Two professors at a math conference stand in front of a blackboard, on which is written the sentence “Only an idiot would believe a sentence like this!” The first professor asks the second, “Do you believe that?” The second answers, “Of course not! Only an idiot would believe a sentence like this!”
Or how about this one….
There Are Two Errors in the
the Title of This Book
Or…
1. No cat has eight tails.
2. A cat has one more tail than no cat.
∴ A cat has nine tails.
Hey Fraiser,
Been a long time. I needed to take a break from the blogging world for a while. How are things going for you?
I’ll take a stab at your riddles:
Concerning the first: Maybe the professor could choose to not believe the sentence (that the sentence is true) for other reasons (for example, maybe it is self-referential non-sense?) and thus escape the dilemma.
Concerning the second: Would it be that the first error is the repetition of “the” while the second error is that there is only one error?
The third is easily understood if “No” and “A” are particular cats.
What is your opinion of semantic paradoxes such as the first and second? I haven’t thought about it much but they seem to be logically impossible since they are contradictions. There appears to be nothing wrong with a proposition such as “All of my words are true.” (i.e. when spoken by God) until one reduces the proposition to *merely* refer to itself in a narrow sense (i.e. “This statement is true”). The former doesn’t present a problem because it presumably refers to other meaningful propositions and, thus, its own validity is related to the validity of those other propositions. When we say the former is “true,” we have in mind the truth-value of all other words of the speaker. The latter rests upon itself. I wonder how this relates to the Trinity? God cannot lie because it is contrary to His nature, but isn’t it also because in order to do so He would have to contradict Himself?
Jonathan,
Glad to see you re-enter the orbit of the blogosphere.
I think your answer to the first riddle is how we should respond to the statement “Only an idiot would believe a sentence like this”, but the question remains what we should say about the guy who disbelieves the sentence only by believing it. He affirms his belief in the sentence when he says, “Of course I don’t believe it, only an idiot would believe a sentence like this.” How we explain his simultaneous belief and disbelief.
You’ve correctly identified the two errors in the title of the book. But the plot thickens. If the second error is that there is no second error then doesn’t it make true the statement “There are two errors in the title of this book.” But if that statement is now true, then it is now false that there are two errors. But if it is now false that there are two errors then there are now two errors. So on ad finitum.
I think you’ve missed the answer to the third riddle entirely. If “No” and “A” are particular cats, then no one would grant a starting premise like:
No cat [a particular cat] has eight tales.
I’ll give you a hint in solving them (at least it’s how I think they are solved). They are all based on equivocation of terms.
[quoting] “No cat [a particular cat] has eight tales.”
My cat has dozens of tales.
I’m still trying to figure out whether the Sentence is true or not (“only an idiot would believe …”).
Maybe if we rewrite it more formally: “Let X be the set of believers of this sentence. Then X is identical to Y, the set of all idiots.” So Professor B says only that he is not a member of Y.
The cat pardox seems a bit like this old tried-and-true:
A ham sandwich is better than nothing.
Nothing is better than eternal happiness.
Therefore, a ham sandwich is better than eternal happiness.
Don’t laugh – we get those every day in politician’s speeches.
I’ve been skeptical of Russel’s paradox for a long time. (“Who shaves the barber?”) The “set of all sets” must be something that cannot exist. If we assume the non-existence of the Uber-set, then Russell’s paradox goes away, and the barber jumps carefully out of the set of which he cannot be a member.
It seems to me that the main reason for the notion of Sets is so that we can compare (and contrast) set A with set B. But an Uber-set would contain all other sets, so nothing useful could be said about it.
Mike,
Great point about those tall-tale cats. I’m not much of a cat lover myself. But does your cat have eight tails?
Yes, the cat paradox is an equivocation on the meaning of “no cats” and thus follows the same pattern as the ham sandwich paradox. Actually none of these are paradoxes. They are simply linguistic plays.
I’ll have to think more about your thoughts on sets.
Thanks for stopping in!
The second professor either didn’t mean “only an idiot would believe it” in a literal sense, in which case he’s really saying something like, “anyone who believes it is mistaken” so there isn’t a logical problem any more. Or else, the second professor did mean “only an idiot would believe it” in a literal sense in which case he has just contradicted himself because first he states he doesn’t believe the proposition, then he affirms the proposition. As it happens the proposition is false, but that doesn’t mean anyone is an idiot for believing it or not believing it, since it is false.
I just stumbled upon this page so don’t judge me at my attempt to take a whack at this:
1) In this, two definitions of the word ‘believe’ are used. One being “do you ACCEPT it as true” and the other being “do you THINK/PERCIEVE it as true”. The 2nd mans response can be taken as he FULLY agrees with the sentence because he’s agreeing with the two different definitions of the word ‘believe’:
A) He can’t “believe” such an obvious fact is written on the board, because (to him) it’s common knowledge that
B) only an idiot would ‘accept it as true’, the statement on the board
2) Yes, there is an error ‘in’ the actual title of the book, but there may be an error ‘in’ the actual book. This statement can’t be taken as true since the statement leaves the reader with the option to pick which definition of the word to use. I.e: ‘Laws’ of nature, ‘Laws’ of the government$
3) Again, word play. ‘No cat’ in the first premise can be taken as two things:
A) ‘No cat’ (a name of a deformed animal with 8 tails) has 8 tails
or
B) No cat (no actual cat in existence) has 8 tails
The second premise can mean:
A) ‘A cat’ (a name of another deformed animal with nine tails) has one more tail then ‘no cat’ (name of first animal)
or
B) a cat (a single cat) has one more tail then no cat (zero cats)
So for
“a cat has nine tails” to be true, one would have to disregard the double meaning of the word ‘cat’ and ‘no’..but one can’t, so it makes the whole thing false..
*shrugs*… I dunno, I’m just talking out my ass. Email me please with the answer. I would like to know