The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A (that replaced ✱9), and with an additional new Appendix B and Appendix C. PM was conceived as a sequel to Russell's 1903 The Principles of Mathematics, but as PM states, this became an unworkable suggestion for practical and philosophical reasons:
PM, according to its introduction, had three aims:
to analyse to the greatest possible extent the ideas and methods of mathematical logic and to minimise the number of primitive notions, axioms, and inference rules; to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; to solve the paradoxes that plagued logic and set theory at the turn of the 20th century, like Russell's paradox.
This third aim motivated the adoption of the theory of types in PM. The theory of types adopts grammatical restrictions on formulas that rule out the unrestricted comprehension of classes, properties, and functions. The effect of this is that formulas such as would allow the comprehension of objects like the Russell set turn out to be ill-formed: they violate the grammatical restrictions of the system of PM.
PM sparked interest in symbolic logic and advanced the subject, popularizing it and demonstrating its power. The Modern Library placed PM 23rd in their list of the top 100 English-language nonfiction books of the 20th century.
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.
Theoretical basis
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". Furthermore in the theory, it is almost immediately observable that interpretations (in the sense of model theory) are presented in terms of truth-values for the behaviour of the symbols "⊢" (assertion of truth), "∾" (logical not), and "V" (logical inclusive OR).
Truth-values: 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.
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, IF-THEN, and "⊃"), "&" (and), "∨" (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 ")".This set is taken from substituting → for ⊃. Symbol strings: The theory will build "strings" of these symbols by concatenation (juxtaposition)., . 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 "well-formed formulas" (wffs). This includes a rule for "substitution"This is the word used by . of strings for the symbols called "variables". 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": The symbol "│" is usually written as a vertical line, here "⊃" means "implies". The symbols A and B are "stand-ins" 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 IF-THEN 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.
Construction 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".Quote from Kleene 1952:45. See discussion LOGICISM at pp. 43–46. Indeed, unlike a Formalist theory that manipulates symbols according to rules of grammar, PM introduces the notion of "truth-values", i.e., truth and falsity in the real-world sense, and the "assertion of truth" almost immediately as the fifth and sixth elements in the structure of the theory (PM 1962:4–36): Variables Uses of various letters 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) asp ⊃ q .=. ∾ p ∨ q Df. (PM 1962:11) and logical product defined as p . q .=. ∾(∾p ∨ ∾q) Df. (PM 1962:12) 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]":In his section 8.5.4 Groping towards metalogic Grattan-Guinness 2000:454ff discusses the American logicians' critical reception of the second edition of PM. For instance Sheffer "puzzled that ' 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).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". Truth-values: "The 'Truth-value' of a proposition is truth if it is true, and falsehood if it is false" (this phrase is due to Gottlob Frege) (PM 1962:7). Assertion-sign: "'⊦. 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). Inference: PMs 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 '⊦. q ' [in other words, the symbols on the left disappear or can be erased]" (PM 1962:9). The use of dots Definitions: These use the "=" sign with "Df" at the right end. Summary of preceding statements: brief discussion of the primitive ideas "∾ p" and "p ∨ q" and "⊦" prefixed to a proposition. Primitive propositions: the axioms or postulates. This was significantly modified in the second edition. Propositional functions: The notion of "proposition" was significantly modified in the second 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. The range of values and total variation Ambiguous assertion and the real variable: This and the next two sections were modified or abandoned in the second 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. Formal implication and formal equivalence Identity Classes and relations Various descriptive functions of relations Plural descriptive functions 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 "not-p", or "p is false," will be represented by "∾p" ". (6) Disjunction: "If p and q are any propositions, 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
The first edition (see discussion 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.