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The Incompleteness of Formal Logic

A Philosophical Criticism of Symbolic Calculus

Daniel J. Castellano (2009-12)

Table of Contents

I. Introduction

II. The Semantics of Logic
[Definitions of concepts]
Signs
Signification and Reference
Formalization
Logic, Logistic and Calculus

III. From Term Logic to Predicate Logic
[Historical introduction]
Terms
Aristotle's Theory of Predication
Logic and Metaphysics
Syllogisms
Boole's Calculus of Logic
Frege and Russell

IV. Modern Predicate Logic
Propositions
Propositional Functions
Predicates
Connectives
Atomic and Well-Formed Formulas
Quantifiers
Axioms
Inferences
Higher Order Logic

V. The Failure of Logicism
Russell's Claims
Atomic Formulas and Free Variables
Representable, Recursive, and Decidable Theories
Strong Undecidability Theorem
Strong Incompleteness Theorem

VI. Set Theory and Universals
Sets and Elements
Classes, Types and Boolean Universals
Zermelo-Fraenkel Theory and Axiom of Choice
Russell's Paradox
Structured Sets
Inadequacy of Set Theory

VII. Improving Predicate Calculus and Set Theory
Commentary on Trotter
Improved Lower Predicate Calculus
New Set Theoretic Axioms

VIII. Psychologism
Linguistic Reaction to Failure of Logicism
Psychological Naturalism of Language
Quine, Wittgenstein, Chomsky
Intensional Logic
Analytic Epistemology Presupposes Metaphysics
Limitations of Empiricism

IX. Conclusion


© 2009-2012 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org

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