Logic question

Quote:

Original post by kSquared Consider the following statements: Premise 1: All people who win every tennis tournament they enter are tennis champions. Premise 2: I've won every tennis tournament I entered. Conclusion: I am a tennis champion. Now, consider the case where I have actually entered zero tournaments. Clearly, in everyday conversation, it would be a lie (or at least misleading) to say that I am a tennis champion. But how come the logic doesn't break down here? This looks like a valid conclusion. Or is it?

If you entered 0 tournaments, in "I've won every tennis tournament I entered."

"I entered" is false(never entered) and "i've won" is false (never won) , therefore depending to how you define conditional combination in your language, entire statement is either undefined or artifically defined to be false.

It's human-language related problem. Human language is ill-suited for logic. Premise 1 is often misinterpreted as "All people who never losed in tennis tournament are tennis champions."
In original text, "every tennis tournament they enter " is added there only to handle special case when you have not entered tennis tournament, to make it be undefined in that case.

It's like if we have variable-length array of bits and "logical and" that takes array of bits, and return 1 if all bits in input is 1. It is undefined on zero-length arrays. Similarly, in human language, result of "I've won every tennis tournament I entered." is undefined if you entered 0 tournaments.

If humans would like logic more, "Premise 1" would be said as as
"All people who entered tennis tournaments and won them all is tennis champions." - it is defined on all cases.
But such statements is uncommon. I'd say, in general, statements that is well-defined for every case is very uncommon. Especially uncommon in laws and rules. Every time i read something law-related (e.g. EULA), i'm is amazed by how even very simple statements is made undefined.

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Original post by Dmytry Quote: Original post by kSquared Consider the following statements: Premise 1: All people who win every tennis tournament they enter are tennis champions. Premise 2: I've won every tennis tournament I entered. Conclusion: I am a tennis champion. Now, consider the case where I have actually entered zero tournaments. Clearly, in everyday conversation, it would be a lie (or at least misleading) to say that I am a tennis champion. But how come the logic doesn't break down here? This looks like a valid conclusion. Or is it? If you entered 0 tournaments, in "I've won every tennis tournament I entered." "I entered" is false(never entered) and "i've won" is false (never won) , therefore depending to how you define conditional combination in your language, entire statement is either undefined or artifically defined to be false. It's human-language related problem. Human language is ill-suited for logic. Premise 1 is often misinterpreted as "All people who never losed in tennis tournament are tennis champions." In fact, in original text, "every tennis tournament they enter " is added there to handle special case when you have not entered tennis tournament. It's like if we have variable-length array of bits and "logical and" that takes array of bits, and return 1 if all bits in input is 1. It is undefined on zero-length arrays. Similarly, in human language, result of "I've won every tennis tournament I entered." is undefined if you entered 0 tournaments. If humans would like logic more, "Premise 1" would be said as as "All people who entered tennis tournaments and won them all is tennis champions." But it is uncommon. I'd say, in general, well-defined statements is uncommon.

So it basically breaks down because the premises are ambiguous. I think you seem to have hit it on the head, because I tried transforming into propositional logic and it was fairly complicated:

Let A[] be the universal quantifier.
Let T(x) be the set of tennis tournaments entered by x.
Let W(x) be the set of tennis tournaments won by x.
Let P(x) be the statement "x is a tennis champion".

Premise 1: A[y is in T(x) -> y is in W(x)] -> P(x).
Premise 2: A[y is in T("me") -> y is in W("me")].
Conclusion: P("me").

Now the question is if A[y is in T(x) -> y is in W(x)] is true when T(x) is the empty set. That is, is it valid to place the universal quantifier ("for all x ...") on a set which is empty? My book says no, such an operation is undefined; is that generally considered the case?

Yes, almost all operations on sets is undefined on empty set.

In human language, "undefined" is often artifically defined to be false or true.

edit:
Something like
"All people who entered tennis tournaments and won them all is tennis champions."
does not include people that haven't entered.

From what I remember from my logic course, the statements are well-defined and valid. Premise 1 has just been given another definition than intended. I've searched but not found if it is so that a "for all x in S, P(x)" is true if S is the empty set. Something interesting then is that the statement "for all x in S, falsum" is true iff S is the empty set. :) But I'm not sure so don't take my word for it. It makes sense to make quantifiers undefined on empty sets, but I don't remember ever seeing any special case on quantifiers like that.

Unfortunately language is logically ill-defined, although that fact probably isn't relevant to this problem.

Note that it's very important to look at these sorts of problems in depth, as when looking at abstract systems, such as algebraic topology, it is important to consider the empty set, although the statements will be well-defined(unless you remember them wrong like me :P).

There's a good illustration of a linguistic contradiction that goes something like: Define A to be the set of all numbers definable in 83(or some other number, it doesn't really matter) or less words. A finite number of words means A is finite.

Now define n to be "the supremum of the set of numbers defined using eighty three or less words plus one". Now, in the proper illustration this statement would be longer and better defined, but I want to get to bed. n is well-defined as A is finite. So does n belong to A or not? Oops, a contradiction!

This is a very difficult situation to try to comprehend and actually falls down on the definition of A and not n as one might expect. When physicists complain about mathematicians studying 'useless' set theory this is the best example to quote in my experience, because it is one that really does require going right back to some very basic axioms, before deciding language is useless in mathematics. It also puts physicists in their place.

If someone can find the contradiction phrased properly that would be nice, it's a very common one. I hope I haven't strayed too far from the OP to have made a boring and irrelevant post.