Many-valued logics. a study of the relationship of propositional calculi and algebraic systems.

This thesis sets out to examine the possibility of devising a theory which will give a unified account of prepositional calculi and algebraic systems. Starting from a historical account of the principal ideas tributary to the main stream of theory from Boole to the present day, it presents a technical- language framework within which it is possible to develop in a uniform format substantial portions of the theories of both sorts of system. The idea of an Interpretation then leads to a discussion of Functional Completeness, and the use of Galois fields in the algebraic representation of functions. Two particular families of systems, the Protomodules and Protorings, are selected for more detailed study. Their principal decision problems are considered, their structure examined, and their relationship to familiar systems of algebra and prepositional calculus displayed. The discussion then specialises again to the use of Galois fields in the solution of computational problems arising in connection with an important class of protorings, the so- called Galois Logics. One of these problems is of sufficient complexity to warrant the use of an automatic digital computer, and details of the computer program are presented in an appendix. Three other appendices are devoted to the presentation of material which evolved as by-products during the contemplation of the main issues; they are concerned with closely related topics, and are given here in support of the thesis rather than as part of the theory.