bracketings and triple negatives

From Language Log: some commentary on yesterday's Doonesbury. The text is as follows:

Some guy: I just don't get it, Jorge. Why do you get the big bucks and not me?
Jorge: We have different work styles, man.
Some guy: Like how?
Jorge: Well, for one thing, I'm not stoned half the time.
Some guy: What are you saying -- that I am?
Jorge: I'm not saying that you're not.
Some guy: Don't try your tricky double negatives on me, señor!
Jorge: Enjoy your break.
Some guy: What?
Jorge: Never mind.

Multiple negatives are indeed tricky. As the post at Language Log points out, a lot of the time a double negative is indeed a negative, and triple negatives often turn out to be positives. The triple negatives they give examples of are generally obtained by taking a double negative which has negative semantics and then negating it again.

And our main character in this particular strip, whose name I don't know, could argue that even if he is stoned half the time, he's also not-stoned half the time. The following conversation could take place:

Jorge: Well, for one thing, [I'm not] [stoned half the time].
Some guy: I'm [not stoned] half the time.

where the same string of words occurs, but with two different meanings. In general there is quite a large number of ways to put parentheses around the words in a string like this, which is a semi-standard tactic for linguistic analysis. (It seems to me that linguists, when talking about syntax, like to draw trees when they want to explain how a whole sentence works, but they do ad-hoc bracketings like the one I did above when they just want to make a simple point about which words go with which words, because trees are annoying to draw.) The number of ways to bracket an n-word sentence completely turns out to be the (n-1)st Catalan number; a lot of problems involving trees (and various other recursive structures!) end up involving Catalan numbers. Of course, most of these bracketings are illogical: anything that looked like

I'm not [stoned half] the time

just isn't going to correspond to a grammatical interpretation of that sentence, because "stoned half" doesn't mean anything. But I suspect that longer sentences have more possible interpretations, on average; it's kind of surprising that a six-word sentence can actually be parsed in multiple ways.

In general, I suspect mathematicians tend to parse natural-language sentences differently than "ordinary people", because we are more used to dealing with subjects where the precise meaning of some statement is what matters, and an interpretation which is off by a little bit might as well be no interpretation at all. (Or perhaps worse than no interpretation -- it is better to not understand a complicated definition and know you don't understand it than to not understand it but think you do. The truly wise are aware of the limits of their knowledge.) The Jargon File, in referring to the speech style of the hacker community (which has some overlap and contact with the mathematical community), states that "...English-speaking hackers almost never use double negatives, even if they live in a region where colloquial usage allows them. The thought of uttering something that logically ought to be an affirmative knowing it will be misparsed as a negative tends to disturb them."