User:Wvbailey/PM

Sandbox for the article Principia Mathematica: ⊃ Ɔ ≡ ⊂ ∪ ∩ ∨ ∧ ∨ ∩ ∪ ⊂ ⊃ Ɔ ≡ ε Λ ℩ ⊓ ⊔ ▪■︰✸ ✹ ✱

Organization

Three volumnes

Second edition (1927) with an important "Introduction To the Second Edition" that introduces the Sheffer stroke, a new chapter ✸8 to replace chapter ✸9 (and thereby remove the distincition between "real" and "apparent" variables).

A 1962 abridged single-volume version to ✸56.

Organization of abridged single-volume version to ✸56

PREFACE

Preface to first edition (dated 1910)

INTRODUCTION TO THE SECOND EDITION

Introduces the Sheffer stroke. Responds to criticisms of Wittgenstein and advances made by Hilbert et. al. by removing the distinction between "real" and "apparent" variable. This is detailed in new section ✸8 to replace chapter ✸9 (but leaves ✸9 in place). PM then abandons the axiom of reducibility:

"The theory of classes is at once simplified in one direction and complicated in another by the assumption that functions only occur through their values and by abandonment of the axiom of reducibility"

In its place PM substitutes a "fully-extensional" notion of a function (i.e. a matrix of values, e.g. for propositional and predicate functions the contemporary notion of truth table introduced by Wittgenstein 1917 and Post 1921). To avoid a "Vicious circle", i.e. created by plugging a function ψ into itself e.g. ψ(ψ(x)), PM stipulates that:

"A function can only appear in a matrix through its values* [*This assumption is fundamental in the following theory. It has its difficulties, but for the moment we ignore them. It takes the place (not quite adequately) of the axiom of reducibility. It is discussed in Appendix C]".

INTRODUCTION

Chapter I. Preliminary Explanations of Ideas and Notations

Chapter II. The theory of Logical Types

This must be read together with the "Introduction to the Second Edition" and Appendix A (new section ✸8 replacing ✸9 of the first edition). Begins with a discussion of the "Vicious-Circle Principle", introduces the notion of the extension of a function (i.e. its values) and "matrix" (contemporary truth table), predicative functions ("it is of the next order above that of its arguments, i.e. of the lowest order compatible with its having that argument", introduces the Axiom of Reducibility. Extensive discussion of "The Contradictions" with 7 examples and detailed analysis.

Chapter III. Incomplete Symbols

"By an 'incomplete' symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts"; examples are given of the derivative d/dx, the definite integral, and the del operator.

PART I. MATHEMATICAL LOGIC

Six sections A - E from ✸1 to ✸43

✸1-✸5 are equivalent to contemporary Propositional logic

✸8 (Appendix A replacing ✸9) - ✸13 are equivalent to contemporary Predicate logic. Both first-order and higher order logic are treated in ✸12 which must be read together with the "Introduction to the Second Edition" aand Appendices A and C.

✸11 adds IDENTITY to the predicate logic.

✸12 introduces the notion of a "description"; but see also ✸30 DESCRIPTIVE FUNCTIONS and following sections.

In sections ✸20 - ✸43 PM defines, for its use, the notions of "classes" (sets) and "relations" (ordered pairs) similar to those found in contemporary set theory. Unfortunately it parallels the treatment of "class" (set) with that of "relation": "A relation . . . will be understood in extension: it may be regarded as the class of couples (x,y) for which some given function ψ(x,y) is true". Treatment of classes is found in ✸20, ✸22, ✸24, and section E "Products and Sums of Classes". Treatment of relations is found in ✸21, ✸23, ✸25, all of section D "Logic of Relations" ✸30 - ✸38 and portions of section E "Products and Sums of Classes".

PART II. PROLEGOMENA TO CARDINAL ARITHMETIC

Section A from ✸50 to ✸56

APPENDIX A

New section ✸8 to replace ✸9 (but ✸9 is left in place)

APPENDIX C: Truth-Functions and others

LIST of DEFINITIONS

Approximately 100 symbols and their usage in an example formula, but without definition nor the location of their first occurrence, nor presented in order of their occurrence.