Horse paradox
From Wikipedia, the free encyclopedia
Several distinct paradoxes have been referred to as the horse paradox, during the history of logic.
[edit] White Horse Not Horse
In Ancient China, logicians such as Gongsun Longzi argued about the claim (Bái mǎ fēi mǎ), literally "white horse is not horse". This is also the title of one of the few surviving works of early Chinese logic, an essay structured as a conversation between two parties, with one party proclaiming truth in the statement and the other questioning.
The argument has been interpreted by A. C. Graham, as playing upon the dual semantic meanings of informal language, in particular the dual interpretations of 'is', being either:
- "Is a member of the class entitled (x)"
- "Is identical to concept (x)"
Thus a white horse is not a horse, because the concept of a white horse is not the same as the concept of a horse.
[edit] I promise you a horse
In medieval western logic a popular sophismata or puzzle-proposition was the claim "I promise you a horse." William of Ockham advanced a version of supposition theory in which the term horse in this claim has merely confused supposition. Thus while I promise you a horse, we cannot "descend to particulars," i.e. The disjunction "I promise you this horse, or I promise you that horse, or I promise you that other horse, etc." is false. However, on Walter Burley's account the term has no personal supposition at all, but only simple supposition. The problem of what claim exactly is being made when one promises a horse was a hotly contested in late medieval logic.
[edit] All horses are the same color
A third "horse paradox" is the following (invalid) proof of the statement "All horses are the same colour":
We use the principle of mathematical induction. As the basis case, we note that in a set containing a single horse, all horses are clearly the same colour. Now assume the truth of the statement for all sets of at most n horses. Let there be n+1 horses in a set. Remove the first horse to get a set of n horses. By the induction assumption, all horses in this set are the same colour. It remains to show that this colour is the same as that of the horse we removed. But this is easy: put back the first horse, take out a different horse and apply the induction principle to this set of n horses. Thus all horses in any set of n+1 horses are the same colour. By the principle of induction, we have established that all horses are the same colour.
The hole in the above "proof" is easy to spot with a little thought: it makes the implicit assumption that the two subsets of horses to which we apply the induction assumption have a common element, but this fails when n = 2.
Thus this is not a paradox, but merely the result of flawed reasoning; it exposes the pitfalls arising from failure to consider special cases for which a general statement may be false.es:Paradoja del caballo he:פרדוקס הסוס hu:Ló-paradoxon pl:paradoks koni
