Nov 15, 2009

Coopertive pragmatics and semantics in proof writing

This post has a scary title.

The cooperative principle basically states that when when people communicate, they try to be helpful by saying the strongest statement feasible.

Example: Look at the puppies to the left.

I could say "There is 1 puppy in this picture.", which translates into 2 different logical statements:

(1) There is only 1 puppy.
(2) There is at least 1 puppy.

The word one here is a quantifier over the set of puppies. Unfortunately one represents the two different quantifications shown one of which is true and one of which is false.

We tend to interpret the sentence to mean (1) because I could have just as easily said "There are two puppies". (2) is a weaker statement and just as much work to say. Thus, "There is 1 puppy in this picture." feels like a lie because its uncooperative.

It recently occurred to me that in math proofs I see this type of ambiguity all the time. I almost did it myself today and strangely I was never taught to not do this. Because the quantifier in meaning number two is essentially the existential quantifier mathematicians may be more prone to intending the second meaning even when it is not really the cooperative one. I going to start making sure I explicitly mark which of the two meanings I mean when writing proofs from now on and take it as a general rule of good proof writing.

Actually this post was just an excuse to show that picture of the two cute puppies. Arent they adorable?