Thursday, May 17, 2007

Monotonicity, for lack of a better title

A few weeks ago I was out at Stanford. While I was there, Johan van Benthem gave a talk about natural logic to some computational linguists. His talk was called "A Brief History of Natural Logic". The talk was good, as always from van Benthem, but the computational linguists seemed a little unimpressed. I think they were hoping that natural logic and generalized quantifiers would be able to help with some issues in textual entailment. It doesn't really look like it will. As a completely ad hominem aside, I find textual entailment to be really boring as an area of inquiry.

One of the interesting little asides that van Benthem gave was about monotonicity. Semantic monotonicity was defined as follows. If some formula \phi(P) is true, and in the model the set assigned to P is a subset of that assigned to Q, then \phi(Q), e.g. from that all cats are mammals, infer that all cats are vertebrates. There are a few quantifiers that exhibit very regular behavior and satisfy, in addition to monotonicity in its arguments, two additional properties, conservativity (taken from the handout, for quantifiers Q and predicates A,B,C: Q AB iff Q A(A\cap B) ) and variety (if A isn't empty, and Q AB for some B, then there is some C such that \lnot Q AC ). These quantifiers are exactly: all, some, no, not all. Neat historical fact: they make up the square of opposition. The neat fact from the talk, which is something I want to move from back to front burner some day is that almost all (maybe all, I forget what exactly van Benthem said) logics that are studied exhibit monotonicity for quantifiers. This is a property that ignores order of logic, applying as much to first-order logic as to higher-order logics. But, since it cuts across these different orders and systems, there isn't a nice way to express it in a general way that holds for the different systems. It is independent of the order of logic being used. Additionally, the monotonicity inferences were known to the medieval logicians (and apparently to ancient Chinese logicians of the Mohist school, according to Liu and Zhang 2007). They were interested in these sorts of inference issues, but they were obscured in the move to the more standard view of logic divided into orders, according to quantifiers. (As another aside, Fred Sommers's work on syllogistic logic was recommended during the talk as a defense of the more traditional approaches to logic that were more concerned with this sort of thing. One point for N.N.) The take away points of the talk seemed to be these. There there was another approach to studying logic that got obscured in modern approaches. That approach got obscured in large part because of the way languages are created and studied now. The other approach catches some important generalizations that get lost on the modern view. And finally, monotonicity inferences are really important. This last one prompted me to be more sensitive to them when I am reading through Articulating Reasons this summer, since I'm curious to what extent they are exemplify Brandom's material inferences.

4 comments:

Jason said...

Note, however, that material inference is not in general monotonic. If I strike this dry, well-made match, it will light ... unless it is done inside a strong magnetic field.

BTW, Brandom had a great deal to say about (non-)monotonicity in his preparatory workshop for the Locke Lectures, esp. in Lecture 10 (of the workshop, not the offical lectures). [Note: Transcriptions of the workshop lectures use to be available online but were taken down after the officially presented Lectures went up.]

Shawn said...

Material inference is not monotonic in side formulas, which is, I think, different than the kind of monotonicity I was talking about. The inference from "this is a cat" to "this is a mammal" is good as is the monotonic inference from "this is a cat" to "this is a vertebrate". I think those sorts of monotonicity inferences are going to be good since they are along the entailment dimension of deontic score-keeping. I think they are also what Brandom (and Sellars before him) took to exhibit a kind of analyticity. I am not sure that material inferences are not monotonic in the sense in the post. This is something I'm not completely at home in though.

Do you have the workshop notes? I suppose I could try to get them from Brandom... I looked at the conference website. There are links to the Locke lectures but the only full text from the workshop that is online is Brandom's. Which one did you mean by lecture 10? There didn't look like 10 lectures in the workshop. I have heard very little about the Prague shindig although it sounded pretty sweet. Brandom says a bit, if I recall, about non-monotonicity in the Locke lectures themselves.

Jason said...

Yeah, I downloaded the lecture transcriptions (12 in all, I recall) while they were still up. Unfortunately it doesn't look like I have them on my box here at home, so I'm not going to be able to forward them on until Monday. The workshop lectures are quite distinct from the official lectures. They're actually transcriptions, so you get to see Brandom riffing on a lot of things he wouldn't normally do in an official lecture setting. You get to see him in action, as it were, and he's upfront about some of the problems his program faces, as well as some of the motivations, that do not get mentioned in the final product. They're also filled with some very interesting discussions of some of the technical matters that have to be dealt with, which get confined to a compact appendix of Lecture 5 of the official presentation.

BTW, as for the Brandom-fest in Prague, I'm working with Jarda Peregrin to put up either streaming or podcast versions of the lectures and follow-up commentary -- hopefully in both audio and video form. This might take a few weeks, however. I'll send you a notice once they go up if you're interested.

Shawn said...

I just realized that in my previous comment where I said "entailment dimension" it should be "commitment dimension". I think it is incorrect as it stands.

If it is alright, I would really like to get copies of those transcripts. There aren't any of the documents for the workshop up except Brandom's presentation, which is available elsewhere. I was a bit disappointed that it wasn't new.