The relationship between logic and grammar has been a source of dispute all through the history of linguistics. Attitudes have ranged all the way from the marriage of grammar, logic, and ontology of the medieval modistae (Thomas of Erfurt) to the atheoretical descriptivism of American structural linguistics (Bloomfield, Harris).
Today the r61e of logic in linguistics is once more expanding. From being read out of linguistic theory as metaphysics semantics is once more a respected part of the grammatical enterprise.
Semantics has been a main ingredient in the development of formal logic from Frege via Tarski to the present time. And applications have been many and varied.
But applications of logic and formal semantics to the study of natural languages were slow in coming. Perhaps for good reasons. The rigid formalism of predicate logic seems to be too blunt an instrument for analyzing the fluid forms of a spoken tongue.
Formal logic is, however, much more than predicate logic. And modelbuilding is much more than mere formalization. One of the first who tried to apply the full range of tools supplied by modern logic to the analysis of natural languages was Hans Reichenbach who included one chapter on 'conversational languages' in his textbook from 1947, Elements of Symbolic Logic. Reichenbach used as his tool higher order logic extended by certain 'pragmatic' operators.
Dating from about the same time is a paper by HaskeU B. Curry, 'Some Logical Aspects of Grammatical Structure'. The paper was written in 1948 but was first published in 1961. In it Curry gives an analysis of the traditional parts of speech in terms of combinatory logic. It is closely related to, but independent of, the categorial analysis of S. Lesniewski and K. Ajdukiewicz.