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Proof that mint tastes bad.

To prove mint tastes bad, we must first prove two other statements:

1. Proof that all horses are the same color.

This will be a proof by induction. We have a group of n horses. For our base case, suppose n = 1. A group of horses containing one horse has exactly one color, and thus all horses in the group have the same color.

Now for the inductive step. If n horses have the same color, prove n + 1 horses have the same color. To the group of n horses of the same color, add a new horse and remove one of the horses. This group still contains n horses, which all have the same color because n horses are the same color. Now re-add the horse to the group and remove a different horse (so neither the new horse nor the re-added horse). This is still a group of n horses, but which contains the originally removed horse and the new horse. Because groups of n horses have the same color, these two horses must be the same color. The horse which is currently removed is also the same color as the new horse because it was previously in a group of n horses with the new horse. Now add back it that removed horse. Because both removed horses are the same color as the new horse, this new group of size n + 1 must all be the same color. QED

2. Proof that not all horses are the same color.

This will be a visual proof.

QED

We are now ready for 3. Proof that mint tastes bad.

Suppose all horses are the same color (proven in bullet point 1) and not all horses are the same color (proven in bullet point 2). Because all horses are the same color (P) is true, the statement all horses are the same color or mint tastes bad (P + Q) is also true. However, the statement not all horses are the same color (!P) is true, so P must be false. Because P is false but P + Q is true, statement Q must be true by disjunctive syllogism. Therefore, mint tastes bad. QED