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Principia MathematicaFirst published Tue May 21, 1996; substantive revision Tue Dec 3, 2013

Principia Mathematica, the landmark work in formal logic written byAlfred North Whitehead and Bertrand Russell, was first published in threevolumes in 1910, 1912 and 1913. A second edition appeared in 1925(Volume 1) and 1927 (Volumes 2 and 3). In 1962 an abbreviated issue(containing only the first 56 chapters) appeared in paperback. In 2011 adigest of the book's main definitions and theorems, originally transcribedby Russell for Rudolf Carnap, was reprinted in The Evolution of PrincipiaMathematica, edited by Bernard Linsky.

Written as a defense of logicism (the thesis that mathematics is in somesignificant sense reducible to logic), the book was instrumental indeveloping and popularizing modern mathematical logic. It also served asa major impetus for research in the foundations of mathematics throughoutthe twentieth century. Along with Aristotle's Organon and Gottlob Frege'sGrundgesetze der Arithmetik, it remains one of the most influential bookson logic ever written.

1. History of Principia Mathematica2. Significance of Principia Mathematica3. Contents of Principia MathematicaBibliographyAcademic ToolsOther Internet ResourcesRelated Entries

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1. History of Principia Mathematica

Logicism is the view that (some or all of) mathematics can be reduced to(formal) logic. It is often explained as a two-part thesis. First, it consists ofthe claim that all mathematical truths can be translated into logical truthsor, in other words, that the vocabulary of mathematics constitutes a propersubset of the vocabulary of logic. Second, it consists of the claim that allmathematical proofs can be recast as logical proofs or, in other words, thatthe theorems of mathematics constitute a proper subset of the theorems oflogic. As Russell writes, it is the logicist's goal “to show that all puremathematics follows from purely logical premises and uses only conceptsdefinable in logical terms” (1959, 74).

The logicist thesis appears to have been first advocated in the lateseventeenth century by Gottfried Leibniz. Later, the idea was defended inmuch greater detail by Gottlob Frege. During the critical movement of the1820s, mathematicians such as Bernard Bolzano, Niels Abel, LouisCauchy and Karl Weierstrass succeeded in eliminating much of thevagueness and many of the contradictions present in the mathematics oftheir day. By the mid- to late-1800s, William Hamilton had gone on tointroduce ordered couples of reals as the first step in supplying a logicalbasis for the complex numbers and Karl Weierstrass, Richard Dedekindand Georg Cantor had all developed methods for founding the irrationalsin terms of the rationals. Using work done by H.G. Grassmann andRichard Dedekind, Guiseppe Peano had then gone on to develop a theoryof the rationals based on his now famous axioms for the natural numbers.By Frege's day, it was thus generally recognized that large parts ofmathematics could be derived from a relatively small set of primitivenotions.

Even so, it was not until 1879, when Frege developed the necessarylogical apparatus, that logicism could finally be said to have become


2 Stanford Encyclopedia of Philosophy

technically plausible. After another five years' work, Frege arrived at thedefinitions necessary for logicising arithmetic and during the 1890s heworked on many of the essential derivations. However, with the discoveryof paradoxes such as Russell's paradox at the turn of the century, itappeared that additional resources would need to be developed if logicismwere to succeed.

By 1902, both Whitehead and Russell had reached this same conclusion.Both men were in the initial stages of preparing second volumes to theirearlier books on related topics: Whitehead's 1898 A Treatise on UniversalAlgebra and Russell's 1903 The Principles of Mathematics. Since theirresearch overlapped considerably, they began collaborating on what wouldeventually become Principia Mathematica. By agreement, Russell workedprimarily on the philosophical parts of the project, including the book'sphilosophically rich Introduction, the theory of descriptions, and the no-class theory (in which set or class terms become meaningful only whenplaced in well-defined contexts), all of which can still be read fruitfullyeven by non-specialists. The two men then collaborated on the technicalderivations. As Russell writes,

Initially, it was thought that the project might take a year to complete.Unfortunately, after almost a decade of difficult work on the part of the

As for the mathematical problems, Whitehead invented most of thenotation, except in so far as it was taken over from Peano; I didmost of the work concerned with series and Whitehead did most ofthe rest. But this only applies to first drafts. Every part was donethree times over. When one of us had produced a first draft, hewould send it to the other, who would usually modify itconsiderably. After which, the one who had made the first draftwould put it into final form. There is hardly a line in all the threevolumes which is not a joint product. (1959, 74)

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two men, Cambridge University Press concluded that publishing Principiawould result in an estimated loss of 600 pounds. Although the pressagreed to assume half this amount and the Royal Society agreed to donateanother 200 pounds, this still left a 100-pound deficit. Only by eachcontributing 50 pounds were the authors able to see their work through topublication (Whitehead, Russell and James, 1910).

Publication itself involved the enormous job of type-setting all threevolumes by hand. In 1911, the printing of the second volume wasinterrupted when Whitehead discovered a difficulty with the symbolism.The result was the insertion (on roman numeral pages) of a long“Prefatory Statement of Symbolic Conventions” at the beginning ofVolume 2.

Today there is not a major academic library anywhere in the world thatdoes not possess a copy of this landmark publication.

2. Significance of Principia Mathematica

Achieving Principia's main goal proved to be a challenge. Primarily atissue were the kinds of assumptions Whitehead and Russell needed tocomplete their project. Although Principia succeeded in providingdetailed derivations of many major theorems in finite and transfinitearithmetic, set theory, and elementary measure theory, two axioms inparticular were arguably non-logical in character: the axiom of infinity andthe axiom of reducibility. The axiom of infinity in effect states that thereexists an infinite number of objects. Arguably it makes the kind ofassumption generally thought to be empirical rather than logical in nature.The axiom of reducibility was introduced as a means of overcoming thenot completely satisfactory effects of the theory of types, the mechanismRussell and Whitehead used to restrict the notion of a well-formedexpression, thereby avoiding Russell's paradox. Although technically



feasible, many critics concluded that the axiom was simply too ad hoc tobe justified philosophically. Kanamori sums up the sentiment of manyreaders: “In traumatic reaction to his paradox Russell had built a complexsystem of orders and types only to collapse it with his Axiom ofReducibility, a fearful symmetry imposed by an artful dodger” (2009,411). In the minds of many, the issue of whether mathematics could bereduced to logic, or whether it could be reduced only to set theory, thusremained open.

In response, Whitehead and Russell argued that both axioms weredefensible on inductive grounds. As they tell us in the Introduction to thefirst volume of Principia,

self-evidence is never more than a part of the reason for acceptingan axiom, and is never indispensable. The reason for accepting anaxiom, as for accepting any other proposition, is always largelyinductive, namely that many propositions which are nearlyindubitable can be deduced from it, and that no equally plausibleway is known by which these propositions could be true if theaxiom were false, and nothing which is probably false can bededuced from it. If the axiom is apparently self-evident, that onlymeans, practically, that it is nearly indubitable; for things havebeen thought to be self-evident and have yet turned out to be false.And if the axiom itself is nearly indubitable, that merely adds tothe inductive evidence derived from the fact that its consequencesare nearly indubitable: it does not provide new evidence of aradically different kind. Infallibility is never attainable, andtherefore some element of doubt should always attach to everyaxiom and to all its consequences. In formal logic, the element ofdoubt is less than in most sciences, but it is not absent, as appearsfrom the fact that the paradoxes followed from premisses whichwere not previously known to require limitations. (1910, 2nd edn

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Whitehead and Russell were also disappointed by the the book's largelyindifferent reception on the part of many working mathematicians. AsRussell writes,

Despite such concerns, Principia Mathematica proved to be remarkablyinfluential in at least three ways. First, it popularized modernmathematical logic to an extent undreamt of by its authors. By using anotation superior to that used by Frege, Whitehead and Russell managed

59)

Both Whitehead and I were disappointed that PrincipiaMathematica was only viewed from a philosophical standpoint.People were interested in what was said about the contradictionsand in the question whether ordinary mathematics had been validlydeduced from purely logical premisses, but they were notinterested in the mathematical techniques developed in the courseof the work. ... Even those who were working on exactly the samesubjects did not think it worth while to find out what PrincipiaMathematica had to say on them. I will give two illustrations:Mathematische Annalen published about ten years after thepublication of Principia a long article giving some of the resultswhich (unknown to the author) we had worked out in Part IV ofour book. This article fell into certain inaccuracies which we hadavoided, but contained nothing valid which we had not alreadypublished. The author was obviously totally unaware that he hadbeen anticipated. The second example occurred when I was acolleague of Reichenbach at the University of California. He toldme that he had invented an extension of mathematical inductionwhich he called 'transfinite induction'. I told him that this subjectwas fully treated in the third volume of the Principia. When I sawhim a week later, he told me that he had verified this. (1959, 86)



to convey the remarkable expressive power of modern predicate logic in away that previous writers had been unable to achieve. Second, byexhibiting so clearly the deductive power of the new logic, Whitehead andRussell were able to show how powerful the idea of a modern formalsystem could be, thus opening up new work in what soon was to be calledmetalogic. Third, Principia Mathematica re-affirmed clear and interestingconnections between logicism and two of the main branches of traditionalphilosophy, namely metaphysics and epistemology, thereby initiating newand interesting work in both of these areas.

As a result, not only did Principia introduce a wide range ofphilosophically rich notions (including propositional function, logicalconstruction, and type theory), it also set the stage for the discovery ofcrucial metatheoretic results (including those of Kurt Gödel, AlonzoChurch, Alan Turing and others). Just as importantly, it initiated atradition of common technical work in fields as diverse as philosophy,mathematics, linguistics, economics and computer science.

Today a lack of agreement remains over the ultimate philosophicalcontribution of Principia, with some authors holding that, with theappropriate modifications, logicism remains a feasible project. Others holdthat the philosophical and technical underpinnings of the project remaintoo weak or too confused to be of great use to the logicist. (For moredetailed discussion, readers should consult Quine (1966a), Quine (1966b),Landini (1998), Landini (2011), Linsky (1999), Linsky (2011), Hale andWright (2001), Burgess (2005), Hintikka (2009) and Gandon (2012).)

There is also lack of agreement over the importance of the second editionof the book, which appeared in 1925 (Volume 1) and 1927 (Volumes 2and 3). The revisions were done by Russell, although Whitehead wasgiven the opportunity to advise. In addition to the correction of minorerrors throughout the original text, changes to the new edition included the

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inclusion of a new Introduction and three new appendices. (Theappendices discuss the theory of quantification, mathematical inductionand the axiom of reducibility, and the principle of extensionalityrespectively.) The book itself was reset more compactly, making pagereferences to the first edition obsolete. Russell continued to makecorrections as late as 1949 for the 1950 printing, the year he and MrsWhitehead finally began to receive royalties.

Today there is still debate over the ultimate value, or even the correctinterpretation, of some of the revisions, revisions that were motivated inlarge part by the work of some of Russell's brightest students, includingLudwig Wittgenstein and Frank Ramsey. Appendix B has beennotoriously problematic. The appendix purports to show howmathematical induction can be justified without use of the axiom ofreducibility; but as Alasdair Urquhart reports,

Linsky (2011) provides helpful discussion, both of the Appendix itself andof the suggestion that by 1925 Russell may have been out of touch withrecent developments in the quickly changing field of mathematical logic.He also addresses the suggestion, made by some commentators, that

The first indication that something was seriously wrong appearedin Gödel's well known essay of 1944, “Russell's MathematicalLogic.” There, Gödel points out that line (3) of the demonstrationof Russell's proposition *89.16 is an elementary logical blunder,while the crucial *89.12 also appears to be highly questionable. Itstill remained to be seen whether anything of Russell's proof couldbe salvaged, in spite of the errors, but John Myhill provided strongevidence of a negative verdict by providing a model-theoreticproof in 1974 that no such proof as Russell's can be given in theramified theory of types without the axiom of reducibility.(Urquhart 2012)



Whitehead may have been opposed to the revisions, or at least indifferentto them, concluding that both charges are likely without foundation.(Whitehead's own comments, published in 1926 in Mind, shed little lighton the issue.)

3. Contents of Principia Mathematica

Principia Mathematica originally appeared in three volumes. Images ofthe title page of the first volume of the first edition and of the cover of thefirst paperback issue may be seen here:

Title page of the first edition of Principia Mathematica, Volume 1(1910)Cover of the first paperback issue of Principia Mathematica to *56(1962).

Together the three volumes are divided into six parts. Volume 1 beginswith a lengthy Introduction containing sections entitled

“Preliminary Explanations of Ideas and Notations,”“The Theory of Logical Types,” and“Incomplete Symbols.”

It also contains Part I, “Mathematical Logic,” which contains sections on

“The Theory of Deduction,”“Theory of Apparent Variables,”“Classes and Relations,”“Logic of Relations,” and“Products and Sums of Classes”,

and Part II, “Prolegomena to Cardinal Arithmetic”, which includessections on

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“Unit Classes and Couples,”“Sub-Classes, Sub-Relations, and Relative Types,”“One-Many, Many-One and One-One Relations,”“Selections,” and“Inductive Relations.”

Volume 2 begins with a “Prefatory Statement of Symbolic Conventions.”It then continues with Part III, “Cardinal Arithmetic,” which itself containssections on

“Definition and Logical Properties of Cardinal Numbers,”“Addition, Multiplication and Exponentiation,” and“Finite and Infinite”.

It also includes Part IV, “Relation-Arithmetic”, which has sections on

“Ordinal Similarity and Relation-Numbers,”“Addition of Relations, and the Product of Two Relations,”“The Principle of First Differences, and the Multiplication andExponentiation of Relations,” and“Arithmetic of Relation-Numbers”;

and the first half of Part V, “Series”, which has sections on

“General Theory of Series,”“On Sections, Segments, Stretches, and Derivatives,” and“On Convergence, and the Limits of Functions.”

Volume 3 continues Part V, with sections on

“Well-Ordered Series,”“Finite and Infinite Series and Ordinals,” and“Compact Series, Rational Series, and Continuous Series.”



It also contains Part VI, “Quantity”, which includes sections on

“Generalization of Number,”“Vector-Families,”“Measurement,” and“Cyclic Families.”

A fourth volume on geometry was begun but never completed (Russell1959, 99).

Overall, the three volumes not only represent a major leap forward withregard to modern logic, they are also rich in early twentieth-centurymathematical developments. To give one example, Whitehead and Russellwere the first to define a series as a set of terms having the properties ofbeing asymmetrical, transitive and connected (1912, 2nd edn, 497). Togive another, it is in Principia that we find the first detailed developmentof a generalized version of Cantor's transfinite ordinals, which the authorscall “relation-numbers.” The resulting “relation-arithmetic” in turn led tosignificant improvements in our understanding of the general notion ofstructure (1912, Part IV).

As T.S. Eliot points out, the book also did a great deal to promote clarityin the use of ordinary language in the early part of the twentieth century:

The book is also not without some self-deprecating humour. As Blackwellpoints out (2011, 158, 160), the authors twice poke fun at the length andtedium of the project's many logical derivations. In Volume 1, the authors

how much the work of logicians has done to make of English alanguage in which it is possible to think clearly and exactly on anysubject. The Principia Mathematica are perhaps a greatercontribution to our language than they are to mathematics (1927,291)

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explain that one cannot list all the non-intensional functions of φ!“because life is too short” (1910, 2nd edn 73); and in Volume 3, after over1,800 pages of dense symbolism, the authors end Part IV, Section D, onCyclic Families, with the comment,

Evidence that the humour originates more with Russell than withWhitehead is perhaps found in not dissimilar remarks that appear inRussell's other writings. Russell's comment when discussing the axiom ofchoice, to the effect that given a collection of sets, it is possible to “pickout a representative arbitrarily from each of them, as is done in a GeneralElection” (1959, 92), is perhaps a case in point.

Contemporary readers (i.e., those who have learned logic in the last fewdecades of the twentieth century or later) will find the book's notationsomewhat antiquated. Readers wanting assistance are advised to consultthe Notation in Principia Mathematica entry in this encyclopedia. Evenso, the book remains one of the great scientific documents of the twentiethcentury.

Bibliography

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–––, 2011, “The Wit and Humour of Principia Mathematica,” in NicholasGriffin, Bernard Linsky and Kenneth Blackwell (2011) PrincipiaMathematica at 100, in Russell (Special Issue), 31(1): 151–160.

Burgess, John P., 2005, “Fixing Frege,” Princeton: Princeton University

z

We have given proofs rather shortly in this Section, particularly inthe case of purely arithmetical lemmas, of which the proofs areperfectly straightforward, but tedious if written out at length.(1913, 2nd edn 461)



Press.Chihara, Charles, 1973, Ontology and the Vicious Circle Principle, Ithaca:

Cornell University Press.Church, Alonzo, 1974, “Russellian Simple Type Theory,” Proceedings

and Addresses of the American Philosophical Association, 47: 21–33.–––, 1978, “A Comparison of Russell's Resolution of the Semantical

Antinomies with that of Tarski,” Journal of Symbolic Logic, 41: 747–760; repr. in A.D. Irvine (ed.), Bertrand Russell: CriticalAssessments, vol. 2, New York and London: Routledge, 1999, 96–112.

Copi, Irving, 1971, The Theory of Logical Types, London: Routledge andKegan Paul.

Eliot, T.S., 1927, “Commentary,” The Monthly Criterion, October 1927.Frege, Gottlob, 1893/1903, Grundgesetze der Arithmetik, Band I (1893),

Band II (1903), Jena: Verlag Hermann Pohle; ed. and trans. by M.Furth in part as The Basic Laws of Arithmetic, Berkeley: Universityof California Press, 1964.

Gabbay, Dov M., and John Woods (eds.), 2009, Handbook of the Historyof Logic: Volume 5 — Logic From Russell to Church, Amsterdam:Elsevier/North Holland.

Gandon, Sébastien, 2012, Russell's Unknown Logicism, New York:Palgrave Macmillan.

Gödel, Kurt, 1944, “Russell's Mathematical Logic,” in Paul ArthurSchilpp (ed.), The Philosophy of Bertrand Russell, 3rd edn, NewYork: Tudor, 1951, 123–153; repr. in Paul Benacerraf and HilaryPutnam (eds), Philosophy of Mathematics, 2nd edn, Cambridge:Cambridge University Press, 1983, 447–469; repr. in David F. Pears(ed.) (1972) Bertrand Russell: A Collection of Critical Essays,Garden City, New York: Anchor Books, 192–226; and repr. in A.D.Irvine (ed.) Bertrand Russell: Critical Assessments, vol. 2, New Yorkand London: Routledge, 1999, 113–134.

Andrew David Irvine


Grattan-Guinness, I., 2000, The Search for Mathematical Roots: 1870-1940, Princeton and Oxford: Princeton University Press.

Griffin, Nicholas, and Bernard Linsky (eds.), 2013, The PalgraveCentenary Companion to Principia Mathematica, London: PalgraveMacmillan.

––– and Kenneth Blackwell (eds.), 2011, Principia Mathematica at 100,Hamilton, ON: Bertrand Russell Research Centre; also published inRussell: The Journal of Bertrand Russell Studies (Special Issue),31(1).

Guay, Alexandre (ed.), 2012, Autour de Principia Mathematica de Russellet Whitehead, Dijon: Editions Universitaires de Dijon.

Hale, Bob, and Crispin Wright, 2001, The Reason's Proper Study, Oxford:Clarendon Press.

Hintikka, Jaakko, 2009, “Logicism,” in A.D. Irvine (ed.), Philosophy ofMathematics, Amsterdam: Elsevier/North Holland, 271–290.

Kanamori, Akihiro, 2009, “Set Theory from Cantor to Cohen,” in A.D.Irvine (ed.), Philosophy of Mathematics, Amsterdam: Elsevier/NorthHolland, 395-459.

Landini, Gregory, 1998, Russell's Hidden Substitutional Theory, NewYork and Oxford: Oxford University Press.

–––, 2011, Russell, London and New York: Routledge.Link, Godehard (ed.), 2004, One Hundred Years of Russell's Paradox,

Berlin and New York: Walter de Gruyter.Linsky, Bernard, 1990, “Was the Axiom of Reducibility a Principle of

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–––, 1999, Russell's Metaphysical Logic, Stanford: CSLI Publications.–––, 2002, “The Resolution of Russell's Paradox in Principia

Mathematica,” Philosophical Perspectives, 16: 395–417.–––, 2003, “Leon Chwistek on the No-Classes Theory in Principia



Mathematica,” History and Philosophy of Logic, 25: 53–71.–––, 2011, The Evolution of Principia Mathematica: Bertrand Russell's

Manuscripts and Notes for the Second Edition, Cambridge:Cambridge University Press.

––– and Kenneth Blackwell, 2006, “New Manuscript Leaves and thePrinting of the First Edition of Principia Mathematica” Russell, 25:141–154.

Mares, Edwin, 2007, “The Fact Semantics for Ramified Type Theory andthe Axiom of Reducibility” Notre Dame Journal of Formal Logic,48: 237–251.

Mayo-Wilson, Conor, 2011, “Russell on Logicism and Coherence,” inNicholas Griffin, Bernard Linsky and Kenneth Blackwell (2011)Principia Mathematica at 100, in Russell (Special Issue), 31(1): 63–79.

Mukhopadhyay, Arnab Kumar, Kumar Mitra and Sanjukta Basu (eds.),2011, Revisiting Principia Mathematica after 100 Years, Kolkata,India: Gangchil.

Murawski, Roman, 2011, “On Chwistek's Philosophy of Mathematics,” inNicholas Griffin, Bernard Linsky and Kenneth Blackwell (2011)Principia Mathematica at 100, in Russell (Special Issue), 31(1): 121–130.

Proops, Ian, 2006, “Russell's Reasons for Logicism,” Journal of theHistory of Philosophy, 44: 267–292.

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Bertrand Russell, Boston: Birkhäuser Press; repr. 1993.Russell, Bertrand, 1903, The Principles of Mathematics, Cambridge:

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Cambridge University Press.–––, 1919, Introduction to Mathematical Philosophy, London: George

Allen & Unwin.–––, 1948, “Whitehead and Principia Mathematica,” Mind, 57: 137–138.–––, 1959, My Philosophical Development, London: George Allen and

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––– and Bertrand Russell, 1910, 1912, 1913, Principia Mathematica, 3vols, Cambridge: Cambridge University Press; 2nd edn, 1925 (Vol.1), 1927 (Vols 2, 3); abridged as Principia Mathematica to *56,Cambridge: Cambridge University Press, 1962. (Page numbers are tothe second edition.)

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Academic Tools

Other Internet Resources

Principia Mathematica: Volume 1 (University of Michigan HistoricalMath Collection)Principia Mathematica: Volume 2 (University of Michigan HistoricalMath Collection)Principia Mathematica: Volume 3 (University of Michigan HistoricalMath Collection)Principia Mathematica: Whitehead and Russell (Stanley Burris,University of Waterloo)

How to cite this entry.Preview the PDF version of this entry at the Friends of the SEPSociety.Look up this entry topic at the Indiana Philosophy OntologyProject (InPhO).Enhanced bibliography for this entry at PhilPapers, with linksto its database.

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Title page of the first edition of PrincipiaMathematica, Volume 1 (1910)

(This image appears courtesy of the Bertrand Russell Archives atMcMaster University.)

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Cover of the first edition of Principia Mathematicato *56 (1962)

(This image appears courtesy of the Bertrand Russell Archives atMcMaster University.)

Copyright © 2014 by the author Andrew David Irvine



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