Friday, May 20, 2005

What the tortoise said to achilles

Howdy! Today I am pleased to learn that somebody is actually reading my blog and has left some useful and encouraging comments. Thanks to all who have participated.

One of the good comments I received was a reference to What The Tortoise Said to Achilles by Lewis Carrol, in response to my question in the previous posting whether logicians have a sense of humor. Incidentally, my hunch says I've seen this article in Douglas Hofstadter's excellent Godel, Escher, Bach: An Eternal Golden Braid.

Anyway, on with the story. Basically, Carrol's article contains a clever and well-constructed argument that reasoning is impossible without using an infinite series of modus ponens. I personally feel this argument itself already shows that logicians are humorous (well ... more precisely, logically humorous). I leave the reader to ponder whether Carrol's article shows that logicians have a sense of humor.

Oh, incidentally, I was quite surprised to learn the well-renowned 'Alice's Adventures in Wonderland' is also written by Lewis Carrol, a logician. Wow!

Anonymous said...

Anthony --

Glad you liked the article! But, I would not characterize Lewis Carroll's Mind article as describing "an infinite series of modus ponens". Each inference rule the tortoise applies is different to the previous inference rules he has applied. Only the first of these rules is Modus Ponens. Indeed, each successive rule he applies requires an additional expression in the antecedant.

Might seem a pedantic quibble, but where would mathematics (and philosophy and computer science) be without pedantry!

-- Peter

Kenny said...

Isn't the point of the article that you get into trouble when you confuse the object language and the metalanguage? The proper resolution might be almost Wittgensteinian - at some point we have to stop appealing to rules in some higher metalanguage to explain what a rule means, and just understand it.

Anonymous said...

He is not using MP at all otherwise he would've accepted the conclusion having accepted the premises (but of course he does not). He is conditionizing the rule MP such that {A,A -> B,B} becomes A -> ((A -> B) -> B) which is not a rule at all. It becomes another premise in the hypothetical to accept.

twidjaja said...

I agree with all of you. I was wrong to say that it requires an infinite series of MPs to prove the conclusion from the premises. [For some reason, I got confused by the discussion I had with a friend, who is also a philosophy student, two years ago.] The characters in the dialogue actually doesn't even grant MP as a rule of inference, that they try to prove the conclusion only from the premises.

Now I think the moral of the story is that:
1. It is not possible to deduce a statement without alluding to metatheorems (in this case, MP). [Not possible to prove something completely within the system - the dialogue also shows the danger of mixing the language and the metalanguage.]
2. That you have grant MP as a rule of deduction.

Thanks for clarifying my confusion.