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.