Principia Mathematica

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 ")".

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" of strings for the symbols called "variables".

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". 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]":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).

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.

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 (not-AND). 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 enumeration" of them to replace "generality" (i.e., the notion of "for all"). PM then "advance[s] to molecular propositions" that are all linked by "the stroke". Definitions give equivalences for "∾", "∨", "⊃", and ".".

Ramified types and the axiom of reducibility

In simple type theory objects are elements of various disjoint "types". Types are implicitly built up as follows. If τ1,...,τm are types then there is a type (τ1,...,τm) that can be thought of as the class of propositional functions of τ1,...,τm (which in set theory is essentially the set of subsets of τ1×...×τm). In particular there is a type () of propositions, and there may be a type ι (iota) of "individuals" from which other types are built. Russell and Whitehead's notation for building up types from other types is rather cumbersome, and the notation here is due to Church.

In the ramified type theory of PM all objects are elements of various disjoint ramified types. Ramified types are implicitly built up as follows. If τ1,...,τm,σ1,...,σn are ramified types then as in simple type theory there is a type (τ1,...,τm,σ1,...,σn) of "predicative" propositional functions of τ1,...,τm,σ1,...,σn. However, there are also ramified types (τ1,...,τm|σ1,...,σn) that can be thought of as the classes of propositional functions of τ1,...τm obtained from propositional functions of type (τ1,...,τm,σ1,...,σn) by quantifying over σ1,...,σn. When n=0 (so there are no σs) these propositional functions are called predicative functions or matrices. This can be confusing because modern mathematical practice does not distinguish between predicative and non-predicative functions, and in any case PM never defines exactly what a "predicative function" actually is: this is taken as a primitive notion.

Russell and Whitehead found it impossible to develop mathematics while maintaining the difference between predicative and non-predicative functions, so they introduced the axiom of reducibility, saying that for every non-predicative function there is a predicative function taking the same values. In practice this axiom essentially means that the elements of type (τ1,...,τm|σ1,...,σn) can be identified with the elements of type (τ1,...,τm), which causes the hierarchy of ramified types to collapse down to simple type theory. (Strictly speaking, PM allows two propositional functions to be different even if they take the same values on all arguments; this differs from modern mathematical practice where one normally identifies two such functions.)

In Zermelo set theory one can model the ramified type theory of PM as follows. One picks a set ι to be the type of individuals. For example, ι might be the set of natural numbers, or the set of atoms (in a set theory with atoms) or any other set one is interested in. Then if τ1,...,τm are types, the type (τ1,...,τm) is the power set of the product τ1×...×τm, which can also be thought of informally as the set of (propositional predicative) functions from this product to a 2-element set {true,false}. The ramified type (τ1,...,τm|σ1,...,σn) can be modelled

Notation

Irvine in the Stanford Encyclopedia of Philosophy 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 "embodies substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism".

Kurt Gödel was harshly critical of the notation: "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." 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 symbol-string 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 elementary parts of the notation (the symbols =⊃≡−ΛVε and the system of dots):

"The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modeled 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 changed Peano's Ɔ to ⊃, and also adopted a few of Peano's later symbols, such as ℩ and ι, and Peano's practice of turning letters upside down.

PM adopts the assertion sign "⊦" from Frege's Begriffsschrift (1879):

"(I)t may be read 'it is true that'"

Thus to assert a proposition p PM writes:

"⊦. p." (PM 1927:92)

An introduction to the notation of "Section A Mathematical Logic" (formulas ✱1–✱5.71)

PM's dots are used in a manner similar to parentheses. Each dot (or multiple dot) represents either a left or right parenthesis or the logical symbol ∧. The number of dots indicates the "depth" of the parentheses, for example, ".", ":" or ":.", "::". However the position of the matching right or left parenthesis is not indicated explicitly in the notation but has to be deduced from some rules that are complex and at times ambiguous. Moreover, when the dots stand for a logical symbol ∧ its left and right operands have to be deduced using similar rules. First one has to decide based on context whether the dots stand for a left or right parenthesis or a logical symbol. Then one has to decide how far the other corresponding parenthesis is: here one carries on until one meets either a larger number of dots, or the same number of dots next that have equal or greater "force", or the end of the line. Dots next to the signs ⊃, ≡,∨, =Df have greater force than dots next to (x), (∃x) and so on, which have greater force than dots indicating a logical product ∧.

Example 1. The line

✱3.4. ⊢ : p . q . ⊃ . p ⊃ q

corresponds to

⊢ ((p ∧ q) ⊃ (p ⊃ q)).

The two dots standing together immediately following the assertion sign indicate that what is asserted is the entire line: since there are two of them, their scope is greater than that of any of the single dots to their right. They are replaced by a left parenthesis standing where the dots are and a right parenthesis at the end of the formula, thus:

⊢ (p . q . ⊃ . p ⊃ q).

(In practice, these outermost parentheses, which enclose an entire formula, are usually suppressed.) The first of the single dots, standing between two propositional variables, represents conjunction. It belongs to the third group and has the narrowest scope. Here it is replaced by the modern symbol for conjunction "∧", thus

⊢ (p ∧ q . ⊃ . p ⊃ q).

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'. 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 truth-values of a propositional or predicate function.

Sheffer stroke: Is the contemporary logical NAND (NOT-AND), 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 true, then and only then p | q evaluates as false." 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 backward 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:

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 is known in contemporary set theory as sets of 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.

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: "⊍", "∸".

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). 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)

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 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, one can ask the following questions about any system such as PM:

whether a contradiction could be derived from the 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.) Russell and Whitehead suspected that the system in PM is incomplete: for example, they pointed out that it does not seem powerful enough to show that the cardinal ℵω exists. However, one can ask if some recursively axiomatizable extension of it is complete and consistent.

Gödel 1930, 1931

In 1930 Gödel's completeness theorem showed that first-order 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 no recursive extension of Principia could be both consistent and complete for arithmetic statements. (As mentioned above, Principia itself was already known to be incomplete for some non-arithmetic statements.) According to the theorem, within every sufficiently powerful recursive 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 Catch-22: 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 quasi-intensional 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".

This new proposal resulted in a dire outcome. An "extensional stance" and restriction to a second-order 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. x1 ∧ x2 ∧ . . . ∧ xn ∧ . . .. Ironically, this change came about as the result of criticism from Ludwig Wittgenstein in his 1919 Tractatus Logico-Philosophicus. As described by Russell in the Introduction to the Second Edition of PM:

There is another course, recommended by Wittgenstein† (†Tractatus Logico-Philosophicus, *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truth-functions, 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 (though often new proofs are required); the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2n > n breaks down unless n is finite."

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:

Gödel 1944

Gödel offered a "critical but sympathetic discussion of the logicistic order of ideas" in his 1944 article "Russell's Mathematical Logic". He wrote:

It is to be regretted that this first comprehensive and thorough-going 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 backward 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.

Contents

Part I Mathematical logic. Volume I ✱1 to ✱43

This section describes the propositional and predicate calculus, and gives the basic properties of classes, relations, and types.

Part II Prolegomena to cardinal arithmetic. Volume I ✱50 to ✱97

This part covers various properties of relations, especially those needed for cardinal arithmetic.

Part III Cardinal arithmetic. Volume II ✱100 to ✱126

This covers the definition and basic properties of cardinals. A cardinal is defined to be an equivalence class of similar classes (as opposed to ZFC, where a cardinal is a special sort of von Neumann ordinal). Each type has its own collection of cardinals associated with it, and there is a considerable amount of bookkeeping necessary for comparing cardinals of different types. PM define addition, multiplication and exponentiation of cardinals, and compare different definitions of finite and infinite cardinals. ✱120.03 is the Axiom of infinity.

Part IV Relation-arithmetic. Volume II ✱150 to ✱186

A "relation-number" is an equivalence class of isomorphic relations. PM defines analogues of addition, multiplication, and exponentiation for arbitrary relations. The addition and multiplication is similar to the usual definition of addition and multiplication of ordinals in ZFC, though the definition of exponentiation of relations in PM is not equivalent to the usual one used in ZFC.

Part V Series. Volume II ✱200 to ✱234 and volume III ✱250 to ✱276

This covers series, which is PM's term for what is now called a totally ordered set. In particular it covers complete series, continuous functions between series with the order topology (though of course they do not use this terminology), well-ordered series, and series without "gaps" (those with a member strictly between any two given members).

Part VI Quantity. Volume III ✱300 to ✱375

This section constructs the ring of integers, the fields of rational and real numbers, and "vector-families", which are related to what are now called torsors over abelian groups.

Comparison with set theory

This section compares the system in PM with the usual mathematical foundations of ZFC. The system of PM is roughly comparable in strength with Zermelo set theory (or more precisely a version of it where the axiom of separation has all quantifiers bounded).

The system of propositional logic and predicate calculus in PM is essentially the same as that used now, except that the notation and terminology has changed.

The most obvious difference between PM and set theory is that in PM all objects belong to one of a number of disjoint types. This means that everything gets duplicated for each (infinite) type: for example, each type has its own ordinals, cardinals, real numbers, and so on. This results in a lot of bookkeeping to relate the various types with each other.

In ZFC functions are normally coded as sets of ordered pairs. In PM functions are treated rather differently. First of all, "function" means "propositional function", something taking values true or false. Second, functions are not determined by their values: it is possible to have several different functions all taking the same values (for example, one might regard 2x+2 and 2(x+1) as different functions on grounds that the computer programs for evaluating them are different). The functions in ZFC given by sets of ordered pairs correspond to what PM call "matrices", and the more general functions in PM are coded by quantifying over some variables. In particular PM distinguishes between functions defined using quantification and functions not defined using quantification, whereas ZFC does not make this distinction.

PM has no analogue of the axiom of replacement, though this is of little practical importance as this axiom is used very little in mathematics outside set theory.

PM emphasizes relations as a fundamental concept, whereas in modern mathematical practice it is functions rather than relations that are treated as more fundamental; for example, category theory emphasizes morphisms or functions rather than relations. (However, there is an analogue of categories called allegories that models relations rather than functions, and is quite similar to the type system of PM.)

Differences between editions

Apart from corrections of misprints, the main text of PM is unchanged between the first and second editions. The main text in Volumes 1 and 2 was reset, so that it occupies fewer pages in each. In the second edition, Volume 3 was not reset, being photographically reprinted with the same page numbering; corrections were still made. The total number of pages (excluding the endpapers) in the first edition is 1,996; in the second, 2,000. Volume 1 has five new additions:

A 54-page introduction by Russell describing the changes they would have made had they had more time and energy. The main change he suggests is the removal of the controversial axiom of reducibility, though he admits that he knows no satisfactory substitute for it. He also seems more favourable to the idea that a function should be determined by its values (as is usual in modern mathematical practice).

Appendix A, numbered as *8, 15 pages, about the Sheffer stroke.

Appendix B, numbered as *89, discussing induction without the axiom of reducibility.

Appendix C, 8 pages, discussing propositional functions.

An 8-page list of definitions at the end, giving a much-needed index to the 500 or so notations used.

In 1962, Cambridge University Press published a shortened paperback edition containing parts of the second edition of Volume 1: the new introduction (and the old), the main text up to *56, and Appendices A and C..

Legacy

Andrew D. Irvine says that PM sparked interest in symbolic logic and advanced the subject by popularizing it; it showcased the powers and capacities of symbolic logic; and it showed how advances in philosophy of mathematics and symbolic logic could go hand-in-hand with tremendous fruitfulness. PM was in part brought about by an interest in logicism, the view on which all mathematical truths are logical truths. Though flawed, PM would be influential in several later advances in meta-logic, including Gödel's incompleteness theorems.

The logical notation in PM was not widely adopted, possibly because its foundations are often considered a form of Zermelo–Fraenkel set theory.

Scholarly, historical, and philosophical interest in PM is great and ongoing, and mathematicians continue to work with PM, whether for the historical reason of understanding the text or its authors, or for furthering insight into the formalizations of math and logic.

The Modern Library placed PM 23rd in their list of the top 100 English-language nonfiction books of the 20th century.