Russell on Propositions and Objects

Dissertation, University of Toronto (Canada) (1993)

Judy Pelham
York University
This thesis discusses Russell's logical work from the period 1903 to 1908, focusing on three well-developed logical systems: The Principles of Mathematics, the substitution theory, and Principia Mathematica. I explore Russell's formal work as well as his philosophy of logic. My thesis is that Russell retained the belief throughout this period that propositions or propositional functions are abstract objects with a definite structure, and that apprehension of that structure is the foundation of human reasoning. ;The first chapter discusses the philosophical and mathematical background to Russell's logicism, and gives an account of his philosophy of logic in The Principles of Mathematics. The second chapter presents Russell's system of logic in The Principles of Mathematics, and the paradoxes that Russell presents there. The relationship of his logical work to his philosophy of logic is discussed, and a section proves the formal completeness of a modern form of Russell's propositional logic. ;The third chapter, on the substitution theory of 1905, explicates one of Russell's attempts to construct a type-free theory which avoids the paradoxes. I consider an axiomatization of the substitution theory presented in an unpublished manuscript of Russell's dated 1905. A modern reconstruction of the substitution theory is given, which presents a recursively defined structure which models Russell's understanding of propositions. The form of the paradox which caused Russell to abandon the substitution theory is presented, and it is shown how the reconstruction of the substitution theory may avoid this paradox using a fixed point construction. ;The fourth chapter presents the evolution of the ramified theory of types. This theory is the solution to the paradoxes Russell developed after abandoning the substitution theory. The different versions of this theory are discussed showing how they ended in the theory of Principia Mathematica
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