Principia Mathematica

The Principia Mathematica is a threevolume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an allnew Appendix C.
PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy,^{1} being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them.
One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.
PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."
The Modern Library placed it 23rd in a list of the top 100 Englishlanguage nonfiction books of the twentieth century.^{2}
Contents
Scope of foundations laid
The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.
A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.
The construction of the theory of PM
As noted in the criticism of the theory by Kurt Gödel (below), unlike a Formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that almost immediately in the theory, interpretations (in the sense of model theory) are presented in terms of truthvalues for the behaviour of the symbols "⊢" (assertion of truth), "~" (logical not), and "V" (logical inclusive OR).
Truthvalues: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw (pure) Formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behave based on the grammar of the theory. Then later, by assignment of "values", a model would specify an interpretation of what the formulas are saying. Thus in the formal Kleene symbol set below, the "interpretation" of what the symbols commonly mean, and by implication how they end up being used, is given in parentheses, e.g., "¬ (not)". But this is not a pure Formalist theory.
The contemporary construction of a formal theory
The following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal system would be constructed as follows:
 Symbols used: This set is the starting set, and other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols "→" (implies, IFTHEN, "⊃"), "&" (and), "V" (or), "¬" (not), "∀" (for all), "∃" (there exists); predicate symbol "=" (equals); function symbols "+" (arithmetic addition), "∙" (arithmetic multiplication), "'" (successor); individual symbol "0" (zero); variables "a", "b", "c", etc.; and parentheses "(" and ")".^{3}
 Symbol strings: The theory will build "strings" of these symbols by concatenation (juxtaposition).^{4}
 Formation rules: The theory specifies the rules of syntax (rules of grammar) usually as a recursive definition that starts with "0" and specifies how to build acceptable strings or "wellformed formulas" (wffs).^{5} This includes a rule for "substitution".^{6} of strings for the symbols called "variables" (as opposed to the other symboltypes).
 Transformation rule(s): The axioms that specify the behaviours of the symbols and symbol sequences.
 Rule of inference, detachment, modus ponens : The rule that allows the theory to "detach" a "conclusion" from the "premises" that led up to it, and thereafter to discard the "premises" (symbols to the left of the line │, or symbols above the line if horizontal). If this were not the case, then substitution would result in longer and longer strings that have to be carried forward. Indeed, after the application of modus ponens, nothing is left but the conclusion, the rest disappears forever.
 Contemporary theories often specify as their first axiom the classical or modus ponens or "the rule of detachment":
 A, A ⊃ B │ B
 The symbol "│" is usually written as a horizontal line, here "⊃" means "implies". The symbols A and B are "standins" for strings; this form of notation is called an "axiom schema" (i.e., there is a countable number of specific forms the notation could take). This can be read in a manner similar to IFTHEN but with a difference: given symbol string IF A and A implies B THEN B (and retain only B for further use). But the symbols have no "interpretation" (e.g., no "truth table" or "truth values" or "truth functions") and modus ponens proceeds mechanistically, by grammar alone.
The logicistic construction of the theory of PM
The theory of PM has both significant similarities, and similar differences, to a contemporary formal theory. Kleene states that "this deduction of mathematics from logic was offered as intuitive axiomatics. The axioms were intended to be believed, or at least to be accepted as plausible hypotheses concerning the world".^{7} Indeed, unlike a Formalist theory that manipulates symbols according to rules of grammar, PM introduces the notion of "truthvalues", i.e., truth and falsity in the realworld sense, and the "assertion of truth" almost immediately as the fifth and sixth elements in the structure of the theory (PM 1962:4–36):
 1. Variables.
 2. Uses of various letters.
 3. The fundamental functions of propositions: "the Contradictory Function" symbolised by "~" and the "Logical Sum or Disjunctive Function" symbolised by "∨" being taken as primitive and logical implication defined (the following example also used to illustrate 9. Definition below) as

 p ⊃ q .=. ~ p ∨ q Df. (PM 1962:11)
 and logical product defined as
 p . q .=. ~(~p ∨ ~q) Df. (PM 1962:12)
 (See more about the confusing "dots" used as both a grammatical device and as to symbolise logical conjunction (logical AND) at the section on notation.)
 4. Equivalence: Logical equivalence, not arithmetic equivalence: "≡" given as a demonstration of how the symbols are used, i.e., "Thus ' p ≡ q ' stands for '( p ⊃ q ) . ( q ⊃ p )'." (PM 1962:7). Notice that to discuss a notation PM identifies a "meta"notation with "[space] ... [space]":^{8}
 Logical equivalence appears again as a definition:
 p ≡ q .=. ( p ⊃ q ) . ( q ⊃ p. ) (PM 1962:12),
 Notice the appearance of parentheses. This grammatical usage is not specified and appears sporadically; parentheses do play an important role in symbol strings, however, e.g., the notation "(x)" for the contemporary "∀x".
 5. Truthvalues: "The 'Truthvalue' of a proposition is truth if it is true, and "falsehood if it is false" (this phrase is due to Frege) (PM 1962:7).
 6. Assertionsign: "'⊦'. p may be read 'it is true that' ... thus '⊦: p .⊃. q ' means 'it is true that p implies q ', whereas '⊦. p .⊃⊦. q ' means ' p is true; therefore q is true'. The first of these does not necessarily involve the truth either of p or of q, while the second involves the truth of both" (PM 1962:92).
 7. Inference: PM 's version of modus ponens. "[If] '⊦. p ' and '⊦ (p ⊃ q)' have occurred, then '⊦ . q ' will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of '⊦. p ' [in other words, the symbols on the left disappear or can be erased]" (PM 1962:9).
 8. The Use of Dots: See the section on notation.
 9. Definitions: These use the "=" sign with "Df" at the right end. See the section on notation.
 10. Summary of preceding statements: brief discussion of the primitive ideas "~ p" and "p ∨ q" and "⊦" prefixed to a proposition.
 11. Primitive propositions: the axioms or postulates. This was significantly modified in the 2nd edition.
 12. Propositional functions: The notion of "proposition" was significantly modified in the 2nd edition, including the introduction of "atomic" propositions linked by logical signs to form "molecular" propositions, and the use of substitution of molecular propositions into atomic or molecular propositions to create new expressions.
 13. The range of values and total variation.
 14. Ambiguous assertion and the real variable: This and the next two sections were modified or abandoned in the 2nd edition. In particular, the distinction between the concepts defined in sections 15. Definition and the real variable and 16 Propositions connecting real and apparent variables was abandoned in the second edition.
 17. Formal implication and formal equivalence.
 18. Identity: See the section on notation. The symbol "=" indicates "predicate" or arithmetic equality.
 19. Classes and relations.
 20. Various descriptive functions of relations.
 21. Plural descriptive functions.
 22. Unit classes.
Primitive ideas
Cf. PM 1962:90–94, for the first edition:
 (1) Elementary propositions.
 (2) Elementary propositions of functions.
 (3) Assertion: introduces the notions of "truth" and "falsity".
 (4) Assertion of a propositional function.
 (5) Negation: "If p is any proposition, the proposition "notp", or "p is false," will be represented by "~p" ".
 (6) Disjunction: "If p and q are any propositons, the proposition "p or q, i.e., "either p is true or q is true," where the alternatives are to be not mutually exclusive, will be represented by "p ∨ q" ".
 (cf. section B)
Primitive propositions (Pp)
The first edition (see discusion relative to the second edition, below) begins with a definition of the sign "⊃"
✸1.01. p ⊃ q .=. ~ p ∨ q. Df.
✸1.1. Anything implied by a true elementary proposition is true. Pp modus ponens
(✸1.11 was abandoned in the second edition.)
✸1.2. ⊦: p ∨ p .⊃. p. Pp principle of tautology
✸1.3. ⊦: q .⊃. p ∨ q. Pp principle of addition
✸1.4. ⊦: p ∨ q .⊃. q ∨ p. Pp principle of permutation
✸1.5. ⊦: p ∨ ( q ∨ r ) .⊃. q ∨ ( p ∨ r ). Pp associative principle
✸1.6. ⊦:. q ⊃ r .⊃: p ∨ q .⊃. p ∨ r. Pp principle of summation
✸1.7. If p is an elementary proposition, ~p is an elementary proposition. Pp
✸1.71. If p and q are elementary propositions, p ∨ q is an elementary proposition. Pp
✸1.72. If φp and ψp are elementary propositional functions which take elementary propositions as arguments, φp ∨ ψp is an elementary proposition. Pp
Together with the "Introduction to the Second Edition", the second edition's Appendix A abandons the entire section ✸9. This includes six primitive propositions ✸9 through ✸9.15 together with the Axioms of reducibility.
The revised theory is made difficult by the introduction of the Sheffer stroke ("") to symbolise "incompatibility" (i.e., if both elementary propositions p and q are true, their "stroke" p  q is false), the contemporary logical NAND (notAND). In the revised theory, the Introduction presents the notion of "atomic proposition", a "datum" that "belongs to the philosophical part of logic". These have no parts that are propositions and do not contain the notions "all" or "some". For example: "this is red", or "this is earlier than that". Such things can exist ad finitum, i.e., even an "infinite eunumeration" of them to replace "generality" (i.e., the notion of "for all").^{9} PM then "advance[s] to molecular propositions" that are all linked by "the stroke". Definitions give equivalences for "~", "∨", "⊃", and ".".
The new introduction defines "elementary propositions" as atomic and molecular positions together. It then replaces all the primitive propositions ✸1.2 to ✸1.72 with a single primitive proposition framed in terms of the stroke:
 "If p, q, r are elementary propositions, given p and p(qr), we can infer r. This is a primitive proposition."
The new introduction keeps the notation for "there exists" (now recast as "sometimes true") and "for all" (recast as "always true"). Appendix A strengths the notion of "matrix" or "predicative function" (a "primitive idea", PM 1962:164) and presents four new Primitive propositions as ✸8.1–✸8.13.
✸88. Multiplicative axiom
✸120. Axiom of infinity
Notation used in PM
One author^{1} observes that "The notation in that work has been superseded by the subsequent development of logic during the 20th century, to the extent that the beginner has trouble reading PM at all"; while much of the symbolic content can be converted to modern notation, the original notation itself is "a subject of scholarly dispute", and some notation "embod[y] substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism".^{10}
Kurt Gödel was harshly critical of the notation:
 "It is to be regretted that this first comprehensive and thoroughgoing presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in ✸1–✸21 of Principia [i.e., sections ✸1–✸5 (propositional logic), ✸8–14 (predicate logic with identity/equality), ✸20(introduction to set theory), and ✸21 (introduction to relations theory)]) that it represents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs".^{11}
This is reflected in the example below of the symbols "p", "q", "r" and "⊃" that can be formed into the string "p ⊃ q ⊃ r". PM requires a definition of what this symbolstring means in terms of other symbols; in contemporary treatments the "formation rules" (syntactical rules leading to "well formed formulas") would have prevented the formation of this string.
Source of the notation: Chapter I "Preliminary Explanations of Ideas and Notations" begins with the source of the notation:
 "The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his Formulario Mathematico [i.e., Peano 1889]. His use of dots as brackets is adopted, and so are many of his symbols" (PM 1927:4).^{12}
PM adopts the assertion sign "⊦" from Frege's 1879 Begriffsschrift:^{13}
 "(I)t may be read 'it is true that'"^{14}
Thus to assert a proposition p PM writes:
 "⊦. p." (PM 1927:92)
(Observe that, as in the original, the left dot is square and of greater size than the period on the right.)
An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)
PM 's dots^{15} are used in a manner similar to parentheses. Later in section ✸14, brackets "[ ]" appear, and in sections ✸20 and following, braces "{ }" appear. Whether these symbols have specific meanings or are just for visual clarification is unclear. More than one dot indicates the "depth" of the parentheses, e.g., ".", ":" or ":.", "::", etc. Unfortunately for contemporary readers, the single dot (but also ":", ":.", "::", etc.) is used to symbolise "logical product" (contemporary logical AND often symbolised by "&" or "∧").
Logical implication is represented by Peano's "Ɔ" simplified to "⊃", logical negation is symbolised by an elongated tilde, i.e., "~" (contemporary "~" or "¬"), the logical OR by "v". The symbol "=" together with "Df" is used to indicate "is defined as", whereas in sections ✸13 and following, "=" is defined as (mathematically) "identical with", i.e., contemporary mathematical "equality" (cf. discussion in section ✸13). Logical equivalence is represented by "≡" (contemporary "if and only if"); "elementary" propositional functions are written in the customary way, e.g., "f(p)", but later the function sign appears directly before the variable without parenthesis e.g., "φx", "χx", etc.
Example, PM introduces the definition of "logical product" as follows:
 ✸3.01. p . q .=. ~(~p v ~q) Df.
 where "p . q" is the logical product of p and q.
 ✸3.02. p ⊃ q ⊃ r .=. p ⊃ q . q ⊃ r Df.
 This definition serves merely to abbreviate proofs.
Translation of the formulas into contemporary symbols: Various authors use alternate symbols, so no definitive translation can be given. However, because of criticisms such as that of Kurt Gödel below, the best contemporary treatments will be very precise with respect to the "formation rules" (the syntax) of the formulas.
The first formula might be converted into modern symbolism as follows:^{16}
 (p & q) =_{df} (~(~p v ~q))
alternately
 (p & q) =_{df} (¬(¬p v ¬q))
alternately
 (p ∧ q) =_{df} (¬(¬p v ¬q))
etc.
The second formula might be converted as follows:
 (p → q → r) =_{df} (p → q) & (q → r)
But note that this is not (logically) equivalent to (p → (q → r)) nor to ((p → q) → r), and these two are not logically equivalent either.
An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)
These sections concern what is now known as Predicate logic, and Predicate logic with identity (equality).

 NB: As a result of criticism and advances, the second edition of PM (1927) replaces ✸9 with a new ✸8 (Appendix A). This new section eliminates the first edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'.^{17} To add to the complexity of the treatment, ✸8 introduces the notion of substituting a "matrix", and the Sheffer stroke:


 Matrix: In contemporary usage, PM 's matrix is (at least for propositional functions), a truth table, i.e., all truthvalues of a propositional or predicate function.
 Sheffer stroke: Is the contemporary logical NAND (NOTAND), i.e., "incompatibility", meaning:
 "Given two propositions p and q, then ' p  q ' means "proposition p is incompatible with proposition q, i.e., if both propositions p and q evaluate as false, then p  q evaluates as true." After section ✸8 the Sheffer stroke sees no usage.

Section ✸10: The existential and universal "operators": PM adds "(x)" to represent the contemporary symbolism "for all x " i.e., " ∀x", and it uses a backwards serifed E to represent "there exists an x", i.e., "(Ǝx)", i.e., the contemporary "∃x". The typical notation would be similar to the following:
 "(x) . φx" means "for all values of variable x, function φ evaluates to true"
 "(Ǝx) . φx" means "for some value of variable x, function φ evaluates to true"
Sections ✸10, ✸11, ✸12: Properties of a variable extended to all individuals: section ✸10 introduces the notion of "a property" of a "variable". PM gives the example: φ is a function that indicates "is a Greek", and ψ indicates "is a man", and χ indicates "is a mortal" these functions then apply to a variable x. PM can now write, and evaluate:
 (x) . ψx
The notation above means "for all x, x is a man". Given a collection of individuals, one can evaluate the above formula for truth or falsity. For example, given the restricted collection of individuals { Socrates, Plato, Russell, Zeus } the above evaluates to "true" if we allow for Zeus to be a man. But it fails for:
 (x) . φx
because Russell is not Greek. And it fails for
 (x) . χx
because Zeus is not a mortal.
Equipped with this notation PM can create formulas to express the following: "If all Greeks are men and if all men are mortals then all Greeks are mortals". (PM 1962:138)
 (x) . φx ⊃ ψx :(x). ψx ⊃ χx :⊃: (x) . φx ⊃ χx
Another example: the formula:
 ✸10.01. (Ǝx). φx . = . ~(x) . ~φx Df.
means "The symbols representing the assertion 'There exists at least one x that satisfies function φ' is defined by the symbols representing the assertion 'It's not true that, given all values of x, there are no values of x satisfying φ'".
The symbolisms ⊃_{x} and "≡_{x}" appear at ✸10.02 and ✸10.03. Both are abbreviations for universality (i.e., for all) that bind the variable x to the logical operator. Contemporary notation would have simply used parentheses outside of the equality ("=") sign:
 ✸10.02 φx ⊃_{x} ψx .=. (x). φx ⊃ ψx Df
 Contemporary notation: ∀x(φ(x) → ψ(x)) (or a variant)
 ✸10.03 φx ≡_{x} ψx .=. (x). φx ≡ ψx Df
 Contemporary notation: ∀x(φ(x) ↔ ψ(x)) (or a variant)
PM attributes the first symbolism to Peano.
Section ✸11 applies this symbolism to two variables. Thus the following notations: ⊃_{x}, ⊃_{y}, ⊃_{x, y} could all appear in a single formula.
Section ✸12 reintroduces the notion of "matrix" (contemporary truth table), the notion of logical types, and in particular the notions of firstorder and secondorder functions and propositions.
New symbolism "φ ! x" represents any value of a firstorder function. If a circumflex "＾" is placed over a variable, then this is an "individual" value of y, meaning that "ŷ" indicates "individuals" (e.g., a row in a truth table); this distinction is necessary because of the matrix/extensional nature of propositional functions.
Now equipped with the matrix notion, PM can assert its controversial axiom of reducibility: a function of one or two variables (two being sufficient for PM 's use) where all its values are given (i.e., in its matrix) is (logically) equivalent ("≡") to some "predicative" function of the same variables. The onevariable definition is given below as an illustration of the notation (PM 1962:166–167):
✸12.1 ⊢: (Ǝ f): φx .≡_{x}. f ! x Pp;

 Pp is a "Primitive proposition" ("Propositions assumed without proof") (PM 1962:12, i.e., contemporary "axioms"), adding to the 7 defined in section ✸1 (starting with ✸1.1 modus ponens). These are to be distinguished from the "primitive ideas" that include the assertion sign "⊢", negation "~", logical OR "V", the notions of "elementary proposition" and "elementary propositional function"; these are as close as PM comes to rules of notational formation, i.e., syntax.
This means: "We assert the truth of the following: There exists a function f with the property that: given all values of x, their evaluations in function φ (i.e., resulting their matrix) is logically equivalent to some f evaluated at those same values of x. (and vice versa, hence logical equivalence)". In other words: given a matrix determined by property φ applied to variable x, there exists a function f that, when applied to the x is logically equivalent to the matrix. Or: every matrix φx can be represented by a function f applied to x, and vice versa.
✸13: The identity operator "=" : This is a definition that uses the sign in two different ways, as noted by the quote from PM:
 ✸13.01. x = y .=: (φ): φ ! x . ⊃ . φ ! y Df
means:
 "This definition states that x and y are to be called identical when every predicative function satisfied by x is also satisfied by y ... Note that the second sign of equality in the above definition is combined with "Df", and thus is not really the same symbol as the sign of equality which is defined."
The notequals sign "≠" makes its appearance as a definition at ✸13.02.
✸14: Descriptions:
 "A description is a phrase of the form "the term y which satisfies φŷ, where φŷ is some function satisfied by one and only one argument."^{18}
From this PM employes two new symbols, a forward "E" and an inverted iota "ɿ". Here is an example:
 ✸14.02. E ! ( ɿy) (φy) .=: ( Ǝb):φy . ≡_{y} . y = b Df.
This has the meaning:
 "The y satisfying φŷ exists," which holds when, and only when φŷ is satisfied by one value of y and by no other value." (PM 1967:173–174)
Introduction to the notation of the theory of classes and relations
The text leaps from section ✸14 directly to the foundational sections ✸20 GENERAL THEORY OF CLASSES and ✸21 GENERAL THEORY OF RELATIONS. "Relations" are what known in contemporary set theory as ordered pairs. Sections ✸20 and ✸22 introduce many of the symbols still in contemporary usage. These include the symbols "ε", "⊂", "∩", "∪", "–", "Λ", and "V": "ε" signifies "is an element of" (PM 1962:188); "⊂" (✸22.01) signifies "is contained in", "is a subset of"; "∩" (✸22.02) signifies the intersection (logical product) of classes (sets); "∪" (✸22.03) signifies the union (logical sum) of classes (sets); "–" (✸22.03) signifies negation of a class (set); "Λ" signifies the null class; and "V" signifies the universal class or universe of discourse.
Small Greek letters (other than "ε", "ι", "π", "φ", "ψ", "χ", and "θ") represent classes (e.g., "α", "β", "γ", "δ", etc.) (PM 1962:188):
 x ε α
 "The use of single letter in place of symbols such as ẑ(φz) or ẑ(φ ! z) is practically almost indispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus ' x ε α' will mean ' x is a member of the class α'". (PM 1962:188)
 α ∪ –α = V
 The union of a set and its inverse is the universal (completed) set.^{19}
 α ∩ –α = Λ
 The intersection of a set and its inverse is the null (empty) set.
When applied to relations in section ✸23 CALCULUS OF RELATIONS, the symbols "⊂", "∩", "∪", and "–" acquire a dot: for example: "⊍", "∸".^{20}
The notion, and notation, of "a class" (set): In the first edition PM asserts that no new primitive ideas are necessary to define what is meant by "a class", and only two new "primitive propositions" called the axioms of reducibility for classes and relations respectively (PM 1962:25).^{21} But before this notion can be defined, PM feels it necessary to create a peculiar notation "ẑ(φz)" that it calls a "fictitious object". (PM 1962:188)
 ⊢: x ε ẑ(φz) .≡. (φx)
 "i.e., ' x is a member of the class determined by (φẑ)' is [logically] equivalent to ' x satisfies (φẑ),' or to '(φx) is true.'". (PM 1962:25)
At least PM can tell the reader how these fictitious objects behave, because "A class is wholly determinate when its membership is known, that is, there cannot be two different classes having he same membership" (PM 1962:26). This is symbolised by the following equality (similar to ✸13.01 above:
 ẑ(φz) = ẑ(ψz) . ≡ : (x): φx .≡. ψx
 "This last is the distinguishing characteristic of classes, and justifies us in treating ẑ(ψz) as the class determined by [the function] ψẑ." (PM 1962:188)
Perhaps the above can be made clearer by the discussion of classes in Introduction to the 2nd Edition, which disposes of the Axiom of Reducibility and replaces it with the notion: "All functions of functions are extensional" (PM 1962:xxxix), i.e.,
 φx ≡_{x} ψx .⊃. (x): ƒ(φẑ) ≡ ƒ(ψẑ) (PM 1962:xxxix)
This has the reasonable meaning that "IF for all values of x the truthvalues of the functions φ and ψ of x are [logically] equivalent, THEN the function ƒ of a given φẑ and ƒ of ψẑ are [logically] equivalent." PM asserts this is "obvious":
 "This is obvious, since φ can only occur in ƒ(φẑ) by the substitution of values of φ for p, q, r, ... in a [logical] function, and, if φx ≡ ψx, the substitution of φx for p in a [logical] function gives the same truthvalue to the truthfunction as the substitution of ψx. Consequently there is no longer any reason to distinguish between functions classes, for we have, in virtue of the above,
 φx ≡_{x} ψx .⊃. (x). φẑ = . ψẑ".
Observe the change to the equality "=" sign on the right. PM goes on to state that will continue to hang onto the notation "ẑ(φz)", but this is merely equivalent to φẑ, and this is a class. (all quotes: PM 1962:xxxix).
Consistency and criticisms
According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.
Beyond the status of the axioms as logical truths, the questions remained:
 whether a contradiction could be derived from the Principia's axioms (the question of inconsistency), and
 whether there exists a mathematical statement which could neither be proven nor disproven in the system (the question of completeness).
Propositional logic itself was known to be consistent, but the same had not been established for Principia's axioms of set theory. (See Hilbert's second problem.)
Gödel 1930, 1931
In 1930, Gödel's completeness theorem showed that firstorder predicate logic itself was complete in a much weaker sense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms (such as those of Principia Mathematica) may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.
Gödel's incompleteness theorems cast unexpected light on these two related questions.
Gödel's first incompleteness theorem showed that Principia could not be both consistent and complete. According to the theorem, within every sufficiently powerful logical system (such as Principia), there exists a statement G that essentially reads, "The statement G cannot be proved." Such a statement is a sort of Catch22: if G is provable, then it is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.
Gödel's second incompleteness theorem (1931) shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the Principia system" cannot be proven in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).
Wittgenstein 1919, 1939
By the second edition of PM, Russell had removed his axiom of reducibility to a new axiom (although he does not state it as such). Gödel 1944:126 describes it this way: "This change is connected with the new axiom that functions can occur in propositions only "through their values", i.e., extensionally . . . [this is] quite unobjectionable even from the constructive standpoint . . . provided that quantifiers are always restricted to definite orders". This change from a quasiintensional stance to a fully extensional stance also restricts predicate logic to the second order, i.e. functions of functions: "We can decide that mathematics is to confine itself to functions of functions which obey the above assumption" (PM 2nd Edition p. 401, Appendix C).
This new proposal resulted in a dire outcome. An "extensional stance" and restriction to a secondorder predicate logic means that a propositional function extended to all individuals such as "All 'x' are blue" now has to list all of the 'x' that satisfy (are true in) the proposition, listing them in a possibly infinite conjunction: e.g. x_{1} V x_{2} V . . . V x_{n} V . . .. Ironically, this change came about as the result of criticism from Wittgenstein in his 1919 Tractatus LogicoPhilosophicus. As described by Russell in the Preface to the 2nd edition of PM:
 "There is another course, recommended by Wittgenstein† (†Tractatus LogicoPhilosophicus, *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truthfunctions, and that a function can only occur in a proposition through its values. . . . [Working through the consequences] it appears that everything in Vol. I remains true . . . the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and wellordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2^{n} > n breaks down unless n is finite." (PM 2nd edition reprinted 1962:xiv, also cf new Appendix C).
In other words, the fact that an infinite list cannot realistically be specified means that the concept of "number" in the infinite sense (i.e. the continuum) cannot be described by the new theory proposed in PM Second Edition.
Wittgenstein in his Lectures on the Foundations of Mathematics, Cambridge 1939 criticised Principia on various grounds, such as:
 It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g., that Principia did not characterise numbers or addition correctly), not as evidence of an error in everyday counting.
 The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g., billions), the formulae would become too long, and some shortcut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on nonfundamental and hence questionable methods such as induction). So again Principia depends on everyday techniques, not vice versa.
Wittgenstein did, however, concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.
Gödel 1944
In his 1944 Russell's mathematical logic, Gödel offers a "critical but sympathetic discussion of the logicistic order of ideas":^{22}
 "It is to be regretted that this first comprehensive and thoroughgoing presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in *1*21 of Principia) that it represents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs . . . The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their definiens . . . it is chiefly the rule of substitution which would have to be proved" (Gödel 1944:124)^{23}
Quotations
 "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful.")
See also
Footnotes
 ^ ^{a} ^{b} Irvine, Andrew D. (1 May 2003). "Principia Mathematica (Stanford Encyclopedia of Philosophy)". Metaphysics Research Lab, CSLI, Stanford University. Retrieved 5 August 2009.
 ^ "The Modern Library's Top 100 Nonfiction Books of the Century". The New York Times Company. 30 April 1999. Retrieved 5 August 2009.
 ^ This set is taken from Kleene 1952:69 substituting → for ⊃.
 ^ Kleene 1952:71, Enderton 2001:15
 ^ Enderton 2001:16
 ^ This is the word used by Kleene 1952:78
 ^ Quote from Kleene 1952:45. See discussion LOGICISM at pages 43–46.
 ^ In his section 8.5.4 Groping towards metalogic GrattainGuiness 2000:454ff discusses the American logicians' critical reception of the second edition of PM. For instance Sheffer "puzzled that ' In order to give an account of logic, we must presuppose and employ logic ' " (p. 452). And Bernstein ended his 1926 review with the comment that "This distinction between the propositional logic as a mathematical system and as a language must be made, if serious errors are to be avoided; this distinction the Principia does not make" (p.454).
 ^ This idea is due to Wittgenstein's Tractatus. See the discussion at PM 1962:xiv–xv)
 ^ http://plato.stanford.edu/entries/pmnotation/
 ^ Kurt Gödel 1944 "Russell's mathematical logic" appearing at page 120 in Feferman et al. 1990 Kurt Gödel Collected Works Volume II, Oxford University Press, NY, ISBN 9780195147216(v.2.pbk.) .
 ^ For comparison, see the translated portion of Peano 1889 in van Heijenoort 1967:81ff. About the only major change I can see is the substitution of ⊃ for Ɔ as used by Peano.
 ^ This work can be found at van Heijenoort 1967:1ff.
 ^ And see footnote, both at PM 1927:92
 ^ The original typography is a square of a heavier weight than the conventional period.
 ^ The first example comes from plato.stanford.edu (loc.cit.).
 ^ page xiii of 1927 appearing in the 1962 paperback edition to ✸56.
 ^ The original typography employs an x with a circumflex rather than ŷ; this continues below
 ^ See the ten postulates of Huntington, in particular postulates IIa and IIb at PM 1962:205 and discussion at page 206.
 ^ The "⊂" sign has a dot inside it, and the intersection sign "∩" has a dot above it; these are not available in the Arial Unicode MS font.
 ^ Wiener 1914 "A simplification of the logic of relations" (van Hejenoort 1967:224ff) disposed of the second of these when he showed how to reduce the theory of relations to that of classes
 ^ Kleene 1952:46.
 ^ Gödel 1944 Russell's mathematical logic in Kurt Gödel: Collected Works Volume II, Oxford University Press, New York, NY, ISBN 019514721 .
References
Primary:
 Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.
 Alfred North Whitehead; Bertrand Russell (February 2009). Principia Mathematica. Volume One. Merchant Books. ISBN 9781603861823.
 Alfred North Whitehead; Bertrand Russell (February 2009). Principia Mathematica. Volume Two. Merchant Books. ISBN 9781603861830.
 Alfred North Whitehead; Bertrand Russell (February 2009). Principia Mathematica. Volume Three. Merchant Books. ISBN 9781603861847.
Secondary:
 Stephen Kleene 1952 Introduction to MetaMathematics, 6th Reprint, NorthHolland Publishing Company, Amsterdam NY, ISBN 0720421039.
 Stephen Cole Kleene; Michael Beeson (March 2009). Introduction to Metamathematics (Paperback ed.). Ishi Press. ISBN 9780923891572.
 Ivor GrattanGuinness (2000) The Search for Mathematical Roots 1870–1940, Princeton University Press, Princeton N.J., ISBN 0691058571 (alk. paper).
 Ludwig Wittgenstein 2009 Major Works: Selected Philosophical Writings, HarperrCollins, NY, NY, ISBN 9780061550249. In particular:

 Tractatus LogicoPhilosophicus (Vienna 1918, original publication in German).
 Jean van Heijenoort editor 1967 From Frege to Gödel: A Source book in Mathematical Logic, 1879–1931, 3rd printing, Harvard University Press, Cambridge MA, ISBN 0674324498 (pbk.)
External links
 Stanford Encyclopedia of Philosophy:
 Principia Mathematica—by A. D. Irvine.
 The Notation in Principia Mathematica—by Bernard Linsky.
 Principia Mathematica online (University of Michigan Historical Math Collection):
 Proposition ✸54.43 in a more modern notation (Metamath)

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