1. Introduction. Since this paper is a written version of one presented as a survey lecture at a meeting of the Association for Symbolic Logic, its form and content have been essentially determined by the form and content of that lecture, and these latter were themselves considerably influenced by a deliberate attempt to avoid the lecture consisting essentially either of a catalogue of results in some fairly wide field or of the presentation of detailed proofs of a few particular results in a very limited field, with, in either case, a consequential serious limitation of the time available for discussion of the motivation for the results and of the connections between them. There is thus no attempt to cover, even in outline, the full range of topics which could reasonably fall within the scope of its title.
The paper consists mainly of a general discussion of some results concerned with finite models of axiomatically defined calculi of propositional form, with particular reference to the occurrence of such models in proofs that certain propositional calculi are decidable. In addition, there is an account of some results concerned with the existence of undecidable propositional calculi and of others connected with the decidability or undecidability of certain structure problems which relate either to sets of propositional calculi or to different presentations of the same calculus.