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.

Notation: Theory of classes
The reader cannot assimilate the "Introduction to the Second Edition" unless they have assimilated the text of the first edition; this is obviously the case because the Introduction does not introduce the necessary symbolism so that it "stands alone". It begins as follows: "According to our present theory [ i.e. the 2nd edition ], all functions of functions [i.e. the 2nd order propositional or predicate calculus ] are extension [i.e. they have or are or produce, matrices (truth tables) ], i.e.
 * φx ≡x ψx ▪ ⊃ ▪ ƒ ( φẑ ) ≡ ƒ ( ψẑ )
 * IF, for all x, [the values of] function φ of x are logically equivalent to some function ψ of the same x, THEN the function ƒ applied to the elements φẑ (i.e. values of ψ evaluated at specific z) will be logically equivalent to the same function ƒ applied to the ψẑ (i.e. values of ψ evaluated at the same z).

2nd edition determines that "there is no longer any reason to distinguish between functions and classes, or we have by virtue of the formula above
 * φx≡x ψx▪ φẑ ) ≡ ψẑ.

From this PM deduces that the symbolism of the text and the reduced symbolism of the second edition are equivalent:
 * ẑ ( φ z ) ≡ ( φ z )

And they conclude that this is a simplification.

Begins without assuming the existence of classes, but provides a notation to represent them. Indeed, in ✸22 CALCULUS OF CLASSES, PM begins with the ironic statement: "In this number we reach what was historically the starting point of symbolic logic."

So what is a "class"?
 * "The characteristics of a class are that it consists of all the terms satisfying some propostional function, so that every propositional function determines a class, and two functions which are formally equivalent (i.e. such that whenever either is true, the other is true also) determine the same class, while conversely two functions which determine the same class are formally equivalent. When two functions are formally equivalent, we shall say that they have the same sextension." PM 1962:187.

x ε ✸20.01 ƒ { ẑ( ψ z ) } ▪ = ︰ (∃ φ) ︰( φ ! x) ≡x ψx︰ ƒ { φ ! z } Df
 * the collection of all the individuals i.e. the specific z, that satisfy the function ψ [?] or is it { ẑ( ψ ! z ) }


 * The abbreviation " ƒ { ẑ( ψ z ) } " is defined as follows:
 * There exists a function φ AND for all x its values (its matrix) ( φ ! x) is logically equivalent to " a function ψ of x AND a function ƒ applied to the matrix φ ! z

✸20.03 Cls = α^ { (∃φ) ▪ α = ẑ ( φ ! z) } Df

Primitive ideas, primitive propositions Pp
⊃ Ɔ ≡ ⊂ ∪ ∩ ∨ ∧ ∨ ∩ ∪ ⊂ ⊃ Ɔ ≡ ε Λ V ℩ ⊓ ⊔ ▪■︰✸ ✹ ✱ ⊢ ⊦ ẑ ŷ ŵ ẋ ⊽ ⊼ ^ ^ ȗ α  ∀ ∃ ƒ


 * Elementary propositions: one without variables
 * Elementary propositional functions: a proposition that contains one or more "undetermined constituents, i.e. a variable . . . such that when a value is assigned to the variable or variables, the resulting value of the expession is an elementary proposition"
 * Assertion: " ⊢ ▪ p. The sign " " is called the assertion-sign; it may be read "it is true that" (although philosophically this is not exactly what it means)"
 * Propositional functions divide propositions into two classes according to certain rules. These are usually named "truth" and "falsehood": "Thus from the formal point of view it is irrelevant what consitutes truth or falsehood: all that matters is that propositions are divided into two classes according to certain rules. . . . Our primitive propositions [define] hypotheses, not truths" (Appendix C "Truth-Functions and Others (cf PM 1962:402-403).


 * But this breaks down with the symbolism φ ! x. And PM concludes that propositions can state facts as well as represent hypotheses. (cf PM 1962:403).


 * PM introduces the notion of "predicate" (cf PM 1965:405), a generalization of a propositional function.