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Michael A.R. Biggs

Editing Wittgenstein’s "Notes on Logic"

Vol. 2

Skriftserie fra Working Papers from theWittgensteinarkivet ved Wittgenstein Archives atUniversitetet i Bergen the University of Bergen

Nr 11, 1996 No 11, 1996

Wittgensteinarkivet ved Universitetet i Bergen

Wittgensteinarkivet er et forskningsprosjekt ved Filosofiskinstitutt ved Universitetet i Bergen. Prosjektet ble startet1. Juni 1990. Dets hovedmålsetting er å gjøre LudwigWittgensteins etterlatte skrifter (Nachlaß) tilgjengelige forforskning. Wittgensteinarkivet produserer derfor en komplett,maskinleselig versjon av Wittgensteins Nachlaß og utviklerprogramvare for presentasjon og analyse av tekstene.

I samarbeid med Wittgenstein Trustees og OxfordUniversity Press arbeides det med sikte på å publisere Nachlaßi elektronisk form. Hele Nachlaß vil først bli publisert ielektronisk faksimile på CD-ROM. Faksimilen vil deretter blisupplert med transkripsjoner fra Wittgensteinarkivet. Imellomtiden har gjesteforskere ved Wittgensteinarkivet adgangtil transkripsjoner og programvare under bearbeidelse samt enfullstendig kopi av Nachlaß.

The Wittgenstein Archives at the University of Bergen

The Wittgenstein Archives is a research project at theDepartment of Philosophy at the University of Bergen, Norway.Established in June 1990, its main purpose is to make LudwigWittgenstein’s collected writings (Nachlaß) available forresearch. To serve this purpose, the Wittgenstein Archives isproducing a machine-readable transcription of Wittgenstein’sNachlaß, and will supply tools for computer-assisted displayand analysis of the textual corpus.

In cooperation with the Wittgenstein Trustees and OxfordUniversity Press, the Nachlaß will be published in electronicform. The entire Nachlaß will first be published as anelectronic facsimile on CD-ROM. This facsimile will then besupplemented by transcriptions prepared at the WittgensteinArchives. In the meantime, visiting scholars to theWittgenstein Archives have access to the texts and toolsprepared by the project as well as to a complete copy ofWittgenstein’s Nachlaß.

Working papers from the Wittgenstein Archivesat the University of Bergen

Editorial Board: Claus HuitfeldtKjell S. JohannessenTore NordenstamAngela Requate

Editorial Address: The Wittgenstein Archives atthe University of Bergen,Harald Hårfagres gt 31,N-5007 Bergen, Norway

tel: +47-55-58 29 50fax: +47-55-58 94 70E-mail: [email protected]

Cover photo: © Knut Erik Tranøy

Wittgensteinarkivet ved Universitetet i Bergen

Michael A.R. Biggs

Editing Wittgenstein’s "Notes on Logic"

Vol. 2

Bergen 1996

ISBN 82-91071-12-8ISBN 82-91071-14-4

ISSN 0803-3137

Skriftserie fraWittgensteinarkivet vedUniversitetet i BergenNr 11, 1996

Copyright:© The Wittgenstein Trustees, the Wittgenstein Archives at the University of Bergenand Michael Biggs

Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Editorial Preface to Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

"Normalised" Transcription of Item 201a-1 . . . . . . . . . . . . . . . . . 5Legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Item 201a-1 (RA1.710.057822 and (RA1.710.057823) . . 7

"Diplomatic" Transcriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Item 201a-1 (RA1.710.057822 and RA1.710.057823) . . . 35Item 201a-2 (RA1.710.057824) . . . . . . . . . . . . . . . . . . . 75Item 201a-3 (also referred to as TSx) . . . . . . . . . . . . . 111Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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Acknowledgements

Wittgenstein’s Trustees: G.E.M. Anscombe, Sir Anthony Kenny,G.H. von Wright and Peter Winch; for their permission to quotefrom Wittgenstein’s Nachlaß.

The University of Bergen; for a Senior Research Fellowship in1994, during part of which period I undertook this research.

The University of Hertfordshire; for their continuing support ofmy research into Wittgenstein’s Nachlaß.

The Wittgenstein Archives at the University of Bergen; for theresearch facilities extended to me during my Fellowship. Inparticular to Peter Philipp and Maria Sollohub for advice ontranslation, to Peter Cripps for detailed work on thetranscriptions, and to Claus Huitfeldt for comments on the draft.

Kenneth Blackwell of The Bertrand Russell Archives atMcMaster University, Canada; for copies from their holdings ofRussell’s Nachlaß, for permission to transcribe and quote fromthem, and for his advice.

Blackwell Publisher Limited, Oxford; for their permission toquote and reproduce diagrams from NL 1961 and NL 1979.

Suhrkamp Verlag, Frankfurt am Main; for advice on thereproduction of diagrams from their publications.

I am especially grateful to Alois Pichler of The WittgensteinArchives at the University of Bergen for his detailed andconstructive criticism of this monograph.

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Preface

This monograph was originally conceived as one volume but ishere presented as two. The first contains a discussion of, anddetailed comparison between, the two published editions ofWittgenstein’s "Notes on Logic". It also contains tables andconcordances by which the published editions may be comparedwith one another, and to four scripts of the work which are stillextant. The second volume provides the reader with atypographical representation of each of these four scripts in itsentirety and made available in published form for the first time.Examination of these scripts supports the arguments concerningprovenance and chronology in the first volume.

The reason for the separation of these two intimately relatedparts is that the second volume is being simultaneouslypublished as an electronic text, in support of the objectives ofThe Wittgenstein Archives at the University of Bergen toprovide primary texts in this format. Readers may thereforeeither avail themselves of the numerous possibilities associatedwith the electronic medium, or use the typographic presentationof the second volume in traditional book form.

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Editorial Preface to Volume 2

Parallel to this paper edition, Wittgenstein’s "Notes on Logic"are also being issued in electronic format. This double-releasemarks the beginning of an ambitious publication plan. In thecourse of the coming years Wittgenstein’s entire Nachlaß will bemade available as machine-readable text. Correctly employed,this medium is sufficiently flexible to satisfy the needs ofeveryone from the casual browser to the most demandingscholar.

The advantages of a well prepared electronic edition arenumerous. Text can be presented, either on a computer screenor as paper printout, in a range of different styles, with orwithout cross-references and footnotes, as a simplified readingversion or with all the variant readings and false starts of anauthor’s original manuscript. The same material can be searchedin seconds for occurrences of a particular term or internallycompared section for section so as to reveal recurrent phrases,and so on.

The foundation of an electronic edition is invariably atranscription of the source material interspersed with codeswhich enable a computer to distinguish the numerous textualfeatures. The Wittgenstein Archives at the University of Bergenuses a "mark-up" language (an encoding syntax plus avocabulary of code names) called MECS-WIT for thetranscription of Wittgenstein’s Nachlaß. This language wasspecially developed to handle the features typical of an opuswhich is largely handwritten and unfinished, such as frequentdeletions, later additions, rearrangement of material etc.

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The transcription of "Notes on Logic" from which the currentpublication is derived was originally prepared by Michael Biggs.It was then proof-read by Michael Biggs and Peter Cripps. Thetranscription of item 201a-1 (McMaster’s catalogue no.sRA1.710.057822 and RA1.710.057823) is presented here in twoformats: "normalised" and "diplomatic". Items 201a-2(McMaster’s catalogue no. RA1.710.057824) and 201a-3 (referredto in this publication as TSx), being essentially variants of 201a-1,are presented in "diplomatic" format only.

The electronic version of this publication will be made availableon the Internet under the URL http://www.hd.uib.no/wab/

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"Normalised" Transcription of Item 201a-1

Legend

Bold Editorial comments and marks

Underlined Underlined with straight solid line

Doubly underlined Underlined with two straight solidlines

Doubly underlined Underlined with three or moreand italicised straight solid lines

Italics Letterspaced

@ Unreadable text

<!> Editorial instruction or intertextualreference - cf diplomatic version

Bold numbers (A1, A2,... B1, B2,...) in the left margin at thebeginning of sections (paragraphs) refer to page numbers in theoriginal, - cf diplomatic version. If the page shift occurs insidea section, its position is marked .

Deleted, overwritten and substituted text has been left out,orthographic errors tacitly corrected.

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Item 201a-1 (RA1.710.057822 and (RA1.710.057823)

A1 Summary

One reason for thinking the old notation wrong is that it isvery unlikely that from every proposition p an infinite numberof other propositions not-not-p, not-not-not-not-p, etc.,should follow.

If only those signs which contain proper names werecomplex then propositions containing nothing but apparentvariables would be simple. Then what about their denials?

The verb of a proposition cannot be „is true” or „is false”,but whatever is true or false must already contain the verb.

Deductions only proceed according to the laws ofdeduction, but these laws cannot justify the deduction.

One reason for supposing that not all propositions whichhave more than one argument are relational propositions is thatif they were, the relations of judgement and inference wouldhave to hold between an arbitrary number of things.

Every proposition which seems to be about a complex canbe analysed into a proposition about its constituents and aboutthe proposition which describes the complex perfectly; i.e., thatproposition which is equivalent to saying the complex exists.

The idea that propositions are names of complexes suggeststhat whatever is not a proper name is a sign for a relation.

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Because spatial complexes1 consist of Things and Relations onlyand the idea of a complex is taken from space.

In a proposition convert all its indefinables into variables;there then remains a class of propositions which is not allpropositions but a type.

A2 There are thus two ways in which signs are similar. Thenames Socrates and Plato are similar: they are both names. Butwhatever they have in common must not be introduced beforeSocrates and Plato are introduced. The same applies tosubject-predicate form etc. Therefore, thing, proposition,subject-predicate form, etc., are not indefinables, i.e., types arenot indefinables.

When we say A judges that etc., then we have tomention a whole proposition which A judges. It will not doeither to mention only its constituents, or its constituents andform, but not in the proper order. This shows that a propositionitself must occur in the statement that it is judged; however, forinstance, „not-p” may be explained. The question „What isnegated” must have a meaning.

To understand a proposition p it is not enough to knowthat p implies „p” is true , but we must also know that ~pimplies „p is false”. This shows the bipolarity of the proposition.W-F = Wahr-Falsch

To every molecular function a WF scheme corresponds.Therefore we may use the WF scheme itself instead of thefunction. Now what the WF scheme does is, it correlates theletters W and F with each proposition. These two letters are thepoles of atomic propositions. Then the scheme correlates anotherW and F to these poles. In this notation all that matters is the

1 you for instance imagine every fact as a spatial complex.

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correlation of the outside poles to the pole of the atomicpropositions. Therefore not-not-p is the same symbol as p.And therefore we shall never get two symbols for the samemolecular function.

A3 The meaning of a proposition is the fact which actuallycorresponds to it.

As the ab functions of atomic propositions are bi-polarpropositions again we can perform ab operations on them. Weshall, by doing so, correlate two new outside poles via the oldoutside poles to the poles of the atomic propositions.

The symbolising fact in a p b is that, say1 a is on the leftof p and b on the right of p; then the correlation of new polesis to be transitive, so that for instance if a new pole a inwhatever way i.e. via whatever poles is correlated to the insidea, the symbol is not changed thereby. It is therefore possible toconstruct all possible ab functions by performing one aboperation repeatedly, and we can therefore talk of all abfunctions as of all those functions which can be obtained byperforming this ab operation repeatedly.

[Note by Bertrand Russell][NB. ab means the same as WF, which means true-false.]

1 This is quite arbitrary but if we such have fixed on whichsides the poles have to stand we must of course stick to ourconvention. If for instance „apb” says p then bpa saysnothing. (It does not say ~p.) But a apb b is the samesymbol as apb the ab function vanishes automatically forhere the new poles are related to the same side of p as the oldones. The question is always: how are the new polescorrelated to p compared with the way the old poles arecorrelated to p.

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Naming is like pointing. A function is like a line dividingpoints of a plane into right and left ones; then „p or not-p” hasno meaning because it does not divide the plane.

But though a particular proposition „p or not-p” has nomeaning, a general proposition „for all p’s, p or not-p” has ameaning because this does not contain the nonsensical function„p or not-p” but the function „p or not-q” just as „for all x’sxRx” contains the function „xRy”.

A4 A proposition is a standard to which all facts behave, withnames it is otherwise; it is thus bi-polarity and sense comes in;just as one arrow behaves to another arrow by being in thesame sense or the opposite, so a fact behaves to a proposition.

The form of a proposition has meaning in the followingway. Consider a symbol „xRy”. To symbols of this formcorrespond couples of things whose names are respectively „x”and „y”. The things x y stand to one another in all sorts ofrelations, amongst others some stand in the relation R, and somenot; just as I single out a particular thing by a particular nameI single out all behaviours of the points x and y with respect tothe relation R. I say that if an x stands in the relation R to a ythe sign „x R y” is to be called true to the fact and otherwisefalse. This is a definition of sense.

In my theory p has the same meaning as not-p butopposite sense. The meaning is the fact. The proper theory ofjudgment must make it impossible to judge nonsense.

It is not strictly true to say that we understand aproposition p if we know that p is equivalent to „p is true” forthis would be the case if accidentally both were true or false.What is wanted is the formal equivalence with respect to theforms of the proposition, i.e., all the general indefinablesinvolved. The sense of an ab function of a proposition is a

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function of its sense. There are only unasserted propositions.Assertion is merely psychological. In not-p, p is exactly thesame as if it stands alone; this point is absolutely fundamental.Among the facts which make „p or q” true there are also factswhich make „p and q” true; if propositions have only meaning,we ought, in such a case, to say that these two propositions areidentical, but in fact, their sense is different for we haveintroduced sense by talking of all p’s and all q’s.Consequently the molecular propositions will only be used incases where there ab function stands under a generality sign orenters into another function such as „I believe that, etc.,”because then the sense enters.

A5 In „a judges p” p cannot be replaced by a proper name.This appears if we substitute „a judges that p is true and not pis false”. The proposition „a judges p” consists of the propername a, the proposition p with its 2 poles, and a being relatedto both of these poles in a certain way. This is obviously not arelation in the ordinary sense.

The ab notation makes it clear that not and or aredependent on one another and we can therefore not use themas simultaneous indefinables. <!> Same objections in the case ofapparent variables to old indefinables, as in the case ofmolecular functions: The application of the ab notation toapparent-variable propositions becomes clear if we consider that,for instance, the proposition „for all x, ϕx” is to be true whenϕx is true for all x’s and false when ϕx is false for some x’s.We see that some and all occur simultaneously in the properapparent variable notation.

A6 The Notation is:

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for (x) ϕx ; a - (x) - a ϕ x b - (∃ x) - b

and

for (∃x) ϕ x : a - (∃x) - a ϕ x b - (x) - b

Old definitions now become tautologous.In aRb it is not the complex that symbolises but the fact

that the symbol a stands in a certain relation to the symbol b.Thus facts are symbolised by facts, or more correctly: that acertain thing is the case in the symbol says that a certain thingis the case in the world.

Judgment, question and command are all on the same level.What interests logic in them is only the unasserted proposition.Facts cannot be named.

A proposition cannot occur in itself. This is the fundamentaltruth of the theory of types.

Every proposition that says something about one thing is asubject-predicate proposition, and so on.

Therefore we can recognize a subject-predicate propositionif we know it contains only one name and one form, etc. Thisgives the construction of types. Hence the type of a propositioncan be recognized by its symbol alone.

A7 What is essential in a correct apparent-variable notation isthis: (1) it must mention a type of propositions; (2) it mustshow which components of a proposition of this type areconstants.

[Components are forms and constituents.]

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Take (ϕ).ϕ!x. Then if we describe the kind of symbols, forwhich ϕ! stands and which, by the above, is enough todetermine the type, then automatically „(ϕ).ϕ!x” cannot befitted by this description, because it contains „ϕ!x” and thedescription is to describe all that symbolizes in symbols of theϕ! kind. If the description is thus complete vicious circles canjust as little occur as for instance in (ϕ).(x)ϕ (where (x)ϕ is asubject-predicate proposition).

B1 First MS.

Indefinables are of two sorts: names, and forms. Propositionscannot consist of names alone; they cannot be classes of names.A name can not only occur in two different propositions, butcan occur in the same way in both.

Propositions [which are symbols having reference to facts]are themselves facts: that this inkpot is on this table may expressthat I sit in this chair.

It can never express the common characteristic of twoobjects that we designate them by the same name but by twodifferent ways of designation, for, since names are arbitrary, wemight also choose different names, and where then would bethe common element in the designations? Nevertheless one isalways tempted, in a difficulty, to take refuge in different waysof designation.

Frege said „propositions are names”; Russell said„propositions correspond to complexes”. Both are false; andespecially false is the statement „propositions are names ofcomplexes”.

It is easy to suppose that only such symbols are complex ascontain names of objects, and that accordingly „(∃x,ϕ).ϕx” or„(∃x,R,y).xRy” must be simple. It is then natural to call the

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first of these the name of a form, the second the name of arelation. But in that case what is the meaning of (e.g.)„~(∃x,y).xRy”? Can we put „not” before a name?

B2 The reason why „~Socrates” means nothing is that „~x” doesnot express a property of x.

There are positive and negative facts: if the proposition„this rose is not red” is true, then what it signifies is negative.But the occurrence of the word „not” does not indicate thisunless we know that the signification of the proposition „thisrose is red” (when it is true) is positive. It is only from both, thenegation and the negated proposition, that we can conclude toa characteristic of the significance of the whole proposition. (Weare not here speaking of negations of general propositions, i.e.of such as contain apparent variables. Negative facts only justifythe negations of atomic propositions.)

Positive and negative facts there are, but not true and falsefacts.

If we overlook the fact that propositions have a sense whichis independent of their truth or falsehood, it easily seems as iftrue and false were two equally justified relations between thesign and what is signified. (We might then say e.g. that „q”signifies in the true way what „not-q” signifies in the falseway). But are not true and false in fact equally justified? Couldwe not express ourselves by means of false propositions just aswell as hitherto with true ones, so long as we know that theyare meant falsely? No! For a proposition is then true when itis as we assert in this proposition; and accordingly if by „q” wemean „not-q”, and it is as we mean to assert, then in the newinterpretation „q” is actually true and not false. But it isimportant that we can mean the same by „q” as by „not-q”, forit shows that neither to the symbol „not” nor to the manner of

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its combination with „q” does a characteristic of the denotationof „q” correspond.

B4 2nd MS.

We must be able to understand propositions which we havenever heard before. But every proposition is a new symbol.Hence we must have general indefinable symbols; these areunavoidable if propositions are not all indefinable.

Whatever corresponds in reality to compound propositionsmust not be more than what corresponds to their several atomicpropositions.

Not only must logic not deal with [particular] things, butjust as little with relations and predicates.

There are no propositions containing real variables.What corresponds in reality to a proposition depends upon

whether it is true or false. But we must be able to understand aproposition without knowing if it is true or false.

What we know when we understand a proposition is this:We know what is the case if the proposition is true, and whatis the case if it is false. But we do not know [necessarily]whether it is true or false.

Propositions are not names.We can never distinguish one logical type from another by

attributing a property to members of the one which we deny tomembers of the other.

Symbols are not what they seem to be. In „aRb”, „R” lookslike a substantive, but is not one. What symbolizes in „aRb” isthat R occurs between a and b. Hence „R” is not the indefinablein „aRb”. Similarly in „ϕx”, „ϕ” looks like a substantive but isnot one; in „~p”, „~” looks like „ϕ” but is not like it. This is thefirst thing that indicates that there may not be logical constants.

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A reason against them is the generality of logic: logic cannottreat a special set of things.

B5 Molecular propositions contain nothing beyond what iscontained in their atoms; they add no material informationabove that contained in their atoms.

All that is essential about molecular functions is their T-Fschema [i.e. the statement of the cases when they are true andthe cases when they are false].

Alternative indefinability shows that the indefinables havenot been reached.

Every proposition is essentially true-false: to understand it,we must know both what must be the case if it is true, andwhat must be the case if it is false. Thus a proposition has twopoles, corresponding to the case of its truth and the case of itsfalsehood. We call this the sense of a proposition.

In regard to notation, it is important to note that not everyfeature of a symbol symbolizes. In two molecular functionswhich have the same T-F schema, what symbolizes must be thesame. In „not-not-p”, „not-p” does not occur; for„not-not-p” is the same as ”p”, and therefore, if „not-p”occurred in „not-not-p”, it would occur in „p”.

Logical indefinables cannot be predicates or relations,because propositions, owing to sense, cannot have predicates orrelations. Nor are „not” and „or”, like judgment, analogous topredicates or relations, because they do not introduce anythingnew.

Propositions are always complex even if they contain nonames.

B6 A proposition must be understood when all its indefinablesare understood. The indefinables in „aRb” are introduced asfollows:

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„a” is indefinable;„b” is indefinable;Whatever „x” and „y” may mean, „xRy” says something

indefinable about their meanings.A complex symbol must never be introduced as a single

indefinable. (Thus e.g. no proposition is indefinable.) For if oneof its parts occurs also in another connection, it must there bere-introduced. And would it then mean the same?

The ways by which we introduce our indefinables mustpermit us to construct all propositions that have sense fromthese indefinables alone. It is easy to introduce „all” and „some”in a way that will make the construction of (say) „(x,y).xRy”possible from „all” and „xRy” as introduced before.

B7 3rd MS.

An analogy for the theory of truth: Consider a black patch onwhite paper; then we can describe the form of the patch bymentioning, for each point of the surface, whether it is white orblack. To the fact that a point is black corresponds a positivefact, to the fact that a point is white (not black) corresponds anegative fact. If I designate a point of the surface (one of Frege’s„truth-values”), this is as if I set up an assumption to be decidedupon. But in order to be able to say of a point that it is black orthat it is white, I must first know when a point is to be calledblack and when it is to be called white. In order to be able tosay that „p” is true (or false), I must first have determinedunder what circumstances I call a proposition true, and therebyI determine the sense of a proposition. The point in which theanalogy fails is this: I can indicate a point of the paper that iswhite and black, but to a proposition without sense nothingcorresponds, for it does not designate a thing (truth-value),

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whose properties might be called „false” or „true”; the verb ofa proposition is not „is true” or „is false”, as Frege believes, butwhat is true must already contain the verb.

The comparison of language and reality is like that ofretinal image and visual image: to the blind spot nothing in thevisual image seems to correspond, and thereby the boundariesof the blind spot determine the visual image as true negationsof atomic propositions determine reality.

B8 Logical inferences can, it is true, be made in accordancewith Frege’s or Russell’s laws of deduction, but this cannotjustify the inference; and therefore they are not primitivepropositions of logic. If p follows from q, it can also be inferredfrom q, and the „manner of deduction” is indifferent.

Those symbols which are called propositions in which„variables occur” are in reality not propositions at all, but onlyschemes of propositions, which only become propositions whenwe replace the variables by constants. There is no propositionwhich is expressed by „x = x”, for „x” has no signification; butthere is a proposition „(x).x = x” and propositions such as„Socrates = Socrates” etc.

In books on logic, no variables ought to occur, but only thegeneral propositions which justify the use of variables. It followsthat the so-called definitions of logic are not definitions, butonly schemes of definitions, and instead of these we ought toput general propositions; and similarly the so-called primitiveideas (Urzeichen) of logic are not primitive ideas, but theschemes of them. The mistaken idea that there are things calledfacts or complexes and relations easily leads to the opinion thatthere must be a relation of questioning to the facts, and then thequestion arises whether a relation can hold between an arbitrarynumber of things, since a fact can follow from arbitrary cases.

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It is a fact that the proposition which e.g. expresses that q

follows from p and p⊃q is this: p.p⊃q.⊃p.q.q.B9 At a pinch, one is tempted to interpret „not-p” as

„everything else, only not p”. That from a single fact p aninfinity of others, not-not-p etc., follow, is hardly credible.Man possesses an innate capacity for constructing symbols withwhich some sense can be expressed, without having the slightestidea what each word signifies. The best example of this ismathematics, for man has until lately used the symbols fornumbers without knowing what they signify or that they signifynothing.

Russell’s „complexes” were to have the useful property ofbeing compounded, and were to combine with this the agreeableproperty that they could be treated like „simples”. But this alonemade them unserviceable as logical types, since there wouldhave been significance in asserting, of a simple, that it wascomplex. But a property cannot be a logical type.

Every statement about apparent complexes can be resolvedinto the logical sum of a statement about the constituents and astatement about the proposition which describes the complexcompletely. How, in each case, the resolution is to be made, isan important question, but its answer is not unconditionallynecessary for the construction of logic.

B10 That „or” and „not” etc. are not relations in the same senseas „right” and „left” etc., is obvious to the plain man. Thepossibility of cross-definitions in the old logical indefinablesshows, of itself, that these are not the right indefinables, and,even more conclusively, that they do not denote relations.

If we change a constituent a of a proposition ϕ(a) into avariable, then there is a class

~p {(∃x).ϕ(x) = p}.

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This class in general still depends upon what, by an arbitraryconvention, we mean by „ϕ(x)”. But if we change into variablesall those symbols whose significance was arbitrarily determined,there is still such a class. But this is now not dependent uponany convention, but only upon the nature of the symbol „ϕ(x)”.It corresponds to a logical type.

Types can never be distinguished from each other by saying(as is often done) that one has these but the other has thatproperties, for this presupposes that there is a meaning inasserting all these properties of both types. But from this itfollows that, at best, these properties may be types, but certainlynot the objects of which they are asserted.

B11 At a pinch, we are always inclined to explanations of logicalfunctions of propositions which aim at introducing into thefunction either only the constituents of these propositions, oronly their forms, etc. etc.; and we overlook that ordinarylanguage would not contain the whole propositions if it did notneed them: However, e.g., „not-p” may be explained, theremust always be a meaning given to the question „what isdenied?”

The very possibility of Frege’s explanations of „not-p” and„if p then q”, from which it follows that not-not-p

denotes the same as p, makes it probable that there is somemethod of designation in which „not-not-p” corresponds tothe same symbol as „p”. But if this method of designationsuffices for logic, it must be the right one.

Names are points, propositions arrows they have sense.The sense of a proposition is determined by the two poles trueand false. The form of a proposition is like a straight line, whichdivides all points of a plane into right and left. The line doesthis automatically, the form of proposition only by convention.

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B12 Just as little as we are concerned, in logic, with the relationof a name to its meaning, just so little are we concerned withthe relation of a proposition to reality, but we want to know themeaning of names and the sense of propositions as weintroduce an indefinable concept „A” by saying: „ A denotessomething indefinable”, so we introduce e.g. the form ofpropositions aRb by saying: „For all meanings of „x” and „y”,„xRy” expresses something indefinable about x and y”.

In place of every proposition „p”, let us write „abp”. Letevery correlation of propositions to each other or of names topropositions be effected by a correlation of their poles „a” and„b”. Let this correlation be transitive. Then accordingly „a-ab-bp”is the same symbol as „abp”. Let n propositions be given. I thencall a „class of poles” of these propositions every class of nmembers, of which each is a pole of one of the n propositions,so that one member corresponds to each proposition. I thencorrelate with each class of poles one of two poles (a and b). Thesense of the symbolizing fact thus constructed I cannot define,but I know it.

If p = not-not-p etc., this shows that the traditionalmethod of symbolism is wrong, since it allows a plurality ofsymbols with the same sense; and thence it follows that, inanalyzing such propositions, we must not be guided by Russell’smethod of symbolizing.

B13 It is to be remembered that names are not things, butclasses: „A” is the same letter as „A”. This has the mostimportant consequences for every symbolic language.

Neither the sense nor the meaning of a proposition is athing. These words are incomplete symbols.

It is impossible to dispense with propositions in which thesame argument occurs in different positions. It is obviouslyuseless to replace ϕ(a,a) by ϕ(a,b).a = b.

21

Since the ab-functions of p are again bi-polar propositions,we can form ab-functions of them, and so on. In this way aseries of propositions will arise, in which in general thesymbolizing facts will be the same in several members. If nowwe find an ab-function of such a kind that by repeatedapplication of it every ab-function can be generated, then we canintroduce the totality of ab-functions as the totality of those thatare generated by application of this function. Such a function is~p∨~q.

B14 It is easy to suppose a contradiction in the fact that on theone hand every possible complex proposition is a simpleab-function of simple propositions, and that on the other handthe repeated application of one ab-function suffices to generateall these propositions. If e.g. an affirmation can be generated bydouble negation, is negation in any sense contained inaffirmation? Does „p” deny „not-p” or assert „p”, or both?And how do matters stand with the definition of „⊃” by „∨”and „ ”, or of „∨” by „ ” and „⊃”? And how e.g. shall weintroduce p|q (i.e. ~p∨~q), if not by saying that this expressionsays something indefinable about all arguments p and q? Butthe ab-functions must be introduced as follows: The functionp|q is merely a mechanical instrument for constructing allpossible symbols of ab-functions. The symbols arising byrepeated application of the symbol „|” do not contain thesymbol „p|q”. We need a rule according to which we can formall symbols of ab-functions, in order to be able to speak of theclass of them; and we now speak of them e.g. as those symbolsof functions which can be generated by repeated application ofthe operation „|”. And we say now: For all p’s and q’s, „p|q”says something indefinable about the sense of those simplepropositions which are contained in p and q.

22

B15 The assertion-sign is logically quite without significance. Itonly shows, in Frege and Whitehead and Russell, that theseauthors hold the propositions so indicated to be true. „ ”therefore belongs as little to the proposition as (say) the numberof the proposition. A proposition cannot possibly assert of itselfthat it is true.

Every right theory of judgment must make it impossible forme to judge that this table penholders the book. Russell’s theorydoes not satisfy this requirement.

It is clear that we understand propositions without knowingwhether they are true or false. But we can only know themeaning of a proposition when we know if it is true or false.What we understand is the sense of the proposition.

The assumption of the existence of logical objects makes itappear remarkable that in the sciences propositions of the form„p∨q”, „p⊃q”, etc. are only then not provisional when „∨” and”⊃” stand within the scope of a generality-sign [apparentvariable].

B16 4th MS.

If we formed all possible atomic propositions, the world wouldbe completely described if we declared the truth or falsehood ofeach. [I doubt this.]

The chief characteristic of my theory is that, in it, p has thesame meaning as not-p.

A false theory of relations makes it easily seem as if therelation of fact and constituent were the same as that of fact andfact which follows from it. But the similarity of the two may beexpressed thus: ϕa.⊃.ϕ,a a = a.

If a word creates a world so that in it the principles of logicare true, it thereby creates a world in which the whole of

23

mathematics holds; and similarly it could not create a world inwhich a proposition was true, without creating its constituents.

Signs of the form „p∨~p” are senseless, but not theproposition „(p).p ∨ ~p”. If I know that this rose is either redor not red, I know nothing. The same holds of all ab-functions.

To understand a proposition means to know what is thecase if it is true. Hence we can understand it without knowingif it is true. We understand it when we understand itsconstituents and forms. If we know the meaning of „a” and „b”,and if we know what „xRy” means for all x’s and y’s, then wealso understand „aRb”.

I understand the proposition „aRb” when I know that eitherthe fact that aRb or the fact that not aRb corresponds to it; butthis is not to be confused with the false opinion that Iunderstand „aRb” when I know that „aRb or not-aRb” is thecase.

B17 But the form of a proposition symbolizes in the followingway: Let us consider symbols of the form „xRy”; to thesecorrespond primarily pairs of objects, of which one has thename „x”, the other the name „y”. The x’s and y’s stand invarious relations to each other, among others the relation Rholds between some, but not between others. I now determinethe sense of „xRy” by laying down: when the facts behave inregard to „xRy” so that the meaning of „x” stands in therelation R to the meaning of „y”, then I say that they [the facts]are „of like sense” [„gleichsinnig”] with the proposition „xRy”;otherwise, „of opposite sense” [„entgegengesetzt”]; I correlatethe facts to the symbol „xRy” by thus dividing them into thoseof like sense and those of opposite sense. To this correlationcorresponds the correlation of name and meaning. Both arepsychological. Thus I understand the form „xRy” when I knowthat it discriminates the behaviour of x and y according as these

24

stand in the relation R or not. In this way I extract from allpossible relations the relation R, as, by a name, I extract itsmeaning from among all possible things.

Strictly speaking, it is incorrect to say: We understand theproposition p when we know that „p” is true ≡ p; for thiswould naturally always be the case if accidentally thepropositions to right and left of the symbol „≡” were both trueor both false. We require not only an equivalence, but a formalequivalence, which is bound up with the introduction of theform of p.

The sense of an ab-function of p is a function of the senseof p.

B18 The ab-functions use the discrimination of facts, which theirarguments bring forth, in order to generate new discriminations.

Only facts can express sense, a class of names cannot. Thisis easily shown.

There is no thing which is the form of a proposition, and noname which is the name of a form. Accordingly we can also notsay that a relation which in certain cases holds between thingsholds sometimes between forms and things. This goes againstRussell’s theory of judgment.

It is very easy to forget that, tho. the propositions of a formcan be either true or false, each one of these propositions canonly be either true or false, not both.

Among the facts which make „p or q” true, there are somewhich make „p and q” true; but the class which makes „p or q”true is different from the class which makes „p and q” true; andonly this is what matters. For we introduce this class, as it were,when we introduce ab-functions.

A very natural objection to the way in which I haveintroduced e.g. propositions of the form xRy is that by itpropositions such as (∃x,y).xRy and similar ones are not

25

explained, which yet obviously have in common with aRb whatcRd has in common with aRb. But when we introducedpropositions of the form xRy we mentioned no one particularproposition of this form; and we only need to introduce(∃x,y).ϕ(x,y) for all ϕ’s in any way which makes the senseof these propositions dependent on the sense of all propositionsof the form ϕ(a,b), and thereby the justification of ourprocedure is proved.

B19 The indefinables of logic must be independent of eachother. If an indefinable is introduced, it must be introduced inall combinations in which it can occur. We cannot thereforeintroduce it first for one combination, then for another; e.g., ifthe form xRy has been introduced, it must henceforth beunderstood in propositions of the form aRb just in the sameway as in propositions such as (∃x,y). xRy and others. Wemust not introduce it first for one class of cases, then for theother; for it would remain doubtful if its meaning was the samein both cases, and there would be no ground for using the samemanner of combining symbols in both cases. In short, for theintroduction of indefinable symbols and combinations ofsymbols the same holds, mutatis mutandis, that Frege has saidfor the introduction of symbols by definitions.

It is a priori likely that the introduction of atomicpropositions is fundamental for the understanding of all otherkinds of propositions. In fact the understanding of generalpropositions obviously depends on that of atomic propositions.

Cross-definability in the realm of general propositions leadsto the quite similar questions to those in the realm ofab-functions.

B20 When we say „A believes p”, this sounds, it is true, as ifhere we could substitute a proper name for „p”; but we can see

26

that here a sense, not a meaning, is concerned, if we say „Abelieves that p is true”; and in order to make the direction ofp even more explicit, we might say „A believes that p is trueand not-p is false”. Here the bi-polarity of p is expressed,and it seems that we shall only be able to express theproposition „A believes p” correctly by the ab-notation; say bymaking „A” have a relation to the poles „a” and „b” of a-p-b.

The epistemological questions concerning the nature ofjudgment and belief cannot be solved without a correctapprehension of the form of the proposition.

The ab-notation shows the dependence of or and not, andthereby that they are not to be employed as simultaneousindefinables.

Not: „The complex sign aRb ” says that a stands in therelation R to b; but that a stands in a certain relation to bsays that aRb.

In philosophy there are no deductions: it is purelydescriptive.

Philosophy gives no pictures of reality.Philosophy can neither confirm nor confute scientific

investigation.B21 Philosophy consists of logic and metaphysics: logic is its

basis.Epistemology is the philosophy of psychology.

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Distrust of grammar is the first requisite for philosophizing.Propositions can never be indefinables, for they are always

complex. That also words like „ambulo” are complex appears inthe fact that their root with a different termination gives adifferent sense.

Only the doctrine of general indefinables permits us tounderstand the nature of functions. Neglect of this doctrineleads to an impenetrable thicket.

Philosophy is the doctrine of the logical form of scientificpropositions (not only of primitive propositions).

The word „philosophy” ought always to designatesomething over or under, but not beside, the natural sciences.

Judgment, command and question all stand on the samelevel; but all have in common the propositional form, whichdoes interests us.

The structure of the proposition must be recognized, therest comes of itself. But ordinary language conceals the structureof the proposition: in it, relations look like predicates, predicateslike names, etc.

Facts cannot be named.B22 It is easy to suppose that „individual”, „particular”,

„complex” etc. are primitive ideas of logic. Russell e.g. says„individual” and „matrix” are „primitive ideas”. This errorpresumably is to be explained by the fact that, by employmentof variables instead of the generality-sign, it comes to seem asif logic dealt with things which have been deprived of allproperties except thing-hood, and with propositions deprived ofall properties except complexity. We forget that the indefinablesof symbols [Urbilder von Zeichen] only occur under thegenerality-sign, never outside it.

Just as people used to struggle to bring all propositions intothe subject-predicate form, so now it is natural to conceive every

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proposition as expressing a relation, which is just as incorrect.What is justified in this desire is fully satisfied by Russell’stheory of manufactured relations.

One of the most natural attempts at solution consists inregarding „not-p” as „the opposite of p”, where then„opposite” would be the indefinable relation. But it is easy tosee that every such attempt to replace the ab-functions bydescriptions must fail.

B23 The false assumption that propositions are names leads usto believe that there must be logical objects: for the meanings oflogical propositions will have to be such things.

A correct explanation of logical propositions must givethem a unique position as against all other propositions.

No proposition can say anything about itself, because thesymbol of the proposition cannot be contained in itself; thismust be the basis of the theory of logical types.

Every proposition which says something indefinable abouta thing is a subject-predicate proposition; every propositionwhich says something indefinable about two things expresses adual relation between these things, and so on. Thus everyproposition which contains only one name and one indefinableform is a subject-predicate proposition, and so on. Anindefinable simple symbol can only be a name, and therefore wecan know, by the symbol of an atomic proposition, whether itis a subject-predicate proposition.

B24 I. Bi-polarity of propositions: sense and meaning, truth andfalsehood.II. Analysis of atomic propositions: general indefinables,predicates, etc.III. Analysis of molecular propositions: ab-functions.IV. Analysis of general propositions.

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V. Principles of symbolism: what symbolizes in a symbol. Factsfor facts.VI. Types.

B25

This is the symbol for~p ∨ ~q

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B26

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Item 201a-1 (RA1.710.057822 and RA1.710.057823)

Transcriber(s): Michael Biggs, Peter CrippsProofreader(s): Peter CrippsComments:Transcriptions of material from The BertrandRussell Archives, McMaster University, Canada;their catalogue no.s:TS section (pp. numbered 1-7) - RA1.710.057822,MS section (pp. numbered 1-23 plus 3 unnumberedfolios) - RA1.710.057823.This item consists of two parts: a typescript of7ff. with corrections in Russell’s hand andadditions in Wittgenstein’s hand, and a manuscriptof 26ff. in Russell’s hand. In the TS part:non-typewriter characters are inserted by hand;fully typed pages have 24 or 25 lines of type;each new sentence is separated from the last bythree spaces. True double spaced blank lines usedoccasionally but not consistently.Hands:s = handwritten insertions that belong to firstpass, ie generally the insertion of symbols whichare not found on the typewriter such as ϕ and ∃,H5 = Ludwig Wittgenstein, or cases where thetranscriber has felt unable to decide whetherLudwig Wittgenstein or Bertrand RussellS1 = Bertrand Russell

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Item 201a-1 Recto Page A11

‹Summary›1

The2 One1 reason for thinking the oldnotation wrong is that it is very unlikely thatfrom every proposition p an infinite number ofother propositions not-not-p, not-not-not-not-p,etc., should follow.

If only th[e|ooo3]se signs which contain propernames were complex then propositions containingnothing but apparent variables would be simple.Then what about their denials?

The verb of a proposition cannot be ”is true”or ”is false”, but whatever is true or false mustalready contain the verb.

The2 [d|DDD3]eductions only proceed accordingto the laws of deduction‹,›4 but these laws cannotjustify the deduction.

The2 One1 reason for supposing that not allpropositions which have more than one argument arerelational propositions is that if 1 theywere‹,›1 the relations of judgement and inferencethat2 would1 have to hold between an arbitrarynumber of things.

Every proposition which seems to be about acomplex can be analysed into a proposition aboutthose2 its 1 cons[i|ttt]ituents and about theproposition which describes a2 the1 complexperfectly; i.e., that proposition which isequivalent to saying a2 the1 complex exists.

The idea that propositions are names ofcomplexes between5 L.W.

4 suggestions5‹s›4 ⟨? ⟩ 6

L.W.4 that whatever is not a proper name is a sign

for a relation. ‹Becaaause spatial complexes*consiiist of Things & Relations only & the idea of acomplex is taken from sp›4

In a proposition convert all its indefinablesinto variables; there then remains a class of

37

propositions which has2 is1 not all propositionsbut a type.⟨* you7 - for instance imaginnne every fact as aspatial complex ⟩ 8

38


There are thus two ways in which signs aresimilar. The names Socrates and Plato are similar:they are both names. But whatever they have incommon must not be introduced before Socrates andPlato are introduced. The same applies tosubject-predicate form etc. Therefore, thing,proposition, subject-predicate form, etc., are notindefinables, i.e., types are not indefinables.

When we say a2 A1 judge‹s›1 is2 that etc.,then we have to mention a whole proposition whicha2 A1 judge‹s›1 is2. It will not do either tomention only its constituent‹s,›1 or itsconstituent‹s›1 and form, but not in the properorder. This shows that a proposition itself mustoccur in the statement that it is judged; however,for instance, ”not-p” may be explained[.|,,,9] ‹p10

must occur in it.›1 5 ⟨[t|TTT9]he question, „What isnegated” must have a meaning ⟩ 11

Always a que[x|ssss]tion that is negated musthave a meaning.5 ⟨Rott! ⟩ 12

To understand a proposition p it is notenough to know that ”5p implies ‹’”›4p” istrue‹’›4, but we must also know that p alsoimplies ‹’›4”not-p” is false‹’›4 5 ~p implies ”pis false”4. This shows the bi 13polarity of theproposition.⟨W-F = Wahr-Falsch ⟩ 6

To every molecular function a 1 [wf|WFWFWF3]scheme corresponds. Therefore we may use the[wf|WFWFWF3] scheme itself instead of the function.Now what the [wf|WFWFWF3] scheme does is, itcorrelates the letters [w|WWW3] and [f|FFF3] with eachproposition. These two letters are the poles ofatomic propositions. Then the scheme 1

corresponds2lates1 another [f|WWW3] and [w|FFF3] tothese poles. In this notation all that it2 mattersis the correlation of the outside poles to the

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pole of its2 the 1 atomic propositions. Thereforenot-not-p is the same symbol as p. ‹And›1

Therefore we shall never get two symbols for thesame molecular functions5.

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The meaning of a proposition is the factwhich actually corresponds to it.

As the ab functions of atomic propositionsare by2i1‹-›1polar propositions again we canperform ab7 operations on them. We [wi|shashasha9]ll‹,›4

b[e|yyy9] doing so, correlate two new outside polesvia the old outside poles to the poles of theatomic propositions.

The symbolising fact in a-p-b is that, say7*a7 is on the left of p7 and b7 on the right ofp7[,|;;;9] then the correlation of new poles is tobe transitive, such2 so1 that for instance 4 if anew pole a7 in whatever way i.e. via whateverpoles 4 is correlated to the inside a7, the symbolis not changed thereby. It is therefore possibleto construct all possible ab7 functions byperforming one ab7 operation repeatedly, and wecan therefore talk of all ab7 functions as of allth[e|ooo9]se functions which can be obtained byperforming this ab7 operation repeatedly.

⟨[Note by B.R.] ⟩ 6[NB. ab7 means the same as [wf|WFWFWF3], which

means true-false.]

Naming is like pointing. A function is like aline dividing points of a plane 4 into right andleft ones; then ‹”›1p or not-p‹”›1 has no meaningbecause it does not divide [a|thethethe9] plane.

But though a particular proposition ”p”5 or a”5not-p” has no meaning‹,›4 a general proposition‹”›4for all p’s, ”5p”5 or ”5not-p” has a meaningbecause this does not contain [a|thethethe9] nonsensicalfunction ‹”›4p [n|ooo]r not-p‹”›4 but [a|thethethe9]function ‹”›4p or ”5not-q” just as ‹”›4for all”5x’s xRx‹”›1 contains the function ‹”›4xRy”.⟨* This is quite arbitrary but if we such havefixed on which sides the poles have to stand we

41

must of course stick to our convention. If forinstance „apb” says p then bpa says nothing7. (Itdoes not say ~p) But a-apb-b is the same symbol asapb ⟩ 8⟨the ab function vanishes automatically) for herethe new poles are ⟩ 14 ⟨related to the same sidedede ofp as the old ones. The question is allways: howare the new poles correlated to p compared withthe way the old poles are correlated tooo p. ⟩ 15

42


A proposition is a standard to which allfacts behave, that2 with1 names it is 1

otherwise; it is then2us1 by2i1‹-›1polarity andsense comes in‹;›1 just as one error2 arrow1

behaves to another error2 arrow1 byyy being in thesame sense or the opposite, so a fact behaves to aproposition.

The form of a proposition has meaning in thefollowing way. Consider a symbol ‹”›4xRy‹”›4. Tosymbols of this form correspond couples of thingswhose names are respectively ‹”›4x‹”›4 and‹”›4y‹”›4. The things x7‹/›4y7 stand to oneanother in all sorts of relations‹,›4 amongstothers some stand in the relatio[j|nnn] of2 R1, andsome not; just as I single out a particular thingby a particular name I single out all behavioursof the point‹s›1 x and y the one between2 withrespect to1 the relation ‹R.›1 of the other2. Isay that if an x stands in the relation of2 R1 toa y the sign ‹”›4x of2 R1 y‹”›4 is to be calledtrue to the fact and otherwise false. This is adefinition of sense.⟨! ⟩ 16

In my theory p has the same meaning as not-pbut opposite snese. The meaning is the fact. Theproper theory of judgment must make it impossibleto judge nonsense.

It is not strictly true to say that weunderstand a proposition p if we know that p isequivalent to ”p is true” for this would be thecase if accidentally both were true or false. Whatis wanted is the formal equivalence with respectto the forms of the proposition[.|,,,9] i.e.,[A|aaa9]ll the general indefinables involved. Thesense of an ab7 function of a propositionnn is afunction of its sense[:|...9] [t|TTT9]here are onlyunasserted propositio‹ns.›4

43


Assertion is mmmerely psychological. If2n1 not-p‹,›1

p10 1 is exactly the same as if it standsalone‹;›1 this point is absolutely fundamental.Among the facts which make ”p or q” true1 thereare also facts which make ”p and q” true‹;›1 ifpropositions do only mean2 have onlymeaning 1‹,›1 we ought‹,›1 to know2 in 1 such acase, to1 say that these two propositions areidentical, but in fact, their sense is differentfor we have introduced sense by talking of all p’sand all q’s. Consequently the molecularpropositions will only be used in cases wherethere ab7 function stands under a generality signor enters into another function such as ‹”›4Ibelieve that, etc.,‹”›4 because then4 the senseenters.

In ”a judges p” p cannot be replaced by aproper name. This appears if we substitute ”ajudges that p is true and not p is false”. Theproposition ”a judges p” consists of the propername a[.|,,,9] [T|ttt9]he proposition p with its 2poles‹,›4 and a7 being related to both of thesepoles in a certain way. This is obviously not the5

‹a›4 relation in the ordinary sense.The ab notation and5 for4 apparent variables5

make‹s›4 it clear that not and or are dependent onone another and we can therefore not use them assimultaneous indefinables. ‹|17›4 Some5 ‹Same›4

objections in the case of app. var. 4 to oldindefinables, [a|AAAA9]s5 as4 18 in the case ofmolecular functions19[,|:::9] [t|TTT9]he applicationof the ab notation to apparently5‹-›4 variablepropositions become‹s›4 clear if we consider that,for instance, the proposition ‹”›4for all ”5x‹,›4

ϕx” is to be true when ϕx is true for all x’s andfalse when ϕx is false for some x’s. We see thatsome and all occur simultaneously in the properapparent variable notation.

44


The Notation is:

for (x) ϕx ; a - (x) - a ϕ x b - (∃ x) - band

for (ϕ5∃x) ϕ x : a - (∃x) - a ϕ x b - (x) - [v|bbb]

Old definitions now become tautologous.In aRb it is not the complex that symbolises

but the fact that the symbol a stands in a certainrelation to the symbol b. Thus facts aresymbolised by facts, or the5 more correctly: thata certain thing is the case in the symbol saysthat a certain thing is the case in the world.

Judgment, question and command are all on thesame level. What interests logic in them is onlythe unasserted proposition. Facts cannot be named.

A proposition cannot occur in itself. This isthe fundamental truth of the theory of types.

Every proposition that says somethingimportant5 about one thing is a subject-predicateproposition, and so on.

Therefore wwwe can recognize asubject-predicate proposition if we know itcontains only one name and one form, etc. Thisgives the construction of types. Hence the type ofa proposition can be recognized by its symbolalone.

45


What is essential in a correctapparent -variable 1 notation is this:- (1) itmust mention a type of propositions; (2) it mustshow which components of a proposition of thistype are constants.@

[Components are forms and constituents.]

Take (ϕ).ϕ!x. Then if we describe the kind7

of symbols20 , for which ϕ! stands 4 ‹&›4 which,by the above, is enough to determine the type,then automatically ”([x|ϕ]).ϕ!x” cannot befi[ll|tttttt]ed by this descri[l|ppp]tion[.|,,,9] ‹becauseit contains7 „ϕ!x” & the description is todescribe all7 that symbolizes in symbols of the ϕ!- kind. If the description is thus7 completevicious circles can just as little occur as if5

for instance in (ϕ).ϕ(x)5 (ϕ).(x)ϕ4 (where (xxx)ϕ isa subject-predicat prop) ›4

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Item 201a-1 Recto Page B1Wittgenstein 1

First MS.Indefinables are of two sorts: names, & forms.Propositions cannot consist of names alone; theycannot be classes of names. A name can not onlyoccur in two different propositions, but canoccur in the same wayyy in both.

Propositions [which are symbols havingreference to facts] are themselves facts: thatthis inkpot is on this table may express that Isit in this chair.

It can never express the commoncharacteristic @ of two objects that we denotedesignate them by the same name but by twodifferent ways of designation, for, since namesare arbitrary, we might ‹also› choose differentnames, & where then would be the common elementin the designations? Nevertheless one is alwaystempted, in a difficulty, to take refuge indifferent ways of designation. @

Frege said ”propositions are names”;Russell said ”propositions correspond tocomplexes”. Both are false; & especially falseis the statement ”propositions are names ofcomplexes”.

It is easy to suppose that only suchsymbols are complex as contain names of objectsobjects, & that accordingly ”(∃x,ϕ).ϕx” or”(∃x, R, y).xRy” must be simple. It is thennatural to call the first of these the name of aform, the second the name of a relation. But inthat case what is the meaning of (e.g.)”~(∃x,y).xRy”? Can we put ”not” before a name?

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Item 201a-1 Recto Page B2Wittg.- 2

The reason why ”~Socrates” means nothing isthat ”~x” does not express a property of x.

There are positive & negative facts: if theproposition ”this rose is not red” is true, thenits what it signifies is negative. But theoccurrence of the word ”not” does not indicatethis unless we know that the signification ofthe proposition ”this rose is red” (when it istrue) is positive. It is only from both, thenegation & the negated proposition, that we canconclude to a characteristic of the significanceof the whole proposition. (We are not herespeaking of negations of general propositions,i.e. of such as contain apparent variables.)Negative facts only justify the negations ofsimpl atomic propositions.)

Positive & negative facts there are, butnot true & false facts.

If we overlook the fact that propositionshave a sense which is independent of their truthor falsehood, it easily seems as if true & falsewere two equally justified relations between thesign & what is signified. (We might then saye.g. that ”q” signifies in the true way what”not-q” signifies in the false way). But are nottrue & false in fact equally justified? Could wenot express ourselves by means of falsepropositions just as well as hitherto with trueones, so long as we know that they are meantfalsely?

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Item 201a-1 Recto Page B33

No! For a proposition is then true when it is aswe assert in this proposition; & accordingly ifby ”q” we mean ”not-q”, & it is as we mean toassert, then in the new interpretation ”q” isactually true & not false. But it is importantthat we can mean the same by ”q” as by ”not-q”,for it shows that neither to the symbol ”not”nor to the manner of its combination with ”q”does a characteristic of the denotation of ”q”correspond.

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2nd MS.We must be able to understand propositions whichwe have never heard before. But everyproposition is a new symbol. Hence we must havegeneral indefinable symbols; these areunavoidable if propositions are not allindefinable.

Whatever corresponds in reality to compoundpropositions must not be more than whatcorresponds to their several atomicpropositions.

Not only must logic not deal with[particular] things, but just as little withrelations & predicates.

There are no propositions containing realvariables.

What corresponds in reality to aproposition depends upon whether it is true orfalse. But we must be able to understand aproposition without knowing if it is true orfalse.

What we know when we understand aproposition is this: We know what is the case ifthe proposition is true, & what is the case ifit is false. But we do not know [necessarily]whether it is true or false.

Propositions are not names.We can never distinguish one logical type

from another by attributing a property tomembers of the one which we deny to members ofthe other.

Symbols are not what they seem to be. In”aRb”, ”R” looks like a substantive, but is notone. What symbolizes in ”aRb” is that R occursbetween a & b. Hence ”R” is not the indefinablein ”aRb”. Similarly in ”ϕx”, ”ϕ” looks like asubstantive but is not one; in ”~p”, ”~” lookslike ”ϕ” but is not like it. This is the first

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thing that indicates that there may not belogical constants. A reason against them is thegenerality of logic: logic cannot treat aspecial set of things.

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Molecular propositions contain nothingbeyond what is contained in their atoms; theyadd no material information above that containedin their atoms.

All that is essential about molecularfunctions is their T-F schema [i.e. thestatement of the cases when they are true & thecases when they are false].

Alternative indefinability shows that theindefinables have not been reached.

Every proposition is essentiallytrue-false: to understand it, we must know bothwhat must be the case if it is true, & what mustbe the case if it is false. Thus a propositionhas two poles, corresponding to the case of itstruth & the case of its falsehood. We call thisthe sense of a proposition.

In regard to notation, it is important tonote that not every feature of a symbolsymbolizes. In two molecular functions whichhave the same T-F schema, what symbolizes mustbe the same. In ”not-not-p”, ”not-p” does notoccur; for ”not-not-p” is the same as ”p”, &therefore, if ”not-p” occurred in ”not-not-p”,it would occur in ”p”.

Logical indefinables cannot be predicatesor relations, because propositions, owing tosense, cannot have predicates or relations. Norare ”not” & ”or”, like judgment, analogous topredicates or relations, because they do notintroduce anything new.

Propositions are always complex even ifthey contain no names.

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A proposition must be understood when allits indefinables are understood. Theindefinables in ”aRb” are introduced as follows:

”a” is indefinable;”b” is indefinable;Whatever ”x” & ”y” may mean, ”xRy” says

something indefinable21 about their meanings.A complex symbol must never be introduced

as a single indefinable. (Thus e.g. noproposition is indefinable.) For if one of itsparts occurs also in another connection, it mustthere be re-introduced. And would it then meanthe same?

The ways by which we introduce ourindefinables must permit us to construct allpropositions that have sense [? meaning] fromthese indefinables alone. It is easy tointroduce ”all” & ”some” in a way that will makethe construction of (say) ”(x,y).xRy” possiblefrom ”all” & ”xRy” as introduced before.

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3rd MS.A comparis An analogy for the theory of truth:Consider a black patch on white paper; then wecan describe the form of the patch bymentioning, for each point of the surface,whether it is white or black. To the fact that apoint is black corresponds a positive fact, tothe fact that a point is white (not black)corresponds a negative fact. If I designate apoint of the surface (one of Frege’s”truth-values”), this is as if I set up anassumption to be decided upon. But in order tobe able to say of a point that it is black orthat it is white, I must first know when a pointis to be called black & when it is to be calledwhite. In order to be able to say that ”p” istrue (or false), I must first have determinedunder what circumstances I call a propositiontrue, & thereby I determine the sense of aproposition. The point [on|ininin] which the analogydepends fails is this: I can indicate a point ofthe paper what is white & black, but to aproposition without sense nothing corresponds,for it does not designate a thing (truth-value),whose properties might be called ”false” or”true”; the verb of a proposition is not ”istrue” or ”is false”, as Frege believes, but whatis true must already contain the verb.

The comparison of language & reality islike that of retinal image & visual image: tothe blind spot nothing in the visual image seemsto correspond, & thereby the boundaries of theblind spot determine the visual image - as truenegations of atomic propositions determinereality.

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Logical inferences can, it is true, be madein accordance with Frege’s or Russell’s laws ofdeduction, but this cannot justify theinference; & therefore they are not primitivepropositions of logic. If p follows from q, itcan also be inferred from q, & the ”manner ofdeduction” is indifferent.

Those symbols which are called propositionsin which ”variables occur” aaare in reality notpropositions at all, but only schemes ofpropositions, which only become propositionswhen we replace the variables by constants.There is no proposition which is expressed by ”x= x”, for ”x” has no signification; but there isa proposition ”(x).x = x” & propositions such as”Socrates = Socrates” etc.

In books on logic, no variables ought tooccur, but only the general propositions whichjustify the use of variables. It follows thatthe so-called definitions of logic are notdefinitions, but only schemes of definitions, &instead of these we ought to put generalpppropositions; & similarly the so-calledprimitive ideas (Urzeichen) of logic are notprimitive ideas, but the schemes of them. Themistaken idea that there are things called factsor complexes & relations easily leads to theopinion that there must be a relation ofquestioning to the facts, & then the questionarises whether a relation can hold between anarbitrary number of things, since a fact canfollow from arbitrary cases. It is a fact thatthe proposition which e.g. expresses that qfollows from p & p⊃q is this: p.p⊃q.⊃p.q.q.

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At a pinch, one is tempted to interpret”not-p” as ”everything else, only not p”. Thatfrom a single fact p an infinity of others,not-not-p etc., follololow, is hardly credible. Manpossesses an innate capacity for constructingsymbols with which some sense can be expressed,without having the slightest idea what each wordsignifies. The best example of this ismathematics, for man has until lately used thesymbols for numbers without knowing what theysignify or that they signify nothing.

Russell’s ”complexes” were to have theuseful property of being compounded, & were tocombine with this the agreeable property thatthey could be treated as like ”simples”. Butthis alone made them unserviceable as logicaltypes, since there would have been significancein asserting, of a simple, that it was complex.But a property cannot be a logical type.

Every statement about apparent complexescan be resolved into the logical sum of astatement about the constituents & a statementabout the proposition which describes thecomplex completely. How, in each case, theresolution is to be made, is an importantquestion, but its answer is not unconditionallynecessary for the construction of logic.

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That ”or” & ”not” etc. are not relations inthe same sense as ”right” & ”left” etc., isobvious to the plain man. The possibility ofcross-definitions in the old logicalindefinables shows, of itself, that these arenot the right indefinables, &, even moreconclusively, that they do not denote relations.

If we change a constituent a of aproposition ϕ(a) into a variable, then there isa class

~p {(∃x).ϕ(x) = p}.

This class in general still depends upon what,by an arbitrary convention, we have mean by”ϕ(x)”. But if we change into variables allthose symbols whose significance was arbitrarilydetermined, there is still such a class. Butthis is now not dependent upon any convention,but only upon the nature of the symbol ”ϕ(x)”.It corresponds to a logical type.

Types can never be distinguished from eachother by saying (as is often done) that one hasth[i|eee]s‹e› @ but the other has th[at|oseoseose]propert[y|iii]‹es›, for this presupposes thatthere is a meaning in asserting all theseproperties of both types. But fffrom this itfollows that, at best, these properties may betypes, but certaincertaincertainly not the objects of whichthey are asserted.

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At a pinch, we are always inclined toexplanations of logical functions ofpropositions which aim at introducing into thefunction either only contain the constituentsof these propositions, or only their forms, etc.etc.etc.etc; & we overlook that ordinary language wouldnot contain the whole propositions if it did notneed them: However, e.g., ”not-p” may beexplained, there must always be a meaning givento the question ”what is denied?”

The very possibility of Frege’sexplanations of ”not-p” & ”if p then q”, fromwhich it follows that ”not-not-p” denotes thesame as p, makes it probable that there is somemethod of designation in which ”not-not-p”corresponds to the same symbol as ”p”. But ifthis method of designation suffices for logic,it must be the righghght one.

Names are points, sentences propositions

arrows - they have sense. The sense of aproposition is determined by the two poles true& false. The form of a proposition is like astraight line, which divides all points of aplane into right & left. The line does thisautomatically, the form of proposition only byconvention.

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Just as little as we are concerned, inlogic, with the relation of a name to itsmeaning, just so little are we concerned withthe relation of a proposition to reality, but wewant to know the meaning of names & the sense ofpropositions - as we introduce an indefinableconcept ”A” by saying: ”’A’ denotes somethingindefinable”, so we introduce e.g. the form ofpropositions aRb by saying: ”For all meanings of”x” & ”y”, ”xRy” expresses something indefinableabout x & y”.

In place of every proposition ”p”, let uswrite @ ”abp”. Let every correlation ofpropositions to each other or of names topropositions be effected by a correlation oftheir poles ”a” & ”b”. Let this correlation betransitive. Then accordingly ”a-ab-bp” is thesame symbol as ”abp”. Let n propositions begiven. I then call a ”class of poles” of thesepropositions every class of n members, of whicheach is a pole of one of the n propositions, sothat one member corresponds to each proposition.I then correlate with each class of poles one oftwo poles (a & b). The sense of the symbolizingfact thus constructed I cannot define, but Iknow it.

If p = not-not-p etc., this shows that thetraditional method of symbolism is wrong, sinceit allows a plurality of symbols with the samesense; & thence it follows that, in analyzingsuch propositions, we must not be guided byRussell’s method of symbolizing.

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It is to be remembered that names are notthings, but classes: ”A” is the same letter as”A”. This hhhas the most important consequencesfor every symbolic language.

Neither the sense nor the meaning of aproposition is a thing. These words areincomplete symbols.

It is impossible to dispense withpropositions in which the same argument occursin different positions. It is obviously uselessto replace ϕ(a,a) by ϕ(a,b).a = b.

Since the ab-functions of p are againbi-polar propositions, we can form ab-functionsof them, & so on. In this way a series ofpropositions will arise, in which in general thesymbolizing facts will be the same in severalmembers. If now we find an ab-function of such akind that by repeated application of it everyab-function can be generated, then we can defineintroduce the totality of ab-functions as thetotality of those that are generated byapplication of this function. Such a function is~p∨~q.

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It is easy to suppose a contradiction inthe fact that on the one hand all every possiblecomplex proposition is a simple ab-function ofsimple propositions, & that on the other handthe repeated application of one ab-functionsuffices to generate all these propositions. Ife.g. an affirmation can be generated by doublenegation, is negation in any sense contained inaffirmation? Does ”p” deny ”not-p” or assert”p”, or both? And how do matters stand with thedefinition of ”⊃” by ”∨” & ”~” ”.”, or of ”∨” by”.” & ”⊃”? And how e.g. shall we introduce p|q(i.e. ~p∨~q), if not by saying that thisexpression says something indefinable about allarguments p & q? But the ab-functions must beintroduced as follows: The function p|q ismerely a mechanical instrument for constructingall possible symbols of ab-functions. Thesymbols arising by repeated application of thesymbol ”|” do not contain the symbol ”p|q”. Weneed a rule according to which we can form allsymbols of ab-functions, in order to be able tospeak of the class of them; & we now speak ofthem e.g. as those symbols of functions whichcan be generated bybyby repeated application of theoperation ”|”. And we say now: For all p’s &q’s, ”p|q” says something indefinable about thesense of those simple propositions which arecontained in p & q.

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The assertion-sign is logically quitewithout significance. It only shows, in Frege& Whitehead & Russell, that these authors holdthe propositions so indicated to be true. ” ”therefore belongs as little to the propositionas (say) the number of the proposition. Aproposition cannot possibly assert of itselfthat it is true.

Every right theory of judgment must make itimpossible for me to judge that this tablepenholders the book. Russell’s theory does notsatisfy this requirement.

It is clear that we understand propositionswithout knowing whether they are true or false.But we can only know the meaning of aproposition when we know if it is true or false.What we understand is the sense of theproposition.

The assumption of the existence of logicalobjects makes it appear remarkable that in thesciences propositions of the form ”p[or|∨∨∨]q”,”p⊃q”, etc. are only then not provisional when”∨” & ”⊃” stand within the scope of agenerality-sign [apparent variable].

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4th MS.If we formed all possible atomic propositions,the world would be completely described if wedeclared the truth or falsehood of each. [Idoubt this.]

The chief characteristic of my theory isthat, in it, p has the same meaning as not-p.

A false theory of relations makes it easilyseem as if the relation of fact & constituentwere the same as that of fact & fact whichfollows from it. But the similarity of the twomay be expressed thus: ϕa.⊃.ϕ,a a = a.

If a word creates a world so that in it theprinciples of logic are true, it thereby createsa world in which the whole of mathematics holds;& similarly it could not create a world in whicha proposition was true, without creating itsconstituents.

Signs of the form ”p∨~p” are senseless, butnot the proposition ”(p).p ∨ ~p”. If I know thatthis rose is either red or not red, I knownothing. The same holds of all ab-functions.

To understand a proposition means to knowwhat is the case if it is true. Hence wwwe canunderstand it without knowing if it is true. Weunderstand it when we understand itsconstituents & forms. If we know the meaning of”a” & ”b”, & if we know what ”xRy” means for allx’s & y’s, then we also understand ”aRb”.

I understand the proposition ”aRb” when Iknow that either the fact that aRb or the factthat not aRb corresponds to it; but this is notto be confused with the false opinion that Iunderstand ”aRb” when I know that ”aRb ornot-aRb” is the case.

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But the form of a proposition symbolizes inthe following way: Let us consider symbols ofthe form ”xRy”; to these correspond primarilypairs of objects, of which one has the name ”x”,the other the name ”y”. The x’s & y’s stand invarious relations to each other, among othersthe relation R holds between some, but notbetween others. I know now determine the senseof ”xRy” by laying down: when the facts behavein regard to ”xRy” so that the meaning of ”x”stands in the relation R to the meaning of ”y”,then I say that they [the facts] are ”of likesense” [”gleichsinnig”] with the proposition”xRy”; otherwise, ”of opposite sense”[entgegengesetzt”]; I correlate the facts to thesymbol ”xRy” by thus dividing them into those oflike sense & those of opposite sense. To thiscorrelation corresponds the correlation of name& meaning. Both are psychological. Thus Iunderstand the form ”xRy” when I know that itdiscriminates the behaviour of x & y accordingas these stand in the relation R or not. In thisway I extract from all possible relations therelation R, as, by a name, I extract its meaningfrom among all possible things.

Strictly speaking, it is incorrect to say:We understand the proposition p when we knowthat ’”p” is true’ ≡ p; for this would naturallyalways be the case if accidentally thepropositions to right & left of the symbol ”≡”were both true or both false. We require notonly an equivalence, but a formal equivalence,which is bound up with the introduction of theform of p.

The sense of an ab-function of p is afunction of the sense of p.

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The ab-functions use the discrimination offacts, which their arguments bring forth, inorder to generate new discriminations.

Only facts can express sense, a class ofnames cannot. This is easily shown.

There is no thing which is the form of aproposition, & no name which is the name of aform. Accordingly we can also not say that arelation which in certain cases holds betweenthings holds sometimes between forms & things.This goes against Russell’s theory of judgment.

It is very easy to forget that, tho’ thepropositions of a form can be either true orfalse, each one of these propositions can onlybe either true or false, not both.

Among the facts which make ”p or q” true,there are some which make ”p & q” true; but theclass which makes ”p or q” true is differentfrom the class which makes ”p & q” true; &only this is what matters. For we introducethis class, as it were, when we introduceab-functions.

A very natural objection to the way inwhich I have introduced e.g. propositions of theform xRy is that by it propositions such as(∃x,y).xRy & similar ones are not explained,which yet obviously have in common with aRb whatcRd has in common with aRb. But when weintroduced propositions of the form xRy wementioned no one particular proposition of thisform; & we only need to introduce (∃x,y).ϕ(x,y)for all ϕ’s in any way which makes the sense ofthese propositions dependent on the sense of allpropositions of the form ϕ(a,b), & thereby thejustification of our procedure is proved.

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The indefinables of logic must beindependent of each other. If an indefinable isintroduced, it must be introduced in allcombinations in which it can occur. We cannottherefore introduce it first for onecombination, then for another; e.g., if the formxRy has been introduced, it must henceforth beunderstood in propositions of the form aRb justin the same way as in propositions such as(∃x,y). xRy & others. We must not introduce itfirst for one class of cases, then for theother; for it would remain doubtful if itsmeaning was the same in both cases, & therewould be no ground for using the same manner ofcombining symbols in both cases. In short, forthe introduction of indefinable symbols &classes combinations of symbols the same holds,mutatis mutandis, that Frege has said for theintroduction of symbols by definitions.

It is aaa priori likely that the introductionof atomic propositions is fundamental for theunderstanding of all other kinds ofpropositions. In fact the understanding ofgeneral propositions obviously depends on thatof atomic propositions.

Cross-definability in the realm of generalpropositions leads to the quite similarquestions to those in the realm of ab-functions.

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When we say ”A believes p”, this sounds, itis true, as if here we could substitute a propername for ”p”; but we can see that here a sense,not a meaning, is concerned, if we say ”Abelieves that ’p’ is true”; & in order to makethe direction of p even more explicit, we mightsay ”A believes that ’p’ is true & ’not-p’ isfalse”. Here the bi-polarity of p is expressed,& it seems that we shall only be able to expressthe proposition ”A believes p” correctly by theab-notation; say by making ”A” have a relationto the poles ”a” & ”b” of a-p-b.

The epistemological questions concerning thenature of judgment & belief cannot be solvedwithout a correct apprehension of the form ofthe proposition.

The ab-notation shows the dependence of or& not, & thereby that they are not to beemployed as simultaneous indefinables.

Not: ”The complex sign ’aRb’” says that astands in the relation R to b; but that ’a’stands in a certain relation to ’b’ says thataRb.

In philosophy there are no deductions: itis purely descriptive.

Philosophy gives no pictures of reality.Philosophy can neither confirm nor confute

scientific investigation.

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Philosophy consists of logic & metaphysics:logic is its basis.

Epistemology is the philosophy ofpsychology.

Distrust of grammar is the first requisitefor philosophizing.

Propositions can never be indefinables, forthey are always complex. That also words like”ambulo” are complex appears in the fact thattheir root with a different termination gives adifferent sense.

Only the doctrine of general indefinablespermits us to understand the nature offunctions. Neglect of this doctrine leads to animpenetrable thicket.

Philosophy is the doctrine of the logicalform of scientific propositions (not only ofprimitive propositions).

The word ”philosophy” ought always todesignate something over or under, but notbeside, the natural sciences.

Judgment, command & question [&|aaa]llstand on the same level; but all have in commonthe propositional form, which does interests us.

The construction structure of the sentenceproposition must be recognized, the rest comesof itself. But ordinary language conceals thestructure of the proposition: in it, relationslook like predicates, predicates like names,etc.

Facts cannot be named.

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It is easy to suppose that ”individual”,”particular”, ”complex” etc. are primitive ideasof logic. Russell e.g. says ”individual” &”matrixxx” are ”primitive ideas”. This errorpresumably is to be explained by the fact that,by employment of variables instead of the

generality-signs, it comes to seem as if logicdealt with things which have been deprived ofall properties except thing-hood, & withpropositions deprived of all properties exceptcomplexity. We forget that the indefinables ofsymbols [Urbilder von Zeichen] only occur underthe generality-sign, never outside it.

Just as people used to struggle to bringall propositions into the subject-predicateform, so now it is natural to conceive everyproposition as expressing a relation, which isjust as incorrect. What is justified in thisdesire is fully satisfied by Russell’s theory ofmanufactured relations.

One of the most natural attempts atsolution consists in regarding ”not-p” as ”theopposite of p”, where then ”opposite” would bethe indefinable relation. But it is easy to seethat every such attempt to replace theab-functions by descriptions must fail.

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The false assumption that propositions arenames leads us to believe that there must belogical objects: for the meanings of logicalpropositions will have to be such things.

A correct explanation of logicalpropositions must give them a unique position asagainst all other propositions.

No proposition can say anything aboutitself, because the symbol of the propositioncannot be contained in itself; this must be thebasis of the theory of logical types.

Every proposition which says somethingindefinable about a thing is a subject-predicateproposition; every proposition which sayssomething indefinable about two things expressesa dual relation between these things, & so on.Thus every proposition which contains only onename & one indefinable form is asubject-predicate proposition, & so on. Anindefinable simple sign symbol can only be aname, & therefore we can know, by the symbol ofan atomic proposition, whether it is asubject-predicate proposition.

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Item 201a-1 Recto Page B24Wittg.-

I. Bi-polarity of propositions: sense & meaning,truth & falsehood.II. Analysis of atomic propositions: generalindefinables, predicates, etc.III. Analysis of molecular fu propositions:ab-functions.IV. Analysis of general propositions22

[IV|VVV]. Principles of symbolism: what symbolizesin a symbol. Facts for facts.V‹I›. Types

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This is the symbol for~p ∨ ~q

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Item 201a-2 (RA1.710.057824)

Transcriber(s): Michael Biggs, Peter CrippsProofreader(s): Peter CrippsComments:Transcriptions of material from The BertrandRussell Archives, McMaster University, Canada;their catalogue no.: RA1.710.057824.This is a typescript of two parts (8ff.+25ff.).The first part is a copy of The Bertrand RussellArchives item no. RA1.710.057822 incorporatingthe handwritten corrections. The second part isa copy of The Bertrand Russell Archives item no.RA1.710.057823. The whole has handwritten Romannumerals in the left margin against eachparagraph indicating the sections of Russell’ssubsequent rearrangement of Wittgenstein’s text(the so-called Costello Version, von Wright’scatalogue no. 201b). Non-typewriter charactersare inserted by hand; fully typed pages have 24or 25 lines of type; each new sentence isseparated from the last by three spaces. Truedouble spaced blank lines used occasionally butnot consistently.Hands:s = handwritten insertions that belong to firstpass, ie generally the insertion of symbolswhich are not found on the typewriter such as ϕand ∃S1 = Bertrand Russell

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Item 201a-2 Recto Page TP

SUMMARY2

‹Notes on Logicby

Ludwig WittgensteinSeptember 1913.›1

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SUMMARY.

‹III› One reason for thinking the old notationwrong is that it is very unlikely that from everyproposition p an infinite number of otherpropositions not-not-p, not-not-not-not-p, etc.,should follow.

‹IV› If only those signs which contain propernames were complex then propositions containingnothing but apparent variables would be simple.Then what about their denials?

‹I› The verb of a proposition cannot be ”is true”or ”is false”, but whatever is true or false mustalready contain the verb.

Deductions only proceed according to the lawsof deduction, but these laws cannot justify thededuction.

‹I› One reason for supposing that not allpropositions which have more than one argument arerelational propositions is that if they were, therelations of judgment and inference would have tohold between an arbitrary number of things.

‹II› Every proposition which seems to be about acomplex can be analysed into a proposition aboutits constituents and about the proposition whichdescribes the complex perfectly; i.e., thatproposition which is equivalent to saying thecomplex exists.

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‹I› The idea that propositions are names ofcomplexes suggests that whatever is not a propername is a sign for a relation. Because spatialcomplexes* * consists of Things and Relations onlyand the idea of a complex is taken from space.

‹VI› In a proposition convert all its indefinablesinto variables; there then remains a class ofpropositions which is not all propositions but atype.

‹VI› There are thus two ways in which signs aresimilar. The names Socrates and Plato are similar:they are both names. But whatever they have incommon must not be introduced before Socrates andPlato are introduced. The same applies to asubject-predicate form etc. Therefore, thing,proposition, subject-predicate form, etc., are notindefinables, i.e., types are not indefinables.

‹I› When we say A judges that etc., then we haveto mention a whole proposition which A judges. Itwill not do either to mention only itsconstituents, or its constituents and form, butnot in the proper order. This shows that aproposition itself must occur in the statementthat it is judged; however, for instance, ”not-p”may be explained, the question, ”What is negated”must have a meaning.

⟨* Russell - for instance imagines every fact as aspatial complex. ⟩ 23

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‹I› To understand a proposition p it is notenough to know that p implies ’”p” is true’, butwe must also know that ~p implies ”p is false”.This shows the bi-polarity of the proposition.

‹III› To every molecular function a WF* schemecorresponds. Therefore we may use the WF schemeitself instead of the function. Now what the WFscheme does is, it correlates the letters W and Fwith each proposition. These two letters are thepoles of atomic propositions. Then the schemecorrelates another W and F to these poles. In thisnotation all that matters is the correlation ofthe outside poles to the poles of the atomicpropositions. Therefore not-not-p is the samesymbol as p. And therefore we shall never get twosymbols for the same molecular function.

‹I› The meaning of a proposition is the factwhich actually corresponds to it.

‹III› As the ab functions of atomic propositionsare bi-polar propositions again we can perform aboperations on them. We shall, by doing so,correlate two new outside poles via the oldoutside poles to the poles of the atomicpropositions.

⟨* W-F = Wahr-Falsch. ⟩ 23

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‹III› The symbolising fact in a-p-b is that, say* ais on the left of p and b on the right of p; thenthe correlation of new poles is to be transitive,so that for instance if a new pole a in whateverway i.e. via whatever poles is correlated to theinside a, the symbol is not changed thereby. It istherefore possible to construct all possible abfunctions by performing one ab operationrepeatedly, and we can therefore talk of all abfunctions as of all those functions which can beobtained by performing this ab operationrepeatedly.

[Note by B.R. ab means the same as WF, whichmeans true-false.]

‹III› Naming is like pointing. A function is like aline dividing points of a plane into right andleft ones; then ”p or not-p” has no meaningbecause it does not divide the plane.

‹III› But though a particular proposition ”p ornot-p” has no meaning, a general proposition ”forall p’s, p or not-p” has a meaning because thisdoes not contain the nonsensical function ”p ornot-p” but the function ”p or not-q” just as ”forall x’s xRx” contains the function ”xRy”.

⟨* This is quite arbitrary but, if we once havefixed on which order the poles have to stand wemust of course stick to our convention. If forinstance ”a p b” says p then b p a says nothing.(It does not say ~p) But a - a p b - b is the samesymbol as apb (here the ab function vanishesautomatically) for here the new poles are relatedto the same side of p as the old ones. Thequestion is always: how are the new polescorrelated to p compared with the way the oldpoles are correlated to ~p. ⟩ 23

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‹I› A proposition is a standard to which allfacts behave, with names it is otherwise; it isthus bi-polarity and sense comes in; just as onearrow behaves to another arrow by being in thesame sense or the opposite, so a fact behaves to aproposition.

‹II› The form of a proposition has meaning in thefollowing way. Consider a symbol ”xRy”. To symbolsof this form correspond couples of things whosenames are respectively ”x” and ”y”. The things x ystand to one another in all sorts of relations,amongst others some stand in the relation R, andsome not; just as I single out a particular thingby a particular name I single out all behavioursof the points x and y with respect to the relationR. I say that if an x stands in the relation R toa y the sign ”xRy” is to be called true to thefact and otherwise false. This is a definition ofsense.

In my theory p has the same meaning as not-pbut opposite sense. The meaning is the fact. Theproper theory of judgment must make it impossibleto judge nonsense.

‹I› It is not strictly true to say that weunderstand a proposition p if we know that p isequivalent to ”p is true” for this would be thecase if accidentally both were true or false. Whatis wanted is the formal equivalence with respectto the forms of the proposition, i.e., all thegeneral indefinables involved. The sense of an abfunction of a proposition is a function of itssense. There are only unasserted propositions.

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‹I› Assertion is merely psychological. In not-p, p isexactly the same as if it stands alone; this pointis absolutely fundamental. Among the facts thatmake ”p or q” true there are also facts which make”p and q” true; if propositions have only meaning,we ought, in such a case, to say that these twopropositions are identical, but in fact, theirsense is different for we have introduced sense bytalking of all p’s and all q’s. Consequently themolecular propositions will only be used in caseswhere their ab function stands under a generalitysign or enters into another function such as ”Ibelieve that, etc”., because then the senseenters.

‹I› In ”a judges p” p cannot be replaced by aproper name. This appears if we substitute ”ajudges that p is true and not p is false”. Theproposition ”a judges p” consists of the propername a, the proposition p with its 2 poles, and abeing related to both of these poles in a certainway. This is obviously not a relation in theordinary sense.

‹III›‹V› The ab notation makes it clear that not andor are dependent on one another and we cantherefore not use them as simultaneousindefinables. Same objections in the case ofapparent variables to the usual1 old indefinables,as in the case of molecular functions. Theapplication of the ab notation to apparentvariable propositions becomes clear if we considerthat, for instance, the proposition ”for all x,ϕx” is to be true when ϕx is true for all x’s andfalse when ϕx is false for some x’s. We see thatsome and all occur simultaneously in the properapparent variable notation.

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‹IV› The notation is:

for (x) ϕx : a - (x) - a ϕx b - (∃ x) - band

for (∃x) ϕx : a - (∃x) - a ϕx b - (x) - b

Old definitions now become tautologous.‹V› In aRb it is not the complex that symbolises

but the fact that the symbol a stands in a certainrelation to the symbol b. Thus facts aresymbolised by facts, or more correctly: that acertain thing is the case in the symbol says thata certain thing is the case in the world.

‹I› Judgment, question and command are all on thesame level. What interests logic in them is onlythe unasserted proposition. Facts cannot be named.

‹VI› A proposition cannot occur in itself. This is thefundamental truth of the theory of types.

Every proposition that says somethingindefinable about one thing is a subject-predicateproposition, and so on.

‹VI› Therefore we can recognise asubject-predicate proposition if we know itcontains only one name and one form, etc. Thisgives the construction of types. Hence the type ofa proposition can be recognized by its symbolalone.

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What is essential in a correctapparent-variable notation is this:- (1) it mustmention a type of propositions; (2) it must showwhich components of a proposition of this type areconstants.

‹VI› [Components are forms and constituents.]

Take (ϕ). ϕ!x. Then if we describe the kindof symbols, for which ϕ! stands and which, by theabove, is enough to determine the type, thenautomatically ”(ϕ). ϕ! x” cannot be fitted by thisdescription, because it CONTAINS ”ϕ!x” and thedescription is to describe ALL that symbolises insymbols of the ϕ! kind. If the description is thuscomplete vicious circles can just as little occuras for instance (ϕ). (X)ϕ (where (X)ϕ is asubject-predicate proposition).

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First MS.

‹II› Indefinables are of two sorts: names, andforms. Propositions cannot consist of names alone;they cannot be classes of names. A name can notonly occur in two different propositions, but canoccur in the same way in both.

Propositions [which are symbols havingreference to facts] are themselves facts: thatthis inkpot is on this table may express that Isit in this chair.

‹V› It can never express the commoncharacteristic of two objects that we designatethem by the same name but by two different ways ofdesignation, for, since names are arbitrary, wemight also choose different names, and where thenwould be the common element in the designations?Nevertheless one is always tempted, in adifficulty, to take refuge in different ways ofdesignation.

‹I› Frege said ”propositions are names”; Russellsaid ”propositions correspond to complexes”. Bothare false; and especially false is the statement”propositions are names of complexes.”

‹IV› It is easy to suppose that only such symbolsare complex as contain names of objects, and thataccordingly ” (∃x,ϕ). ϕx” or ”(∃x,y). x R y” mustbe simple. It is then natural to call the first ofthese the name of a form, the second the name of arelation. But in that case what is the meaning of(e.g.) ”~(∃x,y). x R y”? Can we put ”not” before aname?

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‹III› The reason why ”~Socrates” means nothing isthat ”~x” does not express a property of x.

‹I› There are positive and negative facts: if theproposition ”this rose is not red” is true, thenwhat it signifies is negative. But the occurrenceof the word ”not” does not indicate this unless weknow that the signification of the proposition”this rose is red” (when it is true) is positive.It is only from both, the negation and the negatedproposition, that we can conclude to acharacteristic of the significance of the wholeproposition. (We are not here speaking ofnegations of general propositions, i.e. of such ascontain apparent variables. Negative facts onlyjustify the negations of atomic propositions.)

Positive and negative facts there are, butnot true and false facts.

‹I› If we overlook the fact that propositionshave a sense which is independent of their truthor falsehood, it easily seems as if true and falsewere two equally justified relations between thesign and what is signified. (We might then saye.g. that ”q”24 signifies in the true way what”not-q” signifies in the false way). But are nottrue and false in fact equally justified? Could wenot express ourselves by means of falsepropositions just as well as hitherto with trueones, so long as we know that they are meantfalsely? No! For a proposition is then25 true when

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it is as we assert in this proposition; andaccordingly if by ”q” we mean ”not-q”, and it isas we mean to assert, then in the newinterpretation ”q” is actually true and not false.But it is important that we can mean the same by”q” as by ”not-q”, for it shows that neither tothe symbol ”not” nor to the manner of itscombination with ”q” does a characteristic of thedenotation of ”q” correspond.

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Second MS.

‹II› We must be able to understand propositionswhich we have never heard before. But everyproposition is a new symbol. Hence we must havegeneral indefinable symbols; these are unavoidableif propositions are not all indefinable.

‹III› Whatever corresponds in reality to compoundpropositions must not be more than whatcorresponds to their several atomic propositions.

Not only must logic not deal with[particular] things, but just as little withrelations and predicates.

‹IV› There are no propositions containing realvariables.

‹I› What corresponds in reality to a propositiondepends upon whether it is true or false. But wemust be able to understand a proposition withoutknowing if it is true or false.

What we know when we understand a propositionis this: We know what is the case if theproposition is true, and what is the case if it isfalse. But we do not know [necessarily] whether itis true or false.

Propositions are not names.‹VI› We can never distinguish one logical type

from another by attributing a property to membersof the one which we deny to members of the other.

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‹II› Symbols are not what they seem to be. In ”a Rb”, ”R” looks like a substantive, but is not one.What symbolizes in ”a R b” is that R occursbetween a and b. Hence ”R” is not the indefinablein ”a R b”. Similarly in ”ϕx”, ”ϕ” looks like asubstantive but is not one; in ”~p”, ”~” lookslike ”ϕ” but is not like it. This is the firstthing that indicates that there may not be logicalconstants. A reason against them is the generalityof logic: logic cannot treat a special set ofthings.

‹III› Molecular propositions contain nothing beyondwhat is contained in their atoms; they add nomaterial information above that contained in theiratoms.

All that is essential about molecularfunctions is their T-F schema [i.e. the statementof the cases when they are true and the cases whenthey are false].

‹V› Alternative indefinability shows that theindefinables have not been reached.

‹I› Every proposition is essentially true-false:to understand it, we must know both what must bethe case if it is true, and what must be the caseif it is false. Thus a proposition has two poles,corresponding to the case of its truth and thecase of its falsehood. We call this the sense of aproposition.

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‹V› In regard to notation, it is important tonote that not every feature of a symbolsymbolizes. In two molecular functions which havethe same T-F schema, what symbolizes must be thesame. In ”not-not-p”, ”not-p” does not occur; for”not-not-p” is the same as ”p”, and therefore, if”not-p” occurred in ”not-not-p”, it would occur in”p”.

‹III› Logical indefinables cannot be predicates orrelations, because propositions, owing to sense,cannot have predicates or relations. Nor are ”not”and ”or”, like judgment, analogous to predicatesor relations, because they do not introduceanything new.

‹IV› Propositions are always complex even if theycontain no names.

‹II› A proposition must be understood when all itsindefinables are understood. The indefinables in”a R b” are introduced as follows:

”a” is indefinable;”b” is indefinable;Whatever ”x” and ”y” may mean, ”x R y” says

something indefinable about their meaning.

‹V› A complex symbol must never be introduced asa single indefinable. [Thus e.g. no proposition isindefinable). For if one of its parts occurs alsoin another connection, it must there bere-introduced. And would it then mean the same?

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The ways by which we introduce ourindefinables must permit us to construct allpropositions that have sense [? meaning] fromthese indefinables alone. It is easy to introduce”all” and ”some” in a way that will make theconstruction of (say) ”(x, y). x R y” possiblefrom ”all” and ”x R y” as introduced before.

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3rd. MS.

‹I› An analogy for the theory of truth: Considera black patch on white paper; then we can describethe form of the patch by mentioning, for eachpoint of the surface, whether it is white orblack. To the fact that a point is blackcorresponds a positive fact, to the fact that apoint is white (not black) corresponds a negativefact. If I designate a point of the surface (oneof Frege’s ”truth-values”), this is as if I set upan assumption to be decided upon. But in order tobe able to say of a point that it is black or thatit is white, I must first know when a point is tobe called black and when it is to be called white.In order to be able to say that ”p” is true (orfalse), I must first have determined under whatcircumstances I call a proposition true, andthereby I determine the sense of a proposition.The point in which the analogy fails is this: Ican indicate a point of the paper what is white26

and black, but to a proposition without sensenothing corresponds, for it does not designate athing (truth-value), whose properties might becalled ”false” or ”true”; the verb of aproposition is not ”is true” or ”is false”, asFrege believes, but what is true must alreadycontain the verb.

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‹I› The comparison of language and reality islike that of retinal image and visual image: tothe blind spot nothing in the visual image seemsto correspond, and thereby the boundaries of theblind spot determine the visual image - as truenegations of atomic propositions determinereality.

‹III› Logical inferences can, it is true, be madein accordance with Frege’s or Russell’s laws ofdeduction, but this cannot justify the inference;and therefore they are not primitive propositionsof logic. If p follows from q, it can also beinferred from q, and the ”manner of deduction” isindifferent.

‹IV› Those symbols which are called propositionsin which ”variables occur” are in reality notpropositions at all, but only schemes ofpropositions, which only become propositions whenwe replace the variables by constants. There is noproposition which is expressed by ”x = x”, for ”x”has no signification; but there is a proposition”(x). x = x” and propositions such as ”Socrates =Socrates” etc.

‹IV› In books on logic, no variables ought tooccur, but only the general propositions whichjustify the use of variables. It follows that theso-called definitions of logic are notdefinitions, but only schemes of definitions, andinstead of these we ought to put generalpropositions; and similarly the so-calledprimitive ideas (Urzeichen) of logic are notprimitive ideas, but the schemes of them. Themistaken idea that there

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are things called facts or complexes and relationseasily leads to the opinion that there must be arelation of questioning10 ⟨(?) ⟩ 27 to the facts, andthen the question arises whether a relation canhold between an arbitrary number of things, sincea fact can follow from arbitrary cases. It is afact that the proposition which e.g. expressesthat q follows from p and p ⊃ q is this: p. p ⊃ q.⊃ p.q.q.

‹I› At a pinch, one is tempted to interpret”not-p” as ”everything else, only not p”. Thatfrom a single fact p an infinity of others,not-not-p etc., follow, is hardly credible. Manpossesses an innate capacity for constructingsymbols with which some sense can be expressed,without having the slightest idea what each wordsignifies. The best example of this ismathematics, for man has until lately used thesymbols for numbers without knowing what theysignify or that they signify nothing.

‹II› Russell’s ”complexes” were to have the usefulproperty of being compounded, and were to combinewith this the agreeable property that they couldbe treated like ”simples”. But this alone madethem unserviceable as logical types, since therewould have been significance in asserting, of asimple, that it was complex. But a property cannotbe a logical type.

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‹II› Every statement about apparent complexes canbe resolved into the logical sum of a statementabout the constituents and a statement about theproposition which describes the complexcompletely. How, in each case, the resolution isto be made, is an important question, but itsanswer is not unconditionally necessary for theconstruction of logic.

‹III› That ”or” and ”not” etc. are not relations inthe same sense as ”right” and ”left” etc., isobvious to the plain man. The possibility ofcross-definitions in the old logical indefinablesshows, of itself, that these are not the rightindefinables, and, even more conclusively, thatthey do not denote relations.

‹VI› If we change a constituent a of a propositionϕ(a) into a variable, then there is a class

~p {(∃x). ϕ(x) = p }.

This class in general still depends upon what, byan arbitrary convention, we mean by ”ϕ(x)”. But ifwe change into variables all those symbols whosesignificance was arbitrarily determined, there isstill such a class. But this is now not dependentupon any convention, but only upon the nature ofthe symbol ”ϕ(x)”. It corresponds to a logicaltype.

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‹VI› Types can never be distinguished from eachother by saying (as is often done) that one hasthese but the other has those properties, for thispresupposes that there is a meaning in assertingall these properties of both types. But from thisit follows that, at best, these properties may betypes, but certainly not the objects of which theyare asserted.

‹I› At a pinch we are always inclined toexplanations of logical functions of propositionswhich aim at introducing into the function eitheronly the constituents of these propositions, oronly their form, etc. etc.; and we overlook thatordinary language would not contain the wholepropositions if it did not need them: However,e.g., ”not-p” may be explained, there must alwaysbe a meaning given to the question ”what isdenied?”

‹III› The very possibility of Frege’s explanationsof ”not-p” and ”if p then q”, from which itfollows that ”not-not-p” denotes the same as p,makes it probable that there is some method ofdesignation in which ”not-not-p” corresponds tothe same symbol as ”p”. But if this method ofdesignation suffices for logic, it must be theright one.

‹I› Names are points, propositions arrows - theyhave sense. The sense of a proposition isdetermined by the two poles true and false. Theform of a proposition is like a straight line,which divides all points of a plane into right andleft. The line does this autom at ically, the formof proposition only by convention.

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‹II› Just as little as we are concerned, in logic,with the relation of a name to its meaning, justso little are we concerned with the relation of aproposition to reality, but we want to know themeaning of names and the sense of propositions aswe introduce an indefinable concept ”A” by saying:”’A’ denotes something indefinable”, so weintroduce e.g. the form of propositions a R b bysaying: ”For all meanings of ”x” and ”y”, ”x R y”expresses something indefinable about x and y”.

‹III› In place of every proposition ”p”, let uswrite ”abp”. Let every correlation of propositionsto each other or of names to propositions beeffected by a correlation of their poles ”a” and”b”. Let this correlation be transitive. Thenaccordingly ”a-ab-bp” is the same symbol as ”abp”.Let n propositions be given. I then call a ”classof poles” of these propositions every class of nmembers, of which each is a pole of one of the npropositions, so that one member corresponds toeach proposition. I then correlate with each classof poles one of two poles (a and b). The sense ofthe symbolizing fact thus constructed I cannotdefine, but I know it.

‹III› If p = not-not-p etc., this shows that thetraditional method of symbolism is wrong, since itallows a plurality of symbols with the same sense;and thence it follows that, in analyzing suchpropositions, we must not be guided by Russell’smethod of symbolizing.

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‹V› It is to be remembered that names are notthings, but classes: ”A” is the same letter as”A”. This has the most important consequences forevery symbolic language.

‹I› Neither the sense nor the meaning of aproposition is a thing. These words are incompletesymbols.

‹V› It is impossible to dispense withpropositions in which the same argument occurs indifferent positions. It is obviously useless toreplace ϕ(a, a) by ϕ(a, b). a = b.

‹III› Since the ab-functions of p are againbi-polar propositions, we can form ab-functions ofthem, and so on. In this way a series ofpropositions will arise, in which in general thesymbolizing facts will be the same in severalmembers. If now we find an ab-function of such akind that by repeated application of it everyab-function can be generated, then we canintroduce the totality of ab-functions as thetotality of those that are generated byapplication of this function. Such a function is~p ∨ ~q.

‹III› It is easy to suppose a contradiction in thefact that on the one hand every possible complexproposition is a simple ab-function of simplepropositions, and that on the other hand therepeated application of one ab-function sufficesto generate all these propositions. If e.g. anaffirmation can be generated by double negation,is negation in any sense contained in affirmation?Does ”p” deny ”not-p” or assert ”p”, or both? Andhow do

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matters stand with the definition of ”⊃” by ”∨”and ”.”, or of ”∨” by ”.” and ”⊃”? And how e.g.shall we introduce p[”||||]q (i.e. ~p ∨ ~q), if notby saying that this expression says somethingindefinable about all arguments p and q? But theab-functions must be introduced as follows: Thefunction p[”||||]q is merely a mechanical instrumentfor constructing all possible symbols ofab-functions. The symbols arising by repeatedapplication of the symbol ”|” do not contain thesymbol ”p|q”. We need a rule according to which wecan form all symbols of ab-functions, in order tobe able to speak of the class of them; and we nowspeak of them e.g. as those symbols of functionswhich can be generated by repeated application ofthe operation ”|”. And we say now: For all p’s andq’s, ”p|q” says something indefinable about thesense of those simple propositions which arecontained in p and q.

‹I› The assertion-sign is logically quite withoutsignificance. It only shows, in Frege andWhitehead and Russell, that these authors hold thepropositions so indicated to be true. ” ”therefore belongs as little to the proposition as(say) the number of the proposition. A propositioncannot possibly assert of itself that it is true.

Every right theory of judgment must make itimpossible for me to judge that this tablepenholders the book. Russell’s theory does notsatisfy this requirement.

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‹I› It is clear that we understand propositionswithout knowing whether they are true or false.But we can only know the meaning of a propositionwhen we know if it is true or false. What weunderstand is the sense of the proposition.

‹III› The assumption of the existence of logicalobjects makes it appear remarkable that in thesciences propositions of the form ”p ∨ q”, ”p ⊃q”, etc. are only then not provisional when ”∨”and ”⊃” stand within the scope of agenerality-sign [apparent variable].

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4th. MS.

‹II› If we formed all possible atomicpropositions, the world would be completelydescribed if we declared the truth or falsehood ofeach. [I doubt this.]

‹I› The chief characteristic of my theory isthat, in it, p has the same meaning as not-p.

‹II› A false theory of relations makes it easilyseem as if the relation of fact and constituentwere the same as that of fact and fact whichfollows from it. But the similarity of the two maybe expressed thus: ϕa. ⊃.ϕ,a a = a.

‹II› If a word creates a world so that in it theprinciples of logic are true, it thereby creates aworld in which the whole of mathematics holds; andsimilarly it could not create a world in which aproposition was true, without creating itsconstituents.

‹III› Signs of the form ”p ∨ ~p” are senseless, butnot the proposition ”(p). p ∨ ~p”. If I know thatthis rose is either red or not red, I knownothing. The same holds of all ab-functions.

‹I› To understand a proposition means to knowwhat is the case if it is true. Hence we canunderstand it without knowing if it is true. Weunderstand it when we understand its constituentsand forms. If we know the meaning of ”a” and ”b”,and if we know what ”x R y” means for all x’s andy’s, then we also understand ”a R b”.

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‹I› I understand the proposition ”a R b” when Iknow that either the fact that a R b or the factthat not a R b corresponds to it; but this is notto be confused with the false opinion that Iunderstood ”a R b” when I know that ”a R b or nota R b” is the case.

‹II› But the form of a proposition symbolizes inthe following way: Let us consider symbols of theform ”x R y”; to these correspond primarily pairsof objects, of which one has the name ”x”, theother the name ”y”. The x’s and y’s stand invarious relations to each other, among others therelation R holds between some, but not betweenothers. I nowowow determine the sense of ”x R y” bylaying down: when the facts behave in regard to ”xR y” so that the meaning of ”x” stands in therelation R to the meaning of ”y”, then I say thatthey [the facts] are ”of like sense”[”gleichsinnig”] with the proposition ”x R y”;otherwise, ”of opposite sense” [entgegengesetzttt”];I correlate the facts to the symbol ”x R y” bythus dividing them into those of like sense andthose of opposite sense. To this correlationcorresponds the correlation of name and meaning.Both are psychological. Thus I understand the form”x R y” when I know that it discriminates thebehaviour of x and y according as these stand inthe relation R or not. In this way I extract fromall possible relations the relation R, as, by aname, I extract its meaning from among allpossible things.

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‹I› Strictly speaking, it is incorrect to say: weunderstand the proposition p when we know that’”p” is true’ ≡ p; for this would naturally alwaysbe the case if accidentally the propositions toright and left of the symbol ”≡” were both true orboth false. We require not only an equivalence,but a formal equivalence, which is bound up withthe introduction of the form of p.

‹III› The sense of an ab-function of p is afunction of the sense of p.

‹III› The ab-functions use the discrimination offacts, which their arguments bring forth, in orderto generate new discriminations.

‹V› Only facts can express sense, a class ofnames cannot. This is easily shown.

‹II› There is no thing which is the form of aproposition, and no name which is the name of aform. Accordingly we can also not say that arelation which in certain cases holds betweenthings holds sometimes between forms and things.This goes against Russell’s theory of judgment.

28[It is very easy to forget that, though thepropositions of a form can be either true orfalse, each one of these propositions can only beeither true or false, not both.]28

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‹III› Among the facts which make ”p or q” true,there are some which make ”p and q” true; but theclass which makes ”p or q” true is different fromthe class which makes ”p and q” true; and onlythis is what matters. For we introduce this class,as it were, when we introduce ab-functions.

‹IV› A very natural objection to the way in whichI have introduced e.g. propositions of the form xR y is that by it propositions such as (∃. x. y).x R y and similar ones are not explained, whichyet obviously have in common with a R b what c R dhas in common with a R b. But when we introducepropositions of the form x R y we mentioned no oneparticular proposition of this form; and we onlyneed to introduce (∃ x, y). ϕ(x, y) for all ϕ’s inany way which makes the sense of thesepropositions dependent on the sense of allpropositions of the form ϕ(a, b), and thereby thejustification2ness1 of our procedure is proved.

‹V› The indefinables of logic must be independentof each other. If an indefinable is introduced, itmust be introduced in all combinations in which itcan occur. We cannot therefore introduce it firstfor one combination, then for another; e.g., ifthe form x R y has been introduced, it musthenceforth be understood in propositions of theform a R b just in the same way as in propositionssuch as (∃x, y). x R y and others. We must notintroduce it first for one class of cases, thenfor the other; for it would remain doubtful if itsmeaning was the same in

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both cases, and there would be no ground for usingthe same manner of combining symbols in bothcases. In short, for the introduction ofindefinable symbols and combinations of symbolsthe same holds, mutatis mutandis, that Frege hassaid for the introduction of symbols bydefinitions.

‹III› It is a priori likely that the introductionof atomic propositions is fundamental for theunderstanding of all other kinds of propositions.In fact the understanding of general propositionsobviously depends on that of atomic propositions.

‹IV› Cross-definability in the realm of generalpropositions leads to the2 quite similar questionsto those in the realm of ab-functions.

‹I› When we say ”A believes p”, this sounds, itis true, as if here we could substitute a propername for ”p”; but we can see that here a sense,not a meaning, is concerned, if we say ”A believesthat ’p’ is true”; and in order to make thedirection of p even more explicit, we might say ”Abelieves that ’p’ is true and ’not-p’ is false”.Here the bi-polarity of p is expressed and itseems that we shall only be able to express theproposition ”A believes p” correctly by theab-notation; say by making ”A” have a relation tothe poles ”a” and ”b” of a-p-b.

The epistemological questions concerning thenature of judgment and belief cannot be solvedwithout a correct apprehension of the four10rm1 ofthe proposition.

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‹III› The ab-notation shows the dependence of orand not, and thereby that they are not to beemployed as simultaneous indefinables.

‹V› Not: ”The complex sign ’a R b’” says that astands in the relation R to b; but that ’a’ standsin a certain relation to ’b’ says that a R b.

( 29[ ⟨Preliminary ⟩ 6 In philosophy there are no( deductions: it is purely descriptive.( Philosophy gives no pictures of reality.( Philosophy can neither confirm nor confute( scientific investigation.( Philosophy consists of logic and metaphysics:( logic is its basis.( Epistemology is the philosophy of psychology.( Distrust of grammar is the first requisite( for philosophizing.]29

28[Propositions can never be indefinables,for they are always complex. That also words like”ambulo” are complex appears in the fact thattheir root with a different termination gives adifferent sense.]28

‹II› Only the doctrine of general indefinablespermits us to understand the nature of functions.Neglect of this doctrine leads to an impenetrablethicket.

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( 29[ ⟨Preliminary ⟩ 6( Philosophy is the doctrine of the logical( form of scientific propositions (not only of( primitive propositions).( The word ”philosophy” ought always to( designate something over or under but not beside,( the natural sciences.]29

‹I› Judgment, command and question all stand onthe same level; but all have in common thepropositional form, which does interest us.

‹I› The structure of the proposition must berecognized, the rest comes of itself. But ordinarylanguage conceals the structure of theproposition: in it, relations look likepredicates, predicates like names, etc.

Facts cannot be named.‹VI› It is easy to suppose that ”individual”,

”particular”, ”complex” etc. are primitive ideasof logic. Russell e.g. says ”individual” and”matrix” are ”primitive ideas”. This errorpresumably is to be explained by the fact that, byemployment of variables instead of thegenerality-sign it comes to seem as if logic dealtwith things which have been deprived of allproperties except thing-hood, and withpropositions deprived of all properties exceptcomplexity. We forget that the indefinables ofsymbols [Urbilder von Zeichen] only occur underthe generality-sign, never outside it.

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‹IV› Just as people used to struggle to bring allpropositions into the subject-predicate form, sonow it is natural to conceive every proposition asexpressing a relation, which is just as incorrect.What is justified in this desire is fullysatisfied by Russell’s theory of manufacturedrelations.

‹I› One of the most natural attempts at solutionconsists in regarding ”not-p” as ”the opposite ofp”, where then ”opposite” would be the indefinablerelation. But it is easy to see that every suchattempt to replace the ab-functions bydescriptions must fail.

‹I› The false assumption that propositions arenames leads us to believe that there must belogical objects: for the meanings of logicalpropositions will have to be such things.⟨Preliminary ⟩ 6

A correct explanation of logical propositionsmust give them a unique position as against allother propositions.

No proposition can say anything about itself,because the symbol of the proposition cannot becontained in itself; this must be the basis of thetheory of logical types.

‹VI› Every proposition which says somethingindefinable about a thing is a subject-predicateproposition; every proposition which sayssomething indefinable about two things expresses adual relation between these things, and so on.Thus every proposition which contains only onename and one indefinable form is asubject-predicate proposition, and so on. An

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indefinable simple symbol can only be a name, andtherefore we can know, by the symbol of an atomicproposition, whether it is a subject-predicateproposition.

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Item 201a-3 (also referred to as TSx)

Transcriber(s): Michael Biggs, Peter CrippsProofreader(s): Peter CrippsComments:Transcriptions of material from The WittgensteinArchives at the University of Bergen. Itemreferred to by Michael Biggs as TSx (related to201a). This is a typescript of 15ff. The first 2folios are unnumbered, the foliation then runsfrom 2 to 14. Some logical notation symbols addedby hand, others omitted. References to thepublished text of Tractatus Logico-Philosophicusare added by hand. Fully typed pages have between61 and 65 lines of type; each new sentence isseparated from the last by three spaces. Blanklines between paragraphs.Hands:s = handwritten insertions that belong to firstpass, ie generally the insertion of symbols whichare not found on the typewriter such as ϕ and ∃S2 = handwritten insertions presumably made D.Schwayder.

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Item 201a-3 Recto Page TP

Notes on Logicby

Ludwig WittgensteinSeptember 1913.

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Item 201a-3 Recto Page 1

SUMMARY.

One reason for thinking the old notationwrong is that it is very unlikely that from everyproposition p an infinite number of otherpropositions not-not-p, not-not-not-not-p, etc.,should follow.

If only those signs which contain propernames were complex then propositions containingnothing but apparent variables would be simple.Then what about their denials?

The verb of a proposition cannot be ”istrue” or ”is false”, but whatever is true or falsemust already contain the verb.

Deductions only proceed according to thelaws of deduction, but these laws cannot justifythe deduction.

One reason for supposing that not allpropositions which have more than one argument arerelational propositions is that if they were, therelations of judgment and inference would have tohold between an arbitrary number of things.

Every proposition which seems to be about acomplex can be analysed into a proposition aboutits constituents and about the proposition whichdescribes the complex perfectly; i.e., thatproposition which is equivalent to saying thecomplex exists.

The idea that propositions are names ofcomplexes suggests that whatever is not a proper

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name is a sigggn for a relation. Because spatialcomplexes* consist of Things and Relations onlyand the idea of a complex is taken from space.

In a proposition convert all itsindefinables into variables; there then remains aclass of propositions which is not allpropositions but a type.

There are thus two ways in which signs aresimilar. The names Socrates and Plato are similar:they are both names. But whatever they have incommon must not be introduced before Socrates andPlato are introduced. The same applies to asubject-predicate form etc. Therefore, thing,proposition, subject-predicate form, etc., are notindefinables, i.e., types are not indefinables.

When we say A judges that etc., then we haveto mention a whole proposition which A judges. Itwill not do either to mention only itsconstituents, or its constituents and form, butnot in the proper order. This shows that aproposition itself must occur in the statementthat it is judged; however, for instance, ”not-p”may be explained, the question, ”What is negated”must have a meaning.

To understand a proposition p it is notenough to know that p implies ’”p” is true’, butwe must also know that ~p implies ”p is false”.This shows the bi-polarity of the proposition.

To every molecular function a WF* schemecorresponds. Therefore we may use the WF schemeitself instead of the function. Now what the Wfscheme does is, it correlates the letters W and Fwith each proposition. These two letters are thepoles of atomic propositions. Then the schemecorrelates another W and F to these poles. In this

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notation all that matters is the correlation ofthe outside poles to the poles of the atomicpropositions. Therefore not-not-p is the samesymbol as p. And therefore we shall never get twosymbols for the same molecular function.

⟨* Russell - for instance imagines every fact as aspatial complex.

* W-F = Wahr-Falsch. ⟩ 23

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Item 201a-3 Recto Page 2- 2 -

‹⊗› The meaning of a proposition is the factwhich actually corresponds to it.

As the ab functions of atomic propositionsare bi-polar propositions again we can perform aboperations on them. We shall, by doing so,correlate two new outside poles via the oldoutside poles to the poles of the atomicpropositions.

The symbolising fact in a-p-b is that, say*a is on the left of p and b on the right of p;then the correlation of new poles is to betransitive, so that for instance if a new pole ain whatever way i.e. via whatever poles iscorrelated to the inside a, the symbol is notchanged thereby. It is therefore possible toconstruct all possible ab functions by performingone ab operation repeatedly, and we can thereforetalk of all ab functions as of all those functionswhich can be obtained by performing this aboperation repeatedly.

(Note by B.R. ab means the same as WF, which meanstrue-false.?)

Naming is like pointing. A function is likea line dividing points of a plane into right andleft ones; then ”p or not-p” has no meaningbecause it does not divide the plane.

But though a particular proposition ”p ornot-p” has no meaning, a general proposition ”forall p’s, p or not-p” has a meaning because thisdoes not contain the nonsensical function ”p ornot-p” but the function ”p or not-q” just as ”for

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all x’s xRx” contains the function ”xRy”.

‹⊗› A proposition is a standard to which factsbehave, with names it is otherwise; it is thusbi-polarity and sense comes in; just as one arrowbehaves to another arrow by being in the samesense or the opposite, so a fact behaves to aproposition.

‹⊗› The form of a proposition has meaning in thefollowing way. Consider a symbol ”xRy”. To symbolsof this form correspond couples of things whosenames are respectively ”x” and ”y”. The things x ystand to one another in all sorts of relations,amongst others some stand in the relation R, andsome not; just as I single out a particular thingby a particular name I single out all behavioursof the points x and y with respect to the relationR. I say that if an x stands in the relation R toa y the sign ”xRy” is to be called true to thefact and otherwise false. This is a definition ofsense.

‹ ⊗› In my theory p has the same meaning as not-pbut opposite sense. The meaning is the fact. Theproper theory of judgment must make it impossibleto judge nonsense.

It is not strictly true to say that weunderstand a proposition p ifff we know that p isequivalent to ”p is true” for this would be thecase if accidentally both were true or false. Whatis wanted is the formal equivalence with respectto the forms of the proposition, i.e., all thegeneral indefinables involved. The sense of an abfunction of a proposition is a function of itssense. There are only unasserted propositions.Assertion is merely psychological. In not-p, p isexactly the same as if it stands alone; this pointis absolutely fundamental. Among the facts that

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make ”p or q” true there are also facts which make”p and q” true; if propositions have only meaning,we ought, in such a case, to say that these two

⟨* This is quite arbitrary but, if we once havefixed on which order the poles have to stand wemust of course stick to our convention. If forinstance ”a p b” says p then b p a says nothing.(It does not say p) But a - a p b - b is the samesymbol as apb (here the ab function vanishesautomatically) for here the new poles are relatedto the same side of p as the old ones. Thequestion is always: how are the new polescorrelated to p compared with the way the oldpoles are correlated to p. ⟩ 23

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propositions are identical, but in fact, theirsense is different for we have introduced sense bytalking of all p’s and all q’s. Consequently themolecular propositions will only be used in caseswhere their ab function stands under a generalitysign or enters into another function such as ”Ibelieve that, etc”., because then the senseenters.

In ”a judges p” p cannot be replaced by aproper name. This appears if we substitute ”ajudges that p is true and not p is false”. Theproposition ”a judges p” consists of the propername a, the proposition p with its 2 poles, and abeing related to both of these poles in a certainway. This is obviously not a relation in theordinary sense.

The ab notation makes it clear that not andor are dependent on one another and we cantherefore not use them as simultaneousindefinables. Same objections in the case ofapparent variables to the usual indefinables, asin the case of molecular functions. Theapplication of the ab notation to apparentvariable propositions becomes clear if we considerthat, for instance, the proposition ”for all x, φx” is to be true when φ x is true for all x’s andfalse when φ x is false for some x’s. We see thatsome and all occur simultaneously in the properapparent variable notation. The Notation is:

for (x) φ x : a - (x) - a φ x b - ( x) - b

and

for ( x) x : a - ( x) - a φ x b - (x) - b

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Old definitions now become tautologous.

30[In aRb it is not the complex thatsymbolises but the fact that the symbol a standsin a certain relation to the symbol b. Thus factsare symbolised by facts, or more correctly: that acertain thing is the case in the symbol says thata certain thing is the case in the world.]30

Judgment, question and command are all onthe same level. What interests logic in them isonly the unasserted proposition. Facts cannot benamed.

A proposition cannot occur in itself. Thisis the fundamental truth of the theory of types.

Every proposition that says somethingindefinable about one thing is a subject-predicateproposition, and so on.

Therefore we can recognise asubject-predicate proposition if we know itcontains only one name and one form, etc. Thisgives the construction of types. Hence the type ofa proposition can be recognised by its symbolalone.

What is essential in a correctapparent-variable notation is this:- (1) it mustmention a type of prororopositions; (2) it must showwhich components of a proposition of this type areconstants.

(Components are forms and constituents.)

Tale (φ). [x|φφφ]!x. Then if we describe thekind of symbols, for which φ! stands and which, bythe above, is enough to determine the type, thenautomatically ” (φ). φ! x” cannot be fitted by

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this description, because it CONTAINS ”φ! x” andthe description is to describe ALL that symbolisesin symbols of the φ! kind. If the description isthus complete vicious circles can just as littleoccur as for instance ( φ ). (X) φ (where (XXX) φ isa subject-predicate proposition).

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First MS

Indefinables are of two sorts: names, andforms. Propositions cannot consist of names alone;they cannot be classes of names. A name can notonly occur in two different propositions, but canoccur in the same way in both.

30[Propositions (which are symbols havingreference to facts) are themselves facts: thatthis inkpot is on this table may express that Isit in this chair.]30

It can never express the commoncharacteristic of two objects that we designatethem by the same name but by two different ways ofdesignation, for, since names are arbitrary, wemight also choose different names, and where thenwould be the common element in the designations?Nevertheless one is always tempted, in adifficulty, to take refuge in different ways ofdesignation.

Frege said ”propositions are names”; Russellsaid ”propositions correspond to complexes”. Bothare false; and especially false is the statement”propositions are names of complexes.”

It is easy to suppose that only such symbolsare complex as contain names of objects, and thataccordingly ” (∃x, φ). φx” or ” (∃x,y). x R y”must be simple. It is then natural to call thefirst of these the name of a form, the second thename of a relation. But in that case what is themeaning of (e.g.) ”~(∃x,y). x R y”? Can we put”not” before a name?

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The reason why ”~Socrates” means nothing isthat ”~x” does not express a property of x.

‹⊗› There are positive and negative facts: ifthe proposition ”this rose is not red” is true,then what it signifies is negative. But theoccurrence of the word ”not” does not indicatethis unless we know that the signification of theproposition ”this rose is red” (when it is true)is positive. It is only from both, the negationand the negated proposition, that we can concludeto a characteristic of the significance of thewhole proposition. (We are not here speaking ofnegations of general propositions, i.e. of such ascontain apparent variables. Negative facts onlyjustify the negations of atomic propositions.)

‹⊗› Positive and negative facts there are, butnot true and false facts.

‹⊗› If we overlook the fact that propositionshave a sense which is independent of their truthor falsehood, it easily seems as if true and falsewere two equally justified relations between thesign and what is signified. (We might then saye.g. that ”q” signifies in the true way what”not-q” signifies in the false way). But are nottrue and false in fact equally justified? Could wenot express ourselves by means of falsepropositions just as well as hitherto with trueones, so long as we know that they are meantfalsely? No! For a proposition is then true whenit is as we assert in this proposition; andaccordingly if by ”q” we mean ”not-q”, and it isas we mean to assert, then in the newinterpretation ”q” is actually true and not false.But it is important that we can mean the same by”q” as by ”not-q”, for it shows that neither tothe symbol ”not” nor to the manner of its

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combination with ”q” does a characteristic of thedenotation of ”q” correspond. ‹CCCf. 4.061, 4.062,4.0621›31

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Second MS

We must be able to understand propositionswhich we have never heard before. But everyproposition is a new symbol. Hence we must havegeneral indefinable symbols; these are unavoidableif propositions are not all indefinable. ‹Cf 4.02,4.021, 4.027›31

Whatever corresponds in reality to compoundpropositions must not be more than whatcorresponds to their several atomic propositions.

Not only must logic not deal with(particular) things, but just as little withrelations and predicates.

There are no propositions containing realvariables.

‹⊗› What corresponds in reality to a propositiondepends upon whether it is true or false. But wemust be able to understand a proposition withoutknowing if it is true or false. ‹cf. 4.024›31

‹⊗› What we know when we understand aproposition is this: We know [t|www]hat is the caseif the proposition is true, and what is the caseif it is falssse. But we do not know (necessarily)whether it is true or false. ‹cf. 4.024›31

Propositions are not names. ‹cf. 3.144›31

We can never distinguish one logical typefrom another by attributing a property to membersof the one which we deny to members of the other.

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‹⊗› Symbols are not what they seem to be. In ”aR b”, ”R” looks like a substantive, but is notone. What sssymbolizes in ”a R b” is that R occursbetween a and b. Hence ”R” is not the indefinablein ”a R b”. Similarly in ” φ x” ” φ ” looks like asubstantive but is not one; in ”~p” ”~” looks like” φ ” but is not like it. This is the first thingthat indicates that the re may not be logicalconstants. A reason against them is the generalityof logic: logic cannot treat a special set ofthings. ‹Cf. 3.1432›31

Molecular propositions contain nothingbeyond what is contained in their atoms; they addno material information above that contained intheir atoms.

All that is essential about molecularfunctions is their T-F schema (i.e the statementof the cases when they are true and the cases whenthey are false).

Alternative indefinability shows that theindefinables have not been reached.

‹⊗› 30[Every proposition is essentiallytrue-false: to understand it, we must know bothwhat must be the case if it is true, and what mustbe the case if it is false.]30 Thus a proposition

has two poles, corresponding to the case of itstruth and the case of its falsehood. We call thisthe sense of a proposition.

In regard to notation, it is important tonote that not every feature of a symbolsymbolizes. In two molecular functions which havethe same T-F schema, what symbolizes must be thesame. In ”not-not-p”, ”not-p” does not occur; for”not-not-p” is the same as ”p”, and therefore, if

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”not-p” occurred in ”not-not-p”, it would occur in”p”.

Logical indefinables cannot be predicates orrelations, because propositions, owing to sense,cannot have predicates or relations. Nor are ”not”and ”or”, like judgment, analogous to predicatesor relations, because they do not introduceanything new.

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Propositions are always complex even if theycontain no names.

30[A proposition must be understood when allits indefinables are understood. The indefinablesin ”a R b” are introduced as follows:

”a” is indefinable;”b” is indefinable;Whatever ”x” and ”y” may mean, ”x R y”

says something indefinable about their meaning.]30

A complex symbol must never be introduced asa single indefinable. (Thus e.g. no proposition isindefinable). For if one of its parts occurs alsoin another connection, it must there bere-introduced. And would it then mean the same?

The ways by which we introduce ourindefinables must permit us to construct allpropositions that have sence [? meaning]32 fromthese indefinables alone. It is easy to introduce”all” and ”some” in a way that will make theconstruction of [”|(((]say) ”(x, y) .x R y” possiblefrom ”all” and ”x R y” as introduced before.

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Third MS

An analogy for the theory of truth: Considera black patch on white paper; then we can describethe form of the patch by mentioning, for eachpoint of the surface, whether it is white orblack. To the fact that a point is blackcorresponds a positive fact, to the fact that apoint is white (not black) corresponds a negativefact. If I designate a point of the surface (oneof Frege’s ”truth-values”), this is as if I set upan assumption to be decided upon. But in order tobe able to say of a point that ittt is black or thatit is white, I must first know when a point is tobe called black and when it is to be called white.In order to be able to say that ”p” is true (orfalse), I must first have determined under whatttcircumstances I call a proposition true, andthereby I determine the sense of a proposition.The point in which the analogy fails is this: Ican indicate a point of the paper that is whiteand black, but to a proposition without sensenothing corresponds, for it does not designate athing (truth-value), whose properties might becalled ”false” or ”true”; the verb of aproposition is not ”is true” or ”is false”, asFrege believes, but what is true must alreadycontain the verb. ‹SeeSeeSee 4.063›31

The comparison of language and reality islike that of retinal image and visual image: tothe blind spot nothing in the visual image seemsto correspond, and thereby the boundaries of theblind spot determine the visual image - as truenegations of atomic propositions determinereaaality.

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Logical inferences can, it is true, be madein accordance with Frege’s or Russell’s laws ofdeduction, but this cannot justify the inference;and therefore they are not primitive propositionsof logic. If p follows from q, it can also beinferred from q, and the ”manner of deduction” isindifferent. ‹Cf 5.132›31

Those symbols which are called propositionsin which ”variables occur” are in reality notpropositions at all, but only schemes ofpropositions, which only become propositions whenwe replace the variables by constants. There is noproposition which is expressed by ”x = x”, for ”x”has no significatitition; but there is a proposition”(x). x = x” and propositions such as ”Socrates =Socrates” etc.

In books on logic, no variables ought tooccur, but only the general propositions whichjustify the use of variables. It follows that theso-called definitions of logic are notdefinitions, but only schemes of definitions, andinstead of these we ought to put generalpropositions; and similarly the so-calledprimitive ideas (Urzeichen) of logic are notprimitive ideas, but the schemes of them. Themistaken idea that there are things called factsor complexes and relations easily leads to theopinion that there must be a relation ofquestioning32 ⟨proposition ⟩ 33 to the facts, andthen the question arises whether a relation canhold between an arbitrary number of things, sincea fact can follow from arbitrary cases. It is afact that the proposition which e.g. expressesthat q follows from p and p ⊃ q is this: p.p ⊃ q.⊃p.q q.

At a pinch, one is tempted to interpret”not-p” as ”everything34 else, only not p”. That

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from a single fact p an infinity of others,not-not-p etc., follow, is hardly credible. Manpossesses an innate capacity for constructingsymbols with which some sense can be expressed,without having the slightest idea what each wordsignifies. The best example of this ismathematics, for man has until lately used thesymbols for numbers without knowing what theysignify or that they signify nothing. ‹Cf. 5.43›31

Russell’s ”complexes” were to have theuseful property of being compounded, and were tocombine with this the agreeable

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property that they could be treated like”simples”. But this alone made them unserviceableas logical types, since there would have beensignificance in asserting, of a simple, that itwas complex. But a property cannot be a logicaltype.

Every statement about apparent complexes canbe resolved into the logical sum of a statementabout the constituents and a statement about theproposition which describes the complexcompletely. How, in each case, the resolution isto be made, is an important question, but itsanswer is not unconditionally necessary for theconstruction of logic. ‹Cf 2.0201›31

That ”or” and ”not” etc. are not relationsin the same sense as ”right” and ”left” etc., isobvious to the plain man. The possibility ofcross-definitions in the old logical indefinablesshows, of itself, that these are not the rightindefinables, and, even more conclusively, thatthey do not denote relations. ‹See

31 5.42›31

If we change a constituent a35 of aproposition φ (a) into a variable, then there is aclass

p̂ [( ∃ x). φ (x) = p] .

This class in general still depends upon what, byan arbitrary convention, we mean by ”φ (x)”. Butif we change into variables all those symbolswhose significance was arbitrarily determined,there is still such a class. But this is now notdependent upon any convention, but only upon thenature of the symbol ”φ (x)”. It corresponds to a

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logical type. ‹Cf 3.315›31

Types can never be distinguished from eachother by saying (as is often done) that one hasthese but the other has those properties, for thispresupposes that there is a meaning in assertingall these properties of both types. But from thisit follows that, at best, these properties may betypes, but certainly not the objects of which theyare asserted. ‹[See|CfCfCf36] 4.124›31

At a pinch we are always inclined toexplanations of logical functions of propositionswhich aim at introducing into the functions eitheronly the constituents of these propositions, oronly their form, etc. etc.; and we overlook thatordinary language would not contain the wholepropositions if it did not need them: However,e.g., ”not-p” may be explained, there must alwaysbe a meaning given to the question ”what isdenied?”

The very possibility of Frege’s explanationsof ”not-p” and ”if p then q”, from which itfollows that ”not-not-p” denotes the same as p,makes it probable that there is some method ofdesignation in which ”not-not-p” corresponds tothe same symbol as ”p”. But if this method ofdesignation suffices for logic, it must be theright one.

‹⊗› Names are points, propositions arrows - theyhave sense. The sense of a proposition isdetermined by the two poles true and false. Theform of a proposition is like a straight line,which divides all points of a plane into right andleft. The line does this automatically, the formof proposition only by convention. See 3.144

31

Just as little as we are concerned, in

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logic, with the relation of a name to its meaning,just so little are we concerned with the relationof a proposition to reality, but we want to knowthe meaning of names and the sense of propositionsas we introduce an ind[f|eee]finable concept ”A” bysaying: ”’A’ denotes something indefinable”, so weintroduce e.g. the form of propositions a R b bysaying: ”For all meanings of ”x” and ”y”, ”x R y”expresses something indefinable about x and y”.

In place of every proposition ”p[2|”””], letus write ”ab p”. Let every correlation ofpropositions to each other or of names to

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propositions be effected by a correlation of theirpoles ”a” and ”b”. Let this correlation betransitive. Then accordingly ”a-ab-b p” is thesame symbol as ”ab p”. Let n propositions begiven. I then call a ”class of poles” of thesepropositions every class of n members, of whicheach is a pole of one of the n propositions, sothat one member corresponds to each proposition. Ithen correlate with each class of poles one of twopoles (a and b). The sense of the sssymbolizing factthus constructed I cannot define, but I know it.

If p = not-not-p etc., this shows that thetraditional method of symbolism is wrong, since itallows a plurality of symbols with the same sense;and thence it follows that, in analyzing suchpropositions, we must not be guided by Russell’smethod of symbolizing.

It is to be remembered that names are notthings, but classes: ”A” is the same letter as”A”. This has the most important consequences forevery symbolic language. ‹See 3.203›31

Neither the sense nor the meaning of aproposition is a thing. These words are incompletesymbols.

It is impossible to dispense withpropositions in which the same argument occurs indifferent positions. It is obviously useless toreplace φ (a, a) by φ (a, b). a = b.

Since the ab-functions of p are againbi-polar propositions, we can form ab-functions ofthem, and so on. In this way a series ofpropositions will arise, in which in general the

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symbolizing facts will be the same in severalmembers. If now we find an ab-function of such akind that by repeated application of it everyab-function can be generated, then we canintroduce the totality of ab-functions as thetotality of those that are generated byapplication of this function. Such a function is~p ∨ ~q.

It is easy to suppose a contradiction in thefact that on the one hand every possible complexproposition is a simple ab-function of simplepropositions, and that on the other hand therepeated application of one ab-function sufficesto generate all these propositions. If e.g. anaffirmation can be generated by double negation,is negation in any sense contained in affirmation?Does ”p” deny ”not-p” or assert ”p”, or both? Andhow do matters stand with the definition of ”⊃” by”∨” and ”[.|~]”, or of ”∨” by ”[.|~]” and ”⊃”? Andhow e.g. shall we introduce p/q (i.e. ~p ∨ ~q) ifnot by saying that this expression says somethingindefinable about all arguments p and q? But theab-functions must be introduced as follows: Thefunction p/q is merely a mechanical instrument forconstructing all possible symbols of ab-functions.The symbols arising by repeated application of thesymbol ” | ” do not contain the symbol ”p|q”. Weneed a rule according to which we can form allsymbols of ab-functions, in order to be able tospeak of the class of them; and we now speak ofthem e.g. as those symbols of functions which canbe generated by repeated application of theoperation ” | ”. And we say now: For all p’s andq’s, ”p|q” says something indefinable about thesense of those simple propositions which arecontained in p and q. ‹See 5.44›31

The assertion-sign is logically quitewithout significance. It only shows, in Frege and

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Whitehead and Russell, that these authors hold thepropositions so indicated to be true. ” ”therefore belongs as little to the proposition as(say) the number of the proposition. A propositioncannot possibly assert of itself that it is true.‹See 4.42›31

Every right theory of judgment must make itimpossible for me to judge that this tablepenholders the book. Russell’s theory does notsatisfy this requirement. ‹5.5422›31

‹⊗› It is clear that we understand propositionswithout knowing

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whether they are true or false. But we can onlyknow the meaning of a proposition when we know ifit is true or false. What we understand is thesense of the proposition. ‹Cf 4.024›31

The assumption of the existence of logicalobjects makes it appear remarkable that in thesciences propositions of the form ”p ∨ q”, ”p ⊃q”, etc. are only then not provisional when ” ∨ ”and ” ⊃ ” stand within the scope of agenerality-sign (apparent variable).

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Fourth MS

If we formed all possible atomicpropositions, the world would be completelydescribed if we declared the truth or falsehood ofeach. I doubt this. R? DS. ‹See31 4.26.›31

‹⊗› The chief characteristic of my theory isthat, in it, p has the same meaning as not-p. ‹Cf4.0621›31

A false theory of relations makes it easilyseem as if the relation of fact and constituentwere the same as that of fact and fact whichfollows from it. But the similarity of the two maybe expressed thus: φ a. ⊃ (ϕ,a) a = a.

If a word created32s31 a world so that in itthe principles of logic are true, it therebycreates a world in which the whole of mathematicsholds; and similarly it could not create a worldin which a proposition was true, without creatingits constituents. Cf 5.123

31

Signs of the form ”p ∨ ~p” are senseless,but not the propositions ”(p). p ∨ ~p”. If I knowthat this rose is either red or not red, I knownothing. The same holds of all ab-functions. Cf4.461

31

To understand a proposition means to knowwwwhat is the case if it is true. Hence we canunderstand it without knowing if it is true. Weunderstand it when we understand its constituentsand forms. IIIf we know the meaning of ”a” and ”b”,and if we know what ”x R y” means for all x”s andy’s, then we also understand ”a R b”. Cf 4.024

31

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I understand the proposition ”a R b” when Iknow ttthat either the fact that a R b or the factthat not a R b corresponds to it; but this is notto be confused with the false opinion that Iunderstood ”a R b” when I know that ”a R b or nota R b” is the case.

But the form of a proposition symbolizes inthe following way: Let us consider symbols of theform ”x R y”; to these correspond primarily pairsof objects, of which one has the name ”x”, theother the name ”y”. The x’s and y’s stand invarious relations to eeeach other, among others therelation R holds between some, but not betweenothers. I now determine the sense of ”x R y” bylaying down: when the facts behave in regard to ”xRRR y” so that the meaning of ”x” stands in therelation R to the meaning of ”y”, then I say thatthey (the facts) are ”of like sense”(”gleichsinnig”) with the proposition ”x R y”;otherwise, ”of opposite sense” (entgegengesetzt”);I correlate the facts to the symbol ”x R y” bythus dividing them into those of like sense andthose of opposite sense. To this correlationcorresponds the correlation of name and meaning.Both are psychological. Thus I understand the form”x R y” when I know that it discriminates thebehaviour of x and y according as these stand inthe relation R or not. In this way I extract fromall possible relations the relation R, as, by aname, I extract its meaning from among allpossible things.

Strictly speaking, it is incorrect to say:we understand the proposition p when we know that’”p” is true’ ≡ p; for this would naturally alwaysbe the case if accidentally the propositions toright and left of the symbol ”≡” were both true orboth false. We require not only an equivalence,but a formal equivalence, which is bound up with

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the introduction of the form of p.

The sense of an ab-function of p is afunction of the sense of p. ‹See 5.2341›31

The ab-functions use the discrimination offacts, which their arggguments bring forth, in orderto generate new discriminations.

Only facts can express sense, a class ofnames cannot. This is easily shown. ‹See 3.142›31

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There is no thing which is the form of aproposition, and no name which is the name of aform. Accordingly we can also not say that arelation which in certain cases holds betweenthings holds sometimes between forms and things.This goes against Russell’s theory of judgment.

‹X› It is very easy to forget that, though thepropositions of a form can be either true orfalse, each one of these propositions can only beeither true or false[.|,,,] not both

This was typed in but has excexcexcesses throughit. (D.S.)

Among the facts which make ”p or q” true,there are some which make ”p and q” true; but theclass which makes ”p or q” true is different fromthe class which makes ”p and q” true; and onlythis is what matters. For we introduce this class,as it were, when we introduce ab-functions. ‹Cf5.1241›31

A very natural objection to the way in whichI have introduced e.g. propositions of the form xR y is that by it propositions such as (∃ x. y). xR y and similar ones are not explained, which yetobviously have in common with a R b what c R d hasin commonxxx with a R b. But when we introducedpropositions of the form x R y we mentioned no oneparticular proposition of this form; and we onlyneed to introduce (∃x, y). φ (x, y) for all φ’s inany way which makes the sense of thesepropositions dependent on the sense of allpropositions of the form φ (a, b), and thereby thejustness of our procedure is proved.

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The indefinables of logic must beindependent of each other. If an indefinable isintroduced, it must be introduced in allcombinations in which it can occur. We cannottherefore introduce it first for one combination,then for another; e.g., if the form x R y has beenintroduced, it must henceforth be understood inpropositions of the form a R b just in the sameway as in propositions such as (∃x, y). x R y andothers. We must not introduce it first for oneclass of cases, then for the other; for it wouldremain doubtful if its meaning wwwas the same inboth cases, and there would be no ground for usingthe same manner of combining symbols in bothcases. In short, for the introduction ofindefinable symbols and combinations of symbolsthe same holds, mutatis mutandis, that Frege hassaid for the introduction of symbols bydefinitions. Cf 5.451

31

It is a priori likely that the introductionof atomi[x|c] propositions is fundamental for theunderstanding of all other kinds of propositttions.In fact the understanding of general propositionsobviously depends on that of atomic propositions.

Cross-definability in the realm of generalpropositions leads to quite similar questions tothose in the realm of ab-functions.

When we say ”A believes p”, this sounds, itis true, as if here we could substitute a propername for ”p”; but we can see that here a sense,not a meaning, is concerned, if we say ”A believesthat ’p’ is true”; and in order to make thedirection of p even more explicit, we might say ”Abelieves that ’p’ is true and ’not-p’ is false”.Here the bi-polarity of p is expressed and itseems that we shall only be able to express theproposition ”A believes p” correctly by the

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ab-notation; say by making ”A” have a relation tothe poles ”a” and ”b” of a-p-b.

The epistemological questions concerning thenature of judgment and belief cannot be solvedwithout a correct apprehension of the form of theproposition.

The ab-notation shows the dependence of orand not, and thereby that they are not to beemployed as simultaneous indefinables.

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Not: ”The complex sign ”a R b’” says that astands in the relation R to b; but that ’a’ standsin a certain relation to ’b’ says that a R b.‹3.143›31

In philosophy there are no deductions: it ispurely descriptive.

Philosophy gives no pictures of reality.

Philosophy can neither confirm nor confutescientific investigation. ‹Cf 4.111›31

Philosophy consists of logic andmetaphysics: logic is its basis.

Epistemology is the philosophy ofpsychology. ‹See 4.1126›31

Distrust of grammar is the first requisitefor philosophizing.

Propositions can never be indefinables, forthey are always complex. That also words like”ambulo” are complex appears in the fact thattheir root with a different termination gggives adifferent sense. ‹4.032›31 Crossed out butoriginally typed in. (D.S.)

Only the doctrine of general indefinablespermits us to understand the nature of functions.Neglect of this doctrine leads to an impenetrablethicket.

Philosophy is the doctrine of the logicalform of scientific propositions (not only ofprimitive propositions). ‹Cf 4.113›31

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The word ”ppphilosophy” ought always todesignate something over or under but not beside,the natural sciences. ‹See 4.111›31

‹ › Judgment, command and question all stand onthe same level; but all have in common thepropositional form, which does interest us.

‹ › The structure of the proposition must berecognized, the rest comes of itself. But ordinarylanguage conceals the structure of theproposition: in it, relations look likepredicates, predicates like names, etc. ‹Cf4.002›31

Facts cannot be named. ‹See 3.144›31

It is easy to suppose that ”individual”,”particular”, ”complex” @.. etc. are primitiveideas of logic. Russell e.g. says ”individual” and”matrix” are ”primitive ideas”. This errorpresumably is to be explained by the fact that, byemployment of variables instead of thegenerality-sign, it come s to seem as if logicdealt with things which have been deprived of allproperties except thing-hood, and withpropositions deprived of all properties exceptcomplexity. We forget that the indefinables ofsymbols (Urbilder von Zeichen) only occur underthe generality-sign, never outside it.

Just as people used to struggle to bring allpropositions into the subject-predicate form, sonow it is natural to conceive every proposition asexpressing a relation, which is just as incorrect.What is justified in this desire is fullysatisfied by Russell’s theory of manufacturedrelations.

⟨ wrong termed? ⟩ 33

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One of the most natural attempts at solutionconsists in regarding ”not-p” as ”the opposite ofp”, where then ”opposite” would be the indefinablerelation. But it is easy to see that every suchattempt to replace the ab-functions bydescriptions must fail.

The false assumption that propositions arenames leads us to believe that there must belogical objects: for the meanings of logicalpropositions will have to be such things.

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A correct explanation of logicalpropositions must gggive them a unique position asagainst all other propositions. ‹6.12›31

No proposition can say anything aboutitself, because the sssymbol of the propositioncannot be contained in itself; this must be thebasis of the theory of logical types. ‹Cf 3.332›31

Every proposition which says somettthingindefinable about a thing is a subject-predicateproposition; every proposition which sayssomething indefinable about two things expresses adual relation between these things, and so on.Thus every proposition which contains only onename and one indefinable form is asubject-predicate proposition, and so on. Anindefinable simple symbol can only be a name, andtherefore we can know, by the symbol of an atomicproposition, whether it is a subject-predicateproposition.

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‹

This is the symbol for ~p ∨ ~q ›31

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Notes

1 Insertion by S12 Deletion by S13 Overwriting by S14 Insertion by H55 Deletion by H56 Text in left margin by S17 Underlining by H58 Insertion by H5 in lower margin9 Overwriting by H510 Underlining by S111 Insertion in right margin by H512 Text in right margin by H513 Point of insertion underlined by S114 Insertion by H5 in right margin15 Insertion by H5 in upper margin16 Text in left margin by H517 Cf. MS.; new paragraph?18 Deletion by H5 - deletion cancelled by H519 Underlining with dotted line by H520 Underlining by H5 - underlining cancelled by H521 Cf. MS.; indentation?22 Cf. MS.; inserted line23 Insertion in lower margin24 Cf. MS.; new line?25 Possibly deleted.26 Possibly underlined; doubt.27 Text in right margin by S128 Text vertically or diagonally crossed through by S129 Curved line in left margin by S130 Straight line(s) in left margin by S231 Insertion by S232 Deletion by S233 Text in left margin by S234 Underlining by S235 Underlining cancelled by S236 Overwriting by S2

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Dr Michael Biggs is the Leader of Design Research and PrincipalLecturer in visual communication at the University ofHertfordshire, and was Senior Research Fellow at TheWittgenstein Archives at the University of Bergen during 1994.His PhD thesis "The Illustrated Wittgenstein" is a study of thediagrams in Wittgenstein’s published works (British Libraryreference DX180816).

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Skriftserie fra Wittgensteinarkivet ved Universitetet i Bergen:Working Papers from the Wittgenstein Archives at the University of Bergen:

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