouldn’t they? Of course they would.
Except that our own real-world experience tells us they aren’t, even though this seemingly perfect logical reasoning deduces that they are!
What went wrong then? Maybe George Pólya really is a nutcase. Case closed.
Or maybe not. Maybe, he’s just smarter than us to have fooled us with this. As it happens, he is. Pólya used this reality-contradicting “logic” to expose the, or demonstrate the, fallacy arising out of apparently all-encompassing general statements (laws/rules) when specific cases in which they are demonstrably false aren’t considered. The fallacy in our horse game has been cleverly planted in mathematical induction. Our math equations of if A=C and B=C then A=B are universally valid for all arguments containing three or more elements. However, it just doesn’t apply, indeed it is impossible to apply, when you have less than 3 elements. A=C and ?=C, therefore A=?. Nothing can take the place of ‘?’ as there is nothing to take its place. Therefore, our reasoning is valid only up until we have 3 horses to work with, but the logic breaks down once we are down to two horses. Since we cannot continue the till-now-valid reasoning after reaching two horses, we cannot prove the initial statement, thereby making it a false initial statement. Any theorem based on false assumptions collapses on its face like Nazism.
Hence, all horses in the world are not, as unlikely as it may seem, of the same color.
This type of logic in which the fallacy is intentionally planted in the reasoning is called Falsidical Paradox.
And oddly, I've plagiarized the title of this post from Cormac McCarthy's All the Pretty Horses. Sue me.
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