Two kinds of theoretical framework for syntax are encountered in current linguistics. One emerged from the mathematization of proof theory, and is referred to here as generative-enumerative syntax (GES). A less explored alternative stems from the semantic side of logic, and is here called model-theoretic syntax (MTS). I sketch the outlines of each, and give a capsule summary of some mathematical results pertaining to the latter. I then briefly survey some diverse types of evidence suggesting that in some ways MTS (...) seems better suited to theorizing about the relevant linguistic phenomena. (shrink)

We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes.