Frege - Begriffssgrift

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    46to' and 'stand for' are the two English expressions that have genelallybeen used in the uanslarions of Frege's larer work riat appear in thcpresent volume ('stand for' in the nanslations by ceach, Black, andI-ong and \v}lire). The crman term has, howeverr been placed insquare brackets immediately aftenvalds, in accordance with the prin.ciple of inrerpretive integriry. In rle end, what is mosr importanr is tomake sure tlat each occunence of'bedeuten', however it is translared,is signalledj and this is the policv tlat has been followed here.ri6

    : ' I! ihe fiN svo rdnn)nr of ?Plr. $hich sr\ the li.tr collccti.r of l,rtsc J rvorks lube published in Enshn {th.ush.ot thc n6i book 17 had appe.rcd in 1950: 7],ty ln1952)) Rcdeuona w$ rcndc.ed !s telcrcn.c' rnd b

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    III

    48 Begriffsschrift\i4Et follows here is the Preface, in which F-rcge outlines his motivation,and most of Paft I, which explains his symbolism. A note on pan IIand a surnmary of Part III are provided ar the end.l

    prefaceThe rcognitiofl of a scientific truth gererally passes through severalstages of cenainty. Perhaps firsr guessed ftom a limired number of par-ticular cases, a universal proposirion becomes more and more finnlyestablished by being connected with orher trurhs rhrough chains of infe!-ence whether conclusions &at 6nd confimarion in other ways arcderived ftom it, or whether, conversely, ir is recognized as foltowing fromalready established propositions. It can thus be asked, on rte one hand,by what parh a proposition was gradually reached, and on rhe othe! hand,in what way it is now finally ro be mosr firmly estabtished. The formerquestion possibly Deeds to be answered differently for different peoplejthe latter is more dfinite, and its answer is connecred with the innernature of the prcposition concmed. The fumesr proof is obviously thcpuely losical, whict\ prescinding ftom dre parti-ul@ of rhinss, is basedsoretfolrtre laws on which aU knowledge rests. Accordingly, we dividcall Euths that require justidcationinto iri

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    50 Begriffsschdftrl'i\ lvirh rh( krowle(l8e rhlr .r dcvclopmen. in merhod also tunhers sci-l'lrir. llll,,rr ir(h.c(l rhousht n prsLrable to invent a means by which.!.rvrlu+t ( rD h linrn(l cnsily dl:ln to discover a particular rhing, andaUl.1'rnr Lr ! nrlLr,lviu,(cs in modcm rimes have indeed had their orisinr'r I r"rt,'ovcrrc'rr i'r nrcthod.l,rrl'ri/ r(r, rccol,ruizcd perhaps overestimated the advantages ofur rl]t)i('lrirLc syrnbolism. His conception of a uiversal characteristic,t t nh r\ fhil[o?hicus ot tatioci ato\8 w^s too Erandiose for the attemptro r'crlizc it ro go lurtler tlan the bare pEliminaries. The enrhusiasmthrt scizcd its originator in considerins what an immense increase in rhmcntal power of mankind would aise ftom a symbolism suited ro thingsthemselves let him underestimate the difEolties that such an emerpdsefaces. But even iftlis great aim cannot b achievd at the firsr aremptrone ned not dspair of a slow, srep by step appmach. If a problem inits tull senerality appears insoluble, it has ro be limited provisionally; itcan then, pelhaps, be dealt with by advancing gradualy. Arithmetical,geometrical and chmical symbols can b rgarded as realizations of theIribnizian conceprion in panicuhr nelds. The Besnlsscrri ofered hereadds a new one to these indeed, the one located in the middle, adioin-ing all the others. Frcm here, with the greatest prospect of success, oncan then proceed to 6ll in fie eaps in the existing formula languages,connecr rheir hitherto separate fields into the domain of a single fonnulalanguage and ertend it to fields that have hftherto iacked such a language.I am convinced .\^t my Beyiflsschilt can be successtuIy appliedwhercver a special value has to be placed on the validity of proof, as inr}le case of laying the foundations of the differenrial and int$al calculus.It seems to me to be even easier to extend the domain of rhis formulalanguage to geometry. Onlv a few more s'.rnbols would have to be addedfor the intuitive relations that occur here. In this way one would obtai$a kind of ardl.yrn srrrrr.The transition to the pu,re theory of motion and rhence ro mechanicsand physics misht follow here. In rhe lafter fields, where besides con-ceptual necessiiy, natural necessity prevails,'o a further development ofthe symbolism with rhe advancement ofknowledge is easiest to foresee.But that is no reason to wait until such advancement appeaE to havc

    Ltr_rga-tasCoe+UUsoplr.lo,brcak qre"l)av,eL-of-izordr-gygllrehuman mind, by uncovering i usions thalthrorhlhE"Iiqglblglgscrr On this, s&'l.enderentus Hiton:eh. BentiiEe.N Phil6,zlt. Vol. 3. [iieAd'\ relcFcnc. is to Trendclenb!.g\.:sry On Libnizs P.oject ofa Univebal Chancrcistic',whi.h sccms io hrrc b.$ lnc mrin s.urce oflicgc s undelstnndins ol Lcibniz's corccp-tion Ct Slus,. 1980: cl!. 2, t4.l"' -1fu (;rnnrn \lords hrrc rc l)cnkno$cndigkcir' rnd NdlmotwendiSkcn . r.strctivcly,

    VI

    Begritrsscbrift 5ti,lrcn almosr unavoidabty arise conceming tfie retarions ot conceprsr byrrrcrng Lioughr ftom I dre iainr of ordinary finguisLic means oi-g.plg_'n'n, then my Becnf$"inlr, turrher developed r- tr,*. !i.p.leit;liccome a usetul rool for philosophen. Admittedl% * i, surd;;;iliablIr rh(. case of enemal meanr of represenra!ion. even rhis cannor make,l'ouglrL pru-e dgaini bur rhe deviadonq can, ar least, b limi(ed ro Lhe'rnavoidable and harmless, whilst ar the same dme, iust because rh;rrt of a quite differenr kind from rhose typical oi oiai"".v I"ng"ug.,l)rotcction is provided againsr rhe one_sided influence of one of-*re-serncans of expression.-rhe_very invenrion of |llli, BesilJsrc hnl. it seems ro me, has advancedl,'U'c..1 hope rhai logicians, if Lhey rre nor pur off by firsL impressron.,'i unfamiliariry, wi nor repudiare the innovations to which I was drivenl.v r ne{essiry inherenr in lhe subjecr marrer nself. Tbese deviarionsr,.,m $hdr 's. rmdidonal tud their iulificarion in rhe tacr drar logicIflueno har alwiy( Ioltowed ordinary tansuage ana gramrnar roo ctosei.lI pr(icular, I believe rhat lhe replacement of the conceDrs ,r,r?.r and II'a h. ate by org ne and t'n ion wi| prov irselt in rr,. ion, ru". r, i. lr r.y ro see how rakhg a conLenr as r tuncrion ofu" ",g".e.ri eju." ,i,. /r(' concept formation. !(ahat also deserves notice is G demoistration"l lhe connecdon berween rhe meanings ot rhe $.rd., if. "";:;;. ;;,rhrrc rs, some. all. erc.Only the fotlowins remains to be mentioned in panicular.The restriction, in 56, to a sinsle mode ofinfer."i" i" ,,"tm"a ty *"lct that iA lalins the tound.atin x of slf/.jh a Be?ritsschnri ,r," p;"ii.i*:u.^:l'. Tu1b. as simple as possible if perspicuiry and order are tor\.iLhje!ect. I hrs does noL rule our. /ar?i lran.irions from (everrl iudge_,,,(rLr ro a new onr, whjch are pos(ibte by thre singte mode ofinferenleir,ry in an rndrecr wa'. being convened inro direcl ones lor rhe sake,,1 abbreviaiion. Ir facr, this may be advisable fo. fat".

    "ppfi"aCo""inrhis way, rhen, furtller modes of infrence would arise.-iI realized larer l}lar formutae ( I l.) and (4 t) can be combined inro fie

    F-t-6"=';which makes a few more simplifications possible.'l, Arnhne:rc: as I remarked ar LIe beginning. was rhe sianing poinr of'rrL rrarn ot t})oughr $ar led me ro mv ,Bc8rls{.rnlr. I rherefor; inrendr,' rpply rr ro rhis qcience frrsr. seeking ro provrde fu4her anahsis of its.,,ncepr! and a deeper foundarion for irs rheorems. t an ounce in tfrelbmulac (31) md (4r) dc Mo of rhe mioms rhat Frcse laid down tor hn bsical,l\on, vhnh an bc ambned usins his srmbot for iden,l,v "r-_*,, * app"""j*

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    54 Begriffsschriftthe concept with which r}le judsement is primarily concemed'. The lin.guisric significance of rhe position of the subjecr in the word-order lictin irs mdrA,rs the place where what one particularly wants to draw thcatrenrion of the listener to is put. (See also S9.) This can have the pur.poser for example, of indicating a rclation between this iudgement a'ldothers, thereby facilitating the listener's Erasp of all the interconnections.Now all those features of language rhat rcsult only ftom the interaction ofspeaker and listens where the speaker, for exampler takes the listenerrrexpectations imo account and seeks to put them on the right track evenbefore a sentence is finishd - have no counrerpart in my formula lan-guager since here rhe only thing dlat is relevanr in a iudgement is thalwhich influences irs po$ible conseque ces. Everllhing drat is necessary fola valid inference is fulty exprssed; but what is nor necessary is mosdynor even indicated; ro*ing is left to guexing.Il t]].ls I closely follow thoexample ofthe formula langlage ofmathematics, in which subjct andpredicate can also bc distinguished only by violating it. Imagine a lan'guage in which the proposition 'Archimedes was killed at the capturlof Syracuse' is expressed in rhe folowing way: 'The violent death olArchimedes at rhe capture ofsFacuse is a fact'. Even hereJ if one want!'subject and predicare can be distinguished, but the subject contains th.whole content, and the predicate seFes only to present it as I a iudgc.'J'ent. Su.h d la s aseuouldhaoea l! a shsb predicate fat all judse,nenu,nanely, 'it a fact'. k can be seen that fiere is no question here of sub.iect and predicate in rhe usual sense. O"r Bestf$chnfi is such a lans aNand rhe 'ynbol l- h ns connon ptednak fot all fudsene ts.In my first draft of a formula ianguage I was misld by the exampl.of ordinary language in.o constructing judgements out of subject andpredicate. But I soon convinced myself dlat tltis was an obstacle to mypanicular goal and only led to useless prolixity.54. The following rema*s should suffice, for our purposes, to cx.plain the significance of fie distinctious tlat are made wirh regard tojudsements.A distinction is drawn between unioe$al and particular judgementilthis is not really a distinction between judgements, but btween con.@nrs. One ought to st:ry: 'a judgenent uith a uniue8al content', 'a judg,'me t @ith a tanicular contert r. For these properties belong to the contcnleven when it is ,?o, prcsented as a iudgemnt, but as a proposition (Scas2.)The same applies to negation. In an indirecr way orc says, for examplci'Suppose &at r}le lines lB and CD were not equal'. Here the contentlthat the lines lB and CD are not equal, contains a ncgation, but thhcontcnt, although capable ofbeing judged, is nvertheless not prcscnlcdas a judgement. Negation thcrcfore attaches to thc contcnt, irrcspcctiv!

    Begritrsschrift 55,,1 \vlrerher this appean as a judgernenr or not. I rheiefore hotd ir moreit,t'r,l)riare to regard negarion as a m tk of ^ judgeable co tent.lh. distindion between caregorical, hlToiheticat and disjunctiver'rl|.lrcnts seems ro me ro have only grammarical significance.tsI h. rpodeictic judgement is disringuished from the asserroric in thar,, rrlr(rtcs the existence of universal judeemenrs from which the pro_1, ' rrrn can be inferred, whereas in the case of an asseno.ic judgemenr,,, lL rn indication is lacking. IfI call a proposition neccssary, I thereby!i,! :r lrinr as ro my grounds for judgemenr. Brt since this does ot alfett'r, ,t\n1:fh@l content oJ the judgene t, the apodeictic fom of a iudpenentt" ,r', tictif.ance fu us.lr :r l)roposition is prcscnted as possiblc, then eirher the speaker isr,l[,,,illg from iudgemenr, by indicarina rhar h. knows no iaws from,,1,,, l, rrs negarior would follow; or else he is saying rhat rhe proposi_',,," ,regrrion ir its universal form is false. In de larrer case we have" r, ,r ,\ usually called a partixutar dlfnatiue judser,r, (see Sl2). .rt is| ",.,lnc rhar the Earth will one day collide with ano.her heavenly body,r rrr rrrnrple ofthe first case, and,A cold can rcsult in death, an Llample

    Conditionality\\ ll / and B denoterr judgcable contenrs (S2), ther there are ttel,,ll, ^\,,,i: four possibiliries:I rrLrfti\ lir d will cner$ liom my work as a w|ote.

    '.ltrtryc.\c, in orhcrw.rdsj thcn.Adion ot th. ,adicuta tlimrtirc proposition\ ' l!.ri 'c{ t in death (i c..Sone colds can resuhin drarh.) is rhc univcsal ncgarivc, L '(r' No colds.!n.esnlt nr delth. t adi.ul!. .tdmatn e anrl universat r.rjtive', 1. r," lrl .t ...an" ..n,,.d.,. +...nd1rr...,hi..t1. n.h,;qJ' ,,!,1. l;Jrcc did not dcpa lion rhc t..dniomr view. Ci \l?, ,p ?l .1 bclow.I , r, !i rLc ruDultinrxte paragnptr. rhroughour dri$ section thu verb .bcdeut.n, hrs,, ' , , |.d !s rlcnotc, and rhc noun Bedeurung' !s dcnotarnrn,. Othcr rranstations' " r. tr.ruor ntensclbdrg, Bynun) rond{ tir. ub !s thnd tnr,md re noun as, !r . litrt ri\ ohscurcs drc conneciior bct*ecn Ftust s usc oi rheqr. Sin.. .*and, tri hrvc ! .or.spondirg noun, .nd .melning' is n[ppioprilte il rhe o.e', rl,i.h llcrlouLuns' h lscd (in rhc drnd rrngrartr), ,dcnor.' $d .dcnodln,n,, ' " , f,r h.cn.i.s$ !/hat wc 6a!e in rhis \ection is rrre !./,,t.1urc ot.bcdeut.n,rL r,r' rnc drc r.trtnrn involv.d js drai bsk.en. sisn and ns.fur tincuni,., ,,, ' J 'olv. .h " . .fu..J ,,li..j...linlr.t .",{Lh."""1,-jnh__, ,, Lr')l onwnrls. (For rhe distnr. ,n betwc.n the vcrical lnd horizonlll uscsL L rLtr'ri rn(t n\ cognrt*. st. rhe Intorludion, S.r.) cenairty) nenhcr lncan,nnd, I 'iir '\irll'll ud .signincancc, (or \ignificriion ) .dcqudtty c.prur. r:reqc.sri,. i.,. l lr'.1. 'r ' ,hrf., .n *..io..r .uchr.o,. rr_,1o...,,1., ',,rt.i.',.j. 'rr'. r' !r t, 1 . , . . h.;.. rr h,,1._n,trh,rr,r , 'tu rh.u o.rt rrrt I'nrLri\bc.\Irr$nn$ {r.h rs pron.r njrne! or dctijn.| tr,i lh. li)nncr, n tr..n\ tr!tr. xJrrrrJn.irrt nr \rv. ltcnotc,.r trprc\.nt,or