After my previous post about the "meaning is use" credo, I got an email pointing out papers by Nissim Francez and Roy Dyckhoff which develop a Proof-Theoretic Semantics for natural language, and logic also. A good paper to start with is "Proof-theoretic semantics for a natural language fragment," Linguistics & Philosophy 33:447-477 (2010). Other papers appeared in Studia Logica (2010) and Review of Symbolic Logic (2011).
Proof-theoretic semantics is, as the name implies, offered as an alternative to the usual model-theoretic semantics. While I studied formal semantics (which was always model-theoretic) like every other linguist interested in formal approaches, I have to admit I never liked it very much. So I find these new developments extremely encouraging. Here is a quote from the paper in L&P:
For sentences, replace the received approach of taking their meanings as
truth conditions (in arbitrary models) by an approach taking meanings to
consist of canonical derivability conditions (from suitable assumptions).
Arguments against model-theoretic semantics for natural language are certainly out there (e.g. Michael Dummett), but no one has done much about it for an alternative approach. I am certainly in favor of this new set of ideas; these authors develop a direct proof system for natural language in which the "rules of use" for linguistic elements are used precisely as their definitions. And they also highlight some interesting arguments in favor of these sorts of meanings in a cognitive system.