# 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