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Logical Paradoxes


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Logical Paradoxes

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1. Bonny: My teacher said in class today that all generalities are false.  Do you think that's true?

Charlie: Who knows?  Maybe yes, maybe no.

Bonny: I know.  It's false.  Look.  Suppose it were true.  Then the statement (A) "All generalities are false" would be true.  But (A) itself is a generality.  So if (A) is true, it's true that all generalities re false, so (A) must be false.  So if (A) is true, then it's false.  Well, (A) must be false.  Right?

Charlie: Wrong!  But I don't know why.


2. Charlie: What we need is a bibliography listing all bibliographies.

Bonny: That would be nice.  But how about a bibliography that lists all and only those bibliographies that do not list themselves?

Charlie: Not terribly useful.  But why not?

Bonny: Here's why not.  Such a bibliography either lists itself or it doesn't.  Right?  If it does list itself, then it violates the condition that it list only those bibliographies that don't list themselves.  So it can't list itself.  But if it doesn't list itself, then it violates the condition that it list all those bibliographies that do not list themselves.  So either way the conditions of such a bibliography are violated.  So there cannot be such a bibliography.

Charlie: That's what's wrong with you philosophy majors--you think too much for your own good.

3. Bonny: Ready for another one?

Charlie: No.  But you'll go ahead anyway.

Bonny: OK.  Let's call a number interesting if we can say something special about that number that we can't say about any other number (not counting things such as being identical with themselves, or on greater than the next number, and things like that).  Every low number clearly is interesting: 1 is the lowest number; 2 is the lowest even number; 3 is the number of logic books on my shelf; 4 is the number of offensive backs in football, and so on.  But when we get to extremely large numbers, the situation would seem to be different; for instance, there seems to be nothing interesting about (10^61 + 33).  So some numbers are not interesting.  Right?

Charlie: Right,... on you definition of interestingness.

Bonny: Wrong!  I'm going to prove to you that there are no uninteresting numbers. Imagine two huge bags, A and B, A containing all the interesting numbers, B the uninteresting ones.  If there are no uninteresting numbers, bag B will be empty.  So you think B will not be empty, because you think some numbers are uninteresting.  But if there are any numbers in bag B, there must be a lowest one, right?

Charlie: Right.

Bonny: Well, if that's true, then we can say something about that number that we can't say about any other number, namely, that it is the lowest uninteresting number.  Right?

Charlie: Right.

Bonny: Well, isn't that interesting!

Charlie: What?!?

Bonny: So there can't be a lowest uninteresting number, because that would be interesting.  But if there is no lowest interesting number, then there aren't any.  Q.E.D.

4.  Bonny: Now I'm going to show you that your intuitions about classes are all wet.  For instance, you believe that any items can form a class, don't you, and also that there is a universal class that contains everything?

Charlie: Sure, why not?

Bonny: Well, here's why not.  If any items can form a class, then classes themselves can be items that form a class.  So we can construct a class containing other classes as members (for example, the class of all classes containing exactly ten members), and even construct a class containing itself as a member (for example, the class consisting of itself and the class of states in the Union.

Charlie: That last is a weird class, but why not?

Bonny:Here's why not.  Divide all classes into those that are a member of themselves (for example, the class containing all classes) and those that are not (for example, the class containing all football players and nothing else).  Then the class containing all classes that are a member of themselves would seem to contain itself.  But what about the class containing all classes that do not contain themselves?  Is it a member of itself?  Clearly not, since it is the class of all classes that re not members of themselves.  Well, then, is it not a member of itself?  Again, clearly not, for if it were not, then it would be a class that is not a member of itself, and hence would be a member of itself.  So, if it is a member of itself, it isn't a member of itself, and if it isn't, it is.  clearly, there is no class containing just those classes that are not members of themselves.  Hence, every bunch of items does not form a class, and, incidentally, it therefore can't be true that there is a universal class containing everything.

Charlie: Very clever, but I'll figure out what's wrong... later.

The above logical paradoxes are taken from the following text: 


Tidman, Kahane.  Logic and Philosophy: A Modern Introduction.  Eight Edition. Wadsworth Publishing Company.  1999. p 253-254.

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