Logic and Fixed-Domain Reasoning: Volume 1: An Empirico-Pragmatist Model Theory and Proof Theory

The author develops, incrementally over the course of several chapters, related concepts of logical system over fixed domain, both classical and nonclassical, by means of pragmatism-inspired translations into paradigmatic, infinitary, quantifier-free languages having rigidly designating constants for all elements of a fixed domain. Each concept fully accommodates both the descriptive (model-theoretic) and deductive (proof-theoretic) tasks of any logic. Chapters 1 through 7 concern first-order logics over fixed domain. Final Chapter 8 takes up second-order logics whereby standard names for each and every boolean-valued function are now introduced. Along the way, substitutional theories of first-, alternatively, second-order relations are presented following Russell. Clarification of classification questions (What is a logical vs. mathematical term? A logical law? A logical operation?) is achieved, for logics over fixed domain, in a manner that may be unattainable in the case of variable-domain logics. The author adopts a nonstandard view according to which the fixed-domain logics of Peirce, Zermelo, and Carnap are no mere stepping stones on the path to variable-domain logics but, rather, constitute an alternative conception having certain clarificatory advantages. Consequently, the book should be of interest to both logicians and philosophers of logic.