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Lectures on Logic Carnap's Student Notes 1910- 1914 Publications o/tlze Arclzive C!/Sdelllijic PlztlMOpl,y Ht/i...
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Frege~s
Lectures on Logic Carnap's Student Notes 1910- 1914 Publications o/tlze Arclzive C!/Sdelllijic PlztlMOpl,y Ht/ill/all Library, UmiJer.Jity (!fPit&Jburglz VOLUME
I.
Frege's Leaures on Logic: Carnap's Student Notes, /9/ a-/9/4,
Translated and edited, with introductory essay, by
edited by Erich H. Reck and Steve Awodey
Erich H. Reck and Steve Awodey
2. Carnap Brought Home: The View (rom Jena,
VOLUME
edited by Steve Awodey and Carsten Klein
Based on the German text, edited, with introduction and annotations, by Gottfried Gabriel ",
,. .. "',
At
Opell Court Chir<Jgo and LaSalle, Illinois
\
,
Publications ofdie Archilie ofScientific Philosophy Iltllman Library; Univer,sity rifPituburgh Steve Awodey, Editor
EDITORIAL BOARD James Lennox University of Pittsburgh Richard Creath Arizona State University
John Earman University of Pittsburgh
Michael Friedman Stanford University
Gottfried Gabriel University ofJena
Dana Scott Carnegie Mellon University
Wilfried Sieg Carnegie Mellon University
Mark Wilson University of Pittsburgh
Gereon Wolters University of Constance
t
yv~ l~O
,(~J7 ,l\ C4
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Lihrary ufCongreSJi C.. t~loging-in-Publicllld()1I(Ju ... Frege, Gottloh, 1848-1925. (Lectures. English. selections) Frege's lectures on Logic: Carnap's student notes, 1910-1914 I [J"anslatt'd and edited, with introductory essay hy Erich H. Rcck and Steve Awodey; ba.sed on the German text, edited, with introduction and annotations by Gottfried Gabriel. p. cm. -Wull circle) Incilldf"s bihliographical references and index. ISBN (~812f1-95~6_1 - ISBN 0-8126-9553-4 lpbk. : a(k. paper) I. L'gw, symbniJc lind 1113them,ltical. I. Catnap, Rudolf, 1~91~197l).IL Reek, Erich H., 1959- III. Awodey, Steve, l%lJ1\. Gabrwl. Goufrieri. 194:l- V. Title. VI. Series. B324.'1. F22F.52 2004 160......dc22
To Anna Lucile and Klara Liese/oue
Contents Preface
Xlll
Introduction: Frege's Lectures on Begrifftschrift Gottfried Gabriel
Frege's Lectures on Logic and Their Influence
1
Erieh H. Reek and Steve Awodey
17
Carnap's StndentNotes:
45
Begriffsschrift 1(1910-1911)
49
Appendix A: The Ontological Proof of the Existence of God
79
Appendix B: Numerical Statements abollt Concepts
83
Begriffsschrift II (1913)
87
Logic in Mathematics (1914)
135
Literature Cited
167
, ix
Frege's Lectures on Logic
Carnap's Student Notes, 1910- 1914
Preface
Gottlob Frege (1848-1925) is generally acknowledged to be one of the founders of modern logic, arguably even its main source. Frege presented his pioneering logical system for the first time in Begrifj<schrift (1879), Later an expanded and modified version appeared in Grundgesetze der Arithmetik, Vols. I (1893) and IT (1903), Together with his second book, Die Grundlagen der Arithmetik (1884), a~.d a series ofimportant anicles, including 'Il ckc'i.'iiv(' for the eurly Carnap-th' . n ere. e lena years that ('OIHt'xt or III'o-K a ntianisJtI L b . h" e ten~l()ll bernreen lOgic and life in the , :; '. . , (' f'nsp lto,mphle. and the C 1II11l{ -wIll of eour.'il' only hI' I'ull ' I d yerman youth moveh' . . y reVl'a e to us whe C' '" t Is pt:'flOd and other doeu1llents h' I ' n arnap s diancs from ,I I . ave leen I)roperly , 'd ' a rear y eVldellt thm onel' aga,',) th'" scrutInize . But It is '1 -e eOlltlnental" f' 1 p 11 osopher art' coming to light. roots 0 an "analytie"
I. The Lectures On "Be r'lf
Development of Frege~
'" '
~~sChnftd hJn glc an
IS
the, Context of the Philosophy of Mathematics
In the44ye ' ars, 10 toral. of his tcachin .. semester 1874 to summer jg career at the University of lena (summer h semester 918) F OUr (per week) lecture course 'th h '. re~e rcg~_Ilarly announced a onc~eme.';~~~ from that of 1883/84 ~. t e n.de Begnffsschrift" every winter .Begnlfsscnrift" in the wintf' . dore thiS, Frege had already lechlred on !J("ni 1'1 . r semester 1879/80 . • on 0 lIs Rf'griffssc!Jniji (Freg' 1°791 .10 conjunction with the pubt () . DUl'tolak f '. 4 'l'f . ," C () partIcipants the .
h'
JU,;tlflcation for th"
.
l~~ l~ dlSt:usscd in "Notes for L d . fefge ~. p. 275; Frege 1979 p 255) U Wig Darmstaedter" in
1'o.,t!lUflWII.,· flrilillg.1 (Fr
.r;. This nHllhina"·,, 'h"! no actors! C , ". .... l~' ht't"ll thl:' sub 'ect f. . n arnap s early develo me . . J. U !!,ri.!j~'schrift presentation of the concept of following in a !"t·(I~h·n~·t'. W~1('h (lispt~nses with value~ranges, to that in the Basic Law.,>. At the ~Jt'l!lnnln/!01," Ht'I:~rirrsschrift In he even returns to his earlier terminolo callInl! th~ "eontent strok.e" h t h e concepgy, . tht· "hOrizontal" . . . In substance , thoug, tum or fht' Ha,w' {.o1Os dommates' the horizontal,'s und t d . I f . ,~ , eIS 00 as a specla t11ll'tlOl1 of IIrst level, whosc value for the argument "th IT '" h " e ue IS t e true an d lor a~l, othN ohJects as arguments the false (d. ""Begriffsschrift I," . 73). I he coot.en: of "B~gri~,fsschriftI" corresponds to what Frege~isted under the characteristIc headIng What may 1 regard as the Result of m Work? n as Y F farhackasl9061 p. 184) " ,IS , . . regc1983 " p,200'Frege1979 , . Noneth e \ess there a notlceable dIfference between "Begriffsscbrift I" and th f h' mous writi ,( "I d . ose 0 IS posthungs ntro ucllon to Logic" and "A brief Sluvey f \' I D ''') . h' 0 my ogle a woctnnes 10 ~ .l c h Frcge goes about actually spelling Out the "results" of his o~k. These wrltI~gs belong not to formal logic but to /informal) su h!:itantivf> 10gIe (what would 10 German be called "inhaltliche L('gt'k"j" A full t ( " hi' ' . r~a mellt 0 ~ e at~er ~ Interrupted by his death - Frege embarked on only with the I ulrical . "'" ~ . . I'J Inz'estlgatlOns ("'Thought· " .... N . . , . s. eganon, Compound Thoughts"). Their philosophical onentatIon toward a transcendfntal PI, t ' h ' an .- . .... . a omsm rever erates In ap bOrisUe remark III Begnffsschrl'f't I"·. '"Lomc . IS , not on Iy trans ' even trans-human." g o· -aIlan, b ut
Introduction: Frege's Lectures on Begriffsschrift
5
Novemher 1918; Frege 1976, p, 45; Frege 1980, p, 30). What parts ofthe formal logic, lhe Begriff.~schrift, count toward this "'harvest"? The substantive IObric is devoled to the philosophical analysis of the "basic" logical categories, Thus another of the posthumous writings on suhstantive logic (from the ~ear. 19~5) bears the characteristic title ""My basic logical Insights." Frege begms II WIth the words, "The following may be useful to some as a key to the understanding of my results," but oue should not conclude from this that there are no .... results" in the realm of formal logic. I think we can rightly claim that the text. "Begriffsschrift I," supplemented by certain parts of "BegriffsschriftU;' represents an inventory of what Frege regarded, after the faIlure of the 10gtCIst program, as the result of his work in the field of formal logic. This is in accordance with his statement that his logic is "'in the main" independent of the problems in set theory (Frege 1983, p, 191; Frege 1979, p. 176), since for him, as he already puts it in '''What may I regard as the Result. of my Work?;' "'the extension of a concept or class is not the primary thing for me." . This conjecture is also confirmed by Frege '05 form of presentatIon. In contra'it to the otherwise highly reflective style of Frege's considerations about Russell's paradox, culminating in self-criticism for the careless acce~t~nce of concept-extensions (dasses), 10 the presentation in the posthumous wrltIngs on substantive logic is characterized by what one might call a "dogmatic" pr~ce dure. This i!:i even more thc case in these lectures, which avoid any themattzation or critique of past errors - the antinomy is not mentioned - and instead are concerned to exhibit wbat can be regarded as unquestionably !:iccure. l1 It seems that Frege even expressed this attitude in his lccturing style. He trumped the user-unfriendliness ofthe traditional style of dogmatic exposition at the lectern hy actually turning his back on his students. Car nap describes the lecture course "'Begriffsschrift [" as follows:
He [Fn~gcl seldom looked at the audience. Ordlnarily we saw only his hack. while he drew the strange diagrams of his symhohsm on the hlackboard and explained them. Never did a student ask a question or make a remark, whether during the lecture or afterwards. The possibility of a discussion seemed to be out of the question. (CaTnap 1963, p,5)
Freg~ glo!'sed his elaboration of his substantive logic later with th d I arn trYlOg [(l hrinrr in tht' harvest of my J'f' " (I II' e wor s, t" • Ie etter to ". Dmgler of 17
We can only speculate whether Frege would have embarked on a ne~ exposition of the formal part of his logic once he had completed the substantIve part. But if he had done this, we can now say how it would have looked. It would have contained the following fragIOent of the Basic Laws: 11) the basic laws 1-1II (and perhaps IV); (2) all tbc inference rules witb the exception of those that drop out
" .. This distin('tioll is hPrf' to Iw undf''-s(oo~ I ' h ,. ..h.on y In t. f'- sense ' k IIl'of' 0 II Ilrrnalism. Fr~lJ"t'·s f I I,.' .. th'a t "ClormaI" l OITlC ma es . •• " '=' . IIrma ogw-t e "Beonff h 't'" . '='tuI or"sllh.~tantivf' .. lo.,.;cl'· .(., '=' sse fl t -IS of course a "contentr:>.f'. I IS Interpreted not (' (·tT',u.al cOlltt:'nt" (sefHit' and re-fert'nce) ofth~ . ' . a ~ere ormal calculus), as a "conII. nus remark is also of interest in th . t · slgnfFs Is a~sum~~ throughout. e con ext 0 rege s pohtlcal diary (Frege 1994).
10, Apart from the rele.... ant po~thum()llswritings after 1906, se~ especially the letter to R. Honigswald of26 April to4 Mny 1925. , 11. Carnap confirms this in his autobiography; "I do not rememher, that he I F~egf'l cve,r discussed in his lectures the problem of this antinomy and the questlon of pOSSIble mudifications of his system in order to eliminate it" (Carnap 1963, pp, 4f.).
..
6 Gotdried Gabriel
because of the elimin.t'lOTIO f vaIue-ranges (specIfIcally ' , rul e II,,c, t' BG;lC " La I §48) d(3) II ' WI ,, ; an a the laws derived from basic laws I-lll ( d h IV) b
usmg these rules.
an per aps
y
cept Consid~ring thde fact that Frege systematically avoids any mention of con-extenslOns an value-range i h' I ' , anal f' 5 n 15 ectures, It IS astonishing that in the ogous case 0 expressIons of the form "'the conce t F" h ' in t ti h P e seems to persIst un;~~' ng t ese as n~mes (""Begriffsschrift I," p. 66). Thus he regards "'falls as an expressIon of a first 1 1 1a' . whose value ('or ob)'e t - ev~ Ie t1on~ I.e. as a two-place function .' C 5 as arguments) IS al hal' backwards from the insi ht . ,,~ays a trut -v' ,ue. TIus seems a step g language is heregw'lty f " appare~t, m Ube, Schoenflies" from 1906, that o a counteneit" by", . . . . b' IFregeI98~,p,I92;Fre 1979 I 12 commg ano ]ectfromaconccpt cis-ion o£linguistic expregessl' ,,' Pak' 7!). Frege emphaSIzes that this "impre. on m es It appear "th h i ' , Uon is a third e l e m e n t . at t e re aUon of subsump1983, p, 193, Frege I;~~er~e~;:nt;pon the object and the concept" (Frege del~eptive way of talking wh' p h )h', ut Frege himself gives credence to this ., en e t mks he ea e . I ,Ingu1sucully in categorl'al di n xpress somethmg uscd meta'. seourse. such as "th la' , ulldl~r t he (~onccpt F" in th h' ere tlOn of an object a falling , , eo lect language f' I I ' nbjl'ct (I and thl~ 1)~el1d()-oh)' .. h as a lfSt- eve relation between the eet t e coneeptp" Wh~t ('an Wt~ say On the hasis of the 1 t . , ('OI1('('ptHlIl of the relatio b tw I e~ ures published here about Frege's e een ogle and h . Ill'tWI'I'n logie and arithmet"n , C , mat emaucs. paTticuIarlvJ . c. arnap s statement th F (,'I~t program even after Rus 'ell' d' at rege retained thc 10lTik . . s s Iscovery of th . b~ S epucal sluprise fn,m the start Car a d f' . e antmomy was greeted with h i ' . n p e lllItely " , mamtams. WIth relation to t f' el'turc course "Begriffssehrl'ft I" h .. ' d' , t at at the e d f h In Icatedthatthenewlomc towh' hh h d' n o t e semester Frege . t\'. , Ie e a Introd d nmstructlOIl of the whole of mathem' f "(C uce us, could serve for the re'll, ,I ' h a ICs aroap 196~ 5) B ' a ) (. allne~ e could construct the wholp. of m . , p. : u.t If Frege geometrY-Wlth the help oflol71c th' Id I athemancs_l.e. mcludiTw , M t\'-' IS cou on y m ' h " In athematics" (p. 135), that in arithme '. can, In t e sense of"Logk arc logical. 1:\ This says nothing yet aboutt~~andgeometrythe methodsofproot' this is a mat[er ofthe nature ofth ' e nature of the two disciplines ll~ . e (Uwms Carna 1m F ' , , 109 about this as he sub I ... p ew rege s way of think, "sequent y ",te d d h 1 Mathematics" himself' wh F .n e t e eeture course "'Lome in , ere rege exph·tl 'h ty on the represemability of mathe.n t: I' del y'. ng t at the outset, casts doubt I ,. . a lca 10 uenon th ,,,. f , HI purely lomcal terms "Ber .~' ' . nou II"1 In d et ' "' . e In erence from n to n + ('a I law. Wi[h that, Frege had fri u IOn IS t.reated here as a nonlo m" 1n matht'maticc ar'th~-venupacoreeomp fh' ,"gr,lIll. ' onent 0 IS own IOJncist pro, _ ' ". I mt"tl(.' as well ' 0· Insofar as inlt'rt'llCt>S OlTur in h th C· as geometry, there is logic only Russdl',!oi til rnl of log!i('i~1Il Ii, 0,,, ~rnap. on the other hand, adopted 'AT' , . ' .l. (lilt t lat Included . . Illll~!i (to.g. Carnal) 1(22) and .' . , geometry) from his earliest mamtalllcd It to h's }, I:.! S .\.. .. I ast years (e.g. Carnap >
•• ~~ ,} ,",0
Introdl!{'tion to L
n, S"t" also Frt>ge's 0
PI ),20'lt') . . .
.
.
. " ogtc, Frt"gc 1983, p. 210' Fre!e 19 wn notes for this lecture Co IF' g 79, p. 193. . urse rege 1983 21 f ,pp. 9.~ Frege 1979,
Introduction: Frege's Lectures on Begriffsschrift
7
199~, pp, 137-39), He also does not seem to have regarded the question whether geometry is included or not as Olle of any importance. it may thus have seemed natural 14 tor him to project both these views onto his teacher as well. In any case Carnap later repeated the claim that Frege never gave up his logicism, even when he was specifically asked about this. I;; Meanwhile this claim has heen refuted by the publication of the Posthumous Writings. But Carnap's Obstinacy is easier to understand when we look at the lectures translated here. Apparently Frege's silence ahout the antinomy, in the lectures, led Carnap to the premature conclusion that it presented no problem for him. 16 In fact, though, Frege had already quietly drawn the consequences and eliminated value-ranges. On the other hand, the lectures confirm a conjecture that was already suggested hy some passages in the Posthumous Writings -that Frege did indeed withdraw value-ranges, but continued to regard attributions of number as statements about concepts. And this deserves our attention, particularly with respect [Q the newly awakened interest in f'rege's philosophy of mathematics (as manifested, e.g., in the recent writings of C. Wright, M. Dummett, G. Boolos, and R,C, Heck), At the end of "Begriffsschrift I" Frege returns, in the section "numerical statements about a concept," to an analysis he had tentatively put forward but then rejected as a definition of cardinal numhers in Foundations ofArithmetic (Frege 1884, §§55ff,) - representing attributions of number of the form "the number II uelongs to the conceptF" as second level concepts. Carnap seems on the uasis ofthe lecture to have taken this conception as the core of Frege's logicism (Carnap 1930a, p. 21). He even takes over this idea himself in his presentation of lObricism. and repeats it even as late as 1964 in evident agreement (Carnap lYY3, pp, 1~7f.), Fr~ge's original ohjection to the analysis from the Foundations of Arithmf'lti.: picked up again here amounts, as is well known, to the fact that the 14, Especially given his unreliahle nH~rnory; in the uriginal. unpublished version of Car nap's "lntellt~ctu81Autobiography." ht> notes that this was a reason for giving up studies in less systemntie s6ences: "I would soon give up studies in these other fields, partly because of II love of systematization, connection, and general explanations, and also because of the fact that my memory is quite unusually bad. (Once a psychologist told me that I should take this fact as a blessing in disguise, because a too great familiarity with old ways of thinking is for many an obstacle to finding new ways, andmy sometimes total forgetting of old ways might free me from this obstacle.)" I am grateful to A.W. Cl!TUS for providing this quotation. The manuscript is in the Carnap Papers (Manuscript Collection 1029) at the Special Collections Department of the Young Research Library. University of California at Los Angeles. 15. ASPIRC 086-13-05 (letter to'T.W. Bynum of 4 April 19671. quoted by C. Parsons (1976. p. 274. note 27). See also Bynum (1976, p. 284). 1&. This is thp vit>wtakt>n by T.W. Bynum (1972, p. 48), relying on Carnap, wbowrites, "k, late Hs 191;l-14 he was presenting lind defenlling bis hlgistic [!'oie! programml' in OHlrSt~!,i at Jerm University, '" Also on p. SO, "Tht~re is it widespread myth that Russell's Paradox had left him a disappointed and broken man~ hut actually, at least until lC)14, he helieved his logistic programme had ht*"n carried out successfully," This interpretation was already questioned by Parsons (1976. pp. 274f.). The present puhlication eS6entially cc)rrohorates Parsons's conjectures.
8
Gottfried Gabriel
numbers understood this way can't be regarded as independent objects. For this reason Frege finally ends up introducing the cardinal numbers as conceptextensions and thus as logical objects (Foundations, §68). Once this ronte was m~de impassible by the antinomy, the question remained uppermost in his ~lOd wheth~r the numbers could he understood as objects at all. Fregc puts his hnger on thIS problem most precisely in the "Notes for Ludwig Darmstaedter": Since a statement of number based on counting contains an assertion about a concept, in a logically perfect language a sentence used to make such a statement must contain two parts, first a sign for the concept about which the statement is made, and secondly a sign for a second level concepl. These second level concepts form a series and t~ere is a rule in a~cordance wilh which, if one of these concepts is gIVen, we can speCIfy the next. But still we do not have in them the numhers of arithmetic; we do not have ohjects, but concepts. How tan we get from these concepts [0 the numbers of arithmetic iu a way that tan not be faulted? Or are there no numbers whatever in arithlUetic? Could it he that the numerals are dependent parts of signs for th"s" s""und I"vel concepts? (Frege 1983. p. 277; Frege 1979 pp.
25h[)D
,
•
Th~ idea of introducing numhers as ohjects corresponding, as it were, to con(:(:~ltS of s~~ond level i~ one t~at is, revived in a different form by Frege, despite hl~ skept~clsm about it at thIS POint (1919), in his very late attempt to justif
anthmeuc on the basis of the ....geometrical source of knowledg "H h~ h' b . . . c. ere c repeats ,.IS, aSlc VIew; ....A statement of number contains a.n assertion about a ('o~e~pt (fre~~ lQ8:~, p: 2QB; Frege ~Q79, p. 278), He points out, furthermore, ~l~~t3a nU~lh~r appears In mathemaut·s as an ohjett, e.g. the numlwr :r' ~Freg(' . ' p. 90,. Frege 1979, p. 271). And he concedes that it "seems" tharth(' IU IT _ Ical source of knowled " ' . . I:' ge on Its own (:annot yield us any ohjeers" (Fregv 19B:~ 299; Frege 1979, p. 279). What emerges from this is that ahhough }H~ 11 R) is 'Written:
-r=~P . . .. I dash attached to the The negation ----, P of a formula P is indicated by a vernca horizontal content line:
.,--P , ' a compI'ex formula , with dif.. Wlthm This negation can occur at any pOSItlOn . . F'nally the universal ' f rom d'f£ 1 , ferent meanings resultmg I e rent pOSIuons. quantification: 'I.1')
conjunction existential quantification
(Exchange)
Q-->P-4R . 0 f any nu mberofconditions P, Q, .-.. and similarly for any reordenng
(Transposition) P-4~-4~Q
.. . g one could instead inrN tbe · . an d SimIlarly for any at h er con dit,'onal posloon, e, . conclusion:
,
32
Erich H. Reck and Ste'o'e Awodey
Th.is rule is also taken to allow sirnullaneous cancellation of double negatl'ons as In e.g.: .
Frege's Lectures on Logic and Their Influence
33
P-4 Q (Cut)
~--~-~
Thus a more general instance might look like this:
P-4Q-4R
and similarly for negated Q, etc.
(Collapsing)
Fiually, there is the following rule:
and similarly in the presence of additio n al con d.inons . .III any pOSIuons. ,.
P-4 Q-4 R
P-4~Q-4R
(Negation) -~~~~~~~~~~~
q>(a) (;t'nt~raliza[ioll)
_
V.I 'I' (:t.) wltt'n'.1" lUay 1101
Ins. t'.~.:
f{)lIowin . w~ . " , . t'OndHlons: g Hch tan also be apphed in the case of several
This says thatPis either Q or -,Q, and is used in the Grundgesetze (§51) to prove "propositional extensionality":
(PH Q)
-4
P= Q
The famous theory of extensions provides a term {.r: qJ} of individual type ,for every formula qJ. Since the term {x : qJ} is supposed to represent the extenswn
,
34
Erich H. Reck and Steve Awodey
of the concept represented by q>. these terms are plausibly governed by the axiom:
Axiom V: {x: I'} = {x: 'If} .... 'Ix (I' .... 'If)
The well-known contradiction of Russell arises quite directly from this axiom. Finally, in Grundgesetze Frege also employed a description operator lX.f/', which was supposed to denote the unique individual satisfying the condition expressed by ({'. if there is one, thus formalizing the definite artide, as it occurs in "the x such that cp." This operator is governed by the axiom: Axiom VI: a
= (IX_x = a)
By omitting these three axioms and the corresponding machinery of propositional equality, extensions ofconcepts, and definite descriptions. the logic pre~ scnted hy Frege in the lectures may be characterized as the inferential part of his mature system; i.e, it is that part involved in drawing logical inferences, without tht· constitutive or constructive part, involved in building up logical objects. Lik... modern systems of logic, it can he applied [Q reasoning about various difti.'n'nt domains, hut it has no domain of ""logical objects" of its own to reason ahouL One might say, tentatively, that Frege has cut his system back to a tool for logical inference about other domains, rather than a self-sufficient theory of a domain of independent logical objects. 3. Outline orthe lectures
We now briefly outline the contents of the two logical lectures Begriffsscllrift I and II, to he called Parts [ and II respectively. The third lecture Logic in Mathematics is related to the Nachgelassene Schriften item by the same natTle (Frege 1983, pp. 219-70), and should he compared to it. Pan I contains an exposition of the conceptual notation. motivated hy informal considerations and linguistic intuitions. It makes no mention of axioms or formal deduction, hut instead focuses on expressing mathematical and other statements in the conceptual notation, A few rules of iuference arc given, and some simple arguments are formalized, but the systematic treatment of d.e~uction is given only in Part II, Part I also inclndes a number of topics familIar from Frege's writings, including the doctrine of sense and reference, and the classification of entities into objects, functions, concepts, relarions, second-level functions. etc, Tht:' first few pages of Part I give tbe basic concepts of COmenr and judgsrrokes. ,and the notation for rht> ('onditional and negation as operations 011 sentt'IIn's. f1wse are explained in terms of the possible truth values ofthe m o c.o p l1t'1lt sentences, i.e, negalion ---.P swaps true and false, and the conditional P ~ Q ~xcilides the case wbere P is true and Q is false, It is then sho\'\o'J1 how lhese ran be combined to express the truth-functional operations conjuncIOt'Itt
35
Frege's lectures on logic and Their Influence
lion disjunction. and exclusion ("neither-nor") in .the nowh-fambliliar way. The , , qwte , SImIlar .. IS to a mo d ern t re atment usmg trut ta es. t _ exposition -d etac h m ent consThe rules of exchange, tranSpOSItIon, . ' and cut are nex as sidered in turn, These arc justified by their preservatlon .of truth, under y. . conSIderatIon . 0 f th e trut h -vaIues of the formulas tematlc , mvolved_ Next, general inferences having a common form, like:
1>2--> 12 >2 2>2-->22 >2 3> 2 --> 3' > 2 . h ' d ea of a concept as a function. are considered. These are used to motIvate tel . b- d on . pts relatIons, asequanThe Fregean doctnne of objects, functlons, ~~nc~, d U etc. ,'versal tIues IS presente, n n the notions of saturated an d unsatura t e d e t Thus - negauon . of a general statemen tifieation is introduced as a way to permIt b - Iy by. using . one can adequately express t h e gener . al inference a oye SImp whtle variables to express generality. I>2~X2>2
- that statement expresses t h e gen eralized negation: negatIng for all x, it's not the case that I > 2 ~
I 2
>2
rather than the intended: it's not the case that for all.r,.r > 2 ~ .r 2 > 2 'h . . .. pt~rnHts The U1llwrsal qUllntJlwr t (,-I,atter t (I he t~xprcsst~d as: ---."iX" (.l' > 2 ~
;1. 2
> 2)
. , b comhined with the other logical h ther classical forms of Frege then shows how the quantlfler can e operations to express not only existence, ~ut ~!allso,~ e dO .. one"), and particu. d . Y·· d egauve ( an n . JU gment: umversal af umatIve an n . . ") He arranges these 1Oto I aff(" "and some not . . B if)r~sschrlifit § 12. He t h en ar IrTIlaUve and negatIve some ., " . done 10 egn '.1'the classical "Square of OppOSItIOn as IS U 3 d> 0 Va> d (-,' < lJ -f(a) ;1-8
, , f occu ies nineteen pages in the notefrom eleven listed premIses. The proo .' - ~ h 'ddJe of it. It is clear that i' hook, with the second gap 0 f lour pages. fight.10 , t e mt several intermediate steps of the proof are ml~Smg. e remarks ahout rigor in These detailed examples are followed sOtmtion for achieving it. mathematics and the importance of co~eeptua no a f functions in mathedi Ion of the nature 0 I There follows an cxten de d scuss . h ' . 's no m()rt~ ahout symho s , h ' that ant melle I. . h h mattcs. Frege concludes y saYlOg h mselvcs conclude Wit t c . . 's The lectures t d to YOlilor ' t han hotany is about mlcroscope . , h' he I recommen ' , questions, w Ie . words, .... I have now suggcste d vartous further reflection,"
r
--_.,...
:"'1
_
Erich H. Reck and Steve Awodf!1
38
III. The Influence of Frege's Lectures I. Frege's logical innovations
features of modern logic are already treated at some length in Frege's lectures. We find not o~~y a systematic treatment of propositional logic, including the interdefinahlhty of the propositional connectives and their truth-functional specification, but a~so a comple.te a~cour~t o~what we now call first-order logic, including the fu~ct1onal analysIs of predicatIon, the theory of relations, and the basic ideas of unIversal. and exi~tential quantification. Equality is treated as a hasic, fir5tor~er .1~gtcal r~latlOn. and it."i relation to higher-order quantification through ~elbm.z s L,aw IS presented. Also on the higher-order side, we have a system lOvol~Ing sll~ple types, prcsented as a natural hierarchy of different kinds of rl~II~;tlOnS, ~th types ~ete~mined inductively hy the typcs of their arguments. Fmally, lo~~al d~dUCl1on IS treated by means of a formal system, with axioms and rult's ol1l1rcrente that are deductively complete for the first-ordcr prcdicate calculus: and rna th ematlca ' 1 prools £' •. " and . the rigor orinformallolTica! 0" JS re1ate d to I ht, st ru tly fo~rnnl ehara,cter or such deductions, which can be used to represent .hun. Most of llws(" tOllles treatt'd ' 'ht ,' . : are' . in a qUI'te. mode rn way, Just as t h ey mIg hl ItIUIl ~lHrodllc~ory logw course today. " R(~s~dt~s reOllnding ~I!; how n~ll~h of modern logic was already in Frege, the ~t cture~ al~o s~rve as cV1de~lce Of, hIS continued serious engagement with logic after ht. learm.d of Russell s antInomy and duriug a p ' d ' h' h h b lished nothin on h .> , , ' ' CrIO In W lC e pu , ." . g t e suhJt.
- 2 2
2' > 2 '1::: 2>2
1>2 Both terms fals'"
'---
4
.
h- 3' > 2 • L J>2 .
the lower term fal,c,
. ".
4.
2
10.
One c?mponent ehangcs, namely the nne referred to by the tencr of generality. It is an object. The other componcnt refers to a concept: square root of 4. This conccpt n alv.ays need~ "saturatIOn" by an object. The COI/('CP' is il/Ilced ol"complelio
both true.
-y------_./ The form is always true
11.
sin 1
a" >- 2anda > 2 arc notrcall
sin 2
.. we replace the letter b ' Y prOpOSitions; rather the be . propositions." y a proper name (e.g., I or 2 or ~) . e~me propositions when \\ e call them "quasi-
sin J
sin 4
Here we have a conslant component as welt and one that varies: an ohjeet. Here, too, the constant component "sin" is unsaturated. incomplete. Thefimctio n is in need 01"
In ordinary language we sa y ..something" . instead of a.
completion.
if something squared IS . 4, then its 4 111 power
. IS
16.
sill 2 is an irrational
"
numbcr~ we call it the "valuc" 'varying
I'
In Contrast we have:
L
2: Ih 2' - 4
'----y---.-1 ! t"ouxht.\·.'
in 2" - 4 . , (i _4 " cI complele lhought. IS 1101 a COIll I t h thus",' peet ou~hl . f/lla.l/"propfI.\'itilJr/"
.
of the function for the argument 2.
objcct~ cnnstant function: (
):.
;s the vallie nf our function for the :ugumcnt 2.
,
'----------------.,..-~
2
4
4
"
0
---" 4
!(,
nl\ls 111\· hUl\'linli IS 1.:11IHpkted by II\\' ""r~1I1l1\'nL"
(()n~ shullld 1101 "011//1.\'"
,
.
.
till" vlllm' Ill' Ill\' hllKti(lll with the fUll\'Il\lll ltse!t,)
E very square ' onl a ' Toot ot 4 is the 4 th TOil ' Y combmatlt)n of 2 . t 01 In, su (oncept\'., n0 t 0 1'2 lholl~hIS.
17
16
on lOT .. op of the page, line' 6-6, Acco,ding to f,ege', di,rincri he""een "'m' m caDng ndeutendenl" and" f . [b . h ll d n}"leuers itshouldnotsayherethatagenerallet "" . re erTl~g ezetc ?e e •• . ' .. leT a m the expreSSiOn "0 2 ", 4' (above) refers. 11. Tup of the page line 7' In eN the object name is used instead of the name "square root 4 Frege's conception- For more on thlS ;" Na'hgdamne Sch;'!,en (f«ge 1963, p, 2551; compa« abo the cone>pondi,;g emarks on the eoncept "square root of 1" in the present text (p. 19 belOW)'fP.-hesu,mabf Y lunap' bh . ... t f 4" by means a t e Slgn or th a tevlated Frege's spoken expreSSion square rou 0 . C e square root.
!
~f "~hich c~ntradicts
"~4"
~o~cept lss~e.
66
Winter Semester 1910~1911
67
Frege's Lectures on Logic
Instead of "Wlsaturated"' or "in need of completion" we can also say: "of predicative character." If we say: "All square roots of 4 arc 4th roots of 16," then "square root of 4" seems not to
It suggests itself. then, to regard a concept as a function as well.
he prcdicative; but it only seems that way, since we really have the following: th
Scheme:
;'If something is a square root of 4, then it is a 4 root of 16."
I-c w(n) 'I'(n)
Consequently we should not link logic too closely to everyday language; logic is not only trans-arian, but even trans-human.
,= 8
quare rool of I is a )'d rool of 2.
ur 2
'f'jn\
~lal .
(univer,\"ul negaril'{' judgment)
20 14. Top of the page, lines 1-5~ By "letters of generality [Allgemeinbuchstaben 1." what is meant are Roman letters. The "earlier" refers to p.16, middle of the page, where it says. however;
L
'I~
"
10
•
" In p,j~ni('tlillr, th,· ju(lg1nt'nt stroke is missing, contrary to Frege's (only emphasi-,.edl rule t~llil It mU.'5t hf' the-.re wht'n ft'~f'r81ityis expressed hy means ofa Roman l~tter. Compare (ynuul/l.l"w·(U I. p. 31. on thiS
lSSllt'.
especially note 1.
21
- L
'f'(d)
all
----==:::::::::~:::=-------3-~c~o~ntrary
n" ,\'f/nlt'
-....
\ J opposite contrad(leWry
~t"d21 ~trary
71
70
Winter Semester 1910-1911
all
contrary
Frege's lectUres on Logic
no
~contradictory ~ ~ contmry~ _ _ some not
-r&.-n-r- 2 and 1001--998> 2; in spite of the fact that 3 ,md (1001-998) have the same value, they have different senses. It is only by an
As tbe rnt:aning of a st:ntence we mus't t akc the truth value:
investigation that we find out they arc identical. The two sentences:
word
sense
thought -
"The Morning Star is not luminous"
meaning . is true
and
Truth is not. as. everyday language suggests. u
property of a thought.
b
ut
something
aclditiona1.
'The Evening Star is not luminous"
\ . ". since what is spondence a t' an I.dea with rca lty " .1 " Trutb cannot be defme d as carre .' h cannot be dcfincu, . h h t is suhJectlVe. Trut objective cannol be compared Wit W a .
have different senses. The thought
analyzed, reduced. It is something simple, basic.
'The Morning Star is the Evening Star" is the result of a special insight; in contrast, the sentence:
. , . h ~ Howing fundion: . Our bonlOntallmc stands lor teo function is the frue. th '0 the value of the " the False. ,; If I talo: as its argument the True, e" ",," " "
"The Evening Slar is the Evening Star"
""
difference between the senses. S{)
11
naml,: docs nol ,iust express the
uistinguish, consequently, between the "mc.tning" lind the ..scn.. . t.' ..
thin~ denoled. Ill".1
pnllK'r
"~
.>
~th.c, F~~~~cr true IWr lalse, or somethIng that IS nCl
is true in and of itsclC The tliftcrcllcc between the names thus cI1rrcsponds to a We
('unscqucntly:
n.\lllL'.
24
25
-,,.
75
74
17.
Winter Semester 1910-1911
sign:
proper name
sentence
concept word
sense:
s. of the n.
thought
s. of the c.
meaning:
object
truth value
concept
f~'s
Lectures on logic
nsaturated part: We can divide the sentence 4 > 3 into a saturated an d an u
sentence in indirect speech
thought
4
,; > 3
proper name;
concept word.
~··t· to o.
We can divide thi:; agatn
~ >
3 18.
The sign expresses a sense. The sign n:f(:rs to a meaning.
The proper name expresses the sense of the name. The proper name refers tu the object.
proper name;
etc.
5-1> 3 ~
5-t
~
"A believes that the Morning Star is a planet." In this case we couldn't just substitute in "Evening Star"'; inuccu. fhe thought here
s r _ ~ l ., -'"
function
IS
(not a concept)
precisely lIot the sense, but the meaning.
26 17. 'Top of the page, lines 1-3: In eN, the horizontal lines ofthc table are not drawn. 18. Middle of t~e page. lines 4-5 from top: In used for abbreVIatIOn.
';>3
eN. quotation
marks" 'I" are sometirn e5
27
C
relation.
77
76
Winter Semester 1910 1911
freze's Lectures on Logic
I1Je capifal A... afEngland _
r
?
England;
the capital of (analogous to 5~.g)
I-( level function: argument object
1II
II
a
-.-v-r a'•
2r>J level functiun: 2'''J len:l C\lnco:pt
I' ~ I
~n>O ~n3>O
(-I)' ~ I
2~
2' - I
3'
Ifwe want to ex.press that at most
Otlt'
ohJl:ct falb under a concept, we \loTit..::
e.g .. positive square root of 1: !hese 3 have something III
common:
~~(all
.;" -= 1, has the value: the Trut:, the True ' the False.
;2 has th~ value:
1, 4. 9
(~ I
another example:
';>0
b
~ ,,-+
2'''' len'!lim("l;fln: argument: Iimctinn .wlfi,l/i,'/f ('Ofll""f!f.\
I" lew/limclioll: argulllent: ohwrt I
~2~8J
'--
" .'
~
'
xx &#'*111\
_
0=-
1
i.I'1f /" /1'\'1'1 ('olln'f"
(I and 11:'11 Imd,'rlhi... nllll,:cpl)
,..
d.=ott
1 -- 2 II + 1 2
.;' > 0
29
-------------'
I AppendixA 1...:.:......=:...:......------------
81
80
frege's Lect.um on LOglc
Wincer Semester 1910-1911
Thl! Ontologficul/ PrrJ/!/ot!ht' E.u~t,'nn· of (in" 20.
"(jlxl" is somelimes a l.:oncept name. somcllme~ a prllr n name 19.
C\\slcncc
lh.tr,t":·
h J fCJlurl..'
ICT\,II~-'
"exists" is either a 1" I~yel concept .; Ii\CS n or a 2
e,)neerl
I Ih'
~ II 1
l..
l.jUL'~ll,)n
What
•
21
··ti,'>d" h,n ("
A concept is composed of characteristics:
eg.
'1'::"'=1
~ '"' 0 1 I 'M.juarc root uf I
I
2) ro~itiv(' Ilumhcr
•
F
I
"'tel
-'1'1 ... )
- \(~ I --- l( ~l
r1l1QCIU.:e Ullmot he ITlrhrde([ ,1\
;I
l'hmal'll'fl"tll
t"lllllTpt
i
AI 19. Top of the page, lines 3-5: What is probably meant is that the expression .. ex.ists"~; against its logical nature, combined with a proper name here, not with a concept ~.ahut '"41(;)." .For t.his also IOgJ.c~ly.,~on~enslC~ (unsmntg). See the examples "Africa eXlsts and e also magne eXists 10 Uber dte Grundlagen der Geometrie II (1903). p. 373~ compar. '-Uber Begriff und GegensLand" (1892b), p. 200.
Fr,~ge.
i~ noto~ ··li.n~sticallyimproper (spra~hlich.un~~attha.[~ho.rle
~2
.
. s thc (conjunct1Ve.~ bracket indtcatt; nee is a feature ·concept. -t curved 20. Top of the page, first tOTmu. , lao The f us Th us The no te ""elUSte . combination of the characteristics ,of the on the left. 1 ce of the questton belongs nlore under the sign for eXIstence eriod takes the p a 21. Five lines from the bottom of the pag e·lnCN.ap . 1T1ark.
I I I
I I I
I Appendix B
.••~ ... 't-...~
85
84 Winter Semester 1910-1911
f¥'s lecrures on Logic
Numerical Statements about a Concept:
:3>2: i>2; Pc;'; qJ(S'.';). . '=
The number ofuNeus fallinR under the concept is 0,
.
is
fthis functIOn
. . if for 2 argumen Is the value 0 This function expresses a relatIon, I tion to each other. the True, we say: the two arguments s t a od in that re a
22.
If2 things fall under the cuncept. then they are identical. . t nly one s t ...'>ods in for each obJed." 0g relation. the correspon In
There is at least one ubject that falls, under the concept.
The number ofobjects falling under the concept is J. H~re
we have relations as argumenls. 23.
The numerical statement concerns the kind of satisfaction; it is a 2"d level concept indicating the features of a concept.
Above we had concepts as arguments.
In everyday language we take "two" and "tall"III ~ ' . 'two ta II towers ",0 be adiectives of
Thus there are 2 kind~' OJ,,]"d Ie vel (:oncepts.
equal Slatus.
But: each tower is tall
1here IS y(t another kmd of2nd
. fact • a concept): level function (In
_ ,,(2).
not: each lower is Iwo, rt of the num rhe value of the lundlun IS the 1 rue If I ttlke a prope Y
o~icci.
Plato already realized thai the attrihute "one" docs not apply In the nut fhe concept; for example: the conccp' "chairs in the auditorium" ha,~ the feature (11"
e.g.,
~>
bcr 2 as ar~ument,
I
uniqueness.
BI
~hould
82
ofunique~e" (d. p.
22. Top o:the page, line 3, Ifwe follow F,ege', u'ua! rep,e,entation 29; also Grundgesetze I. §22). the upper part of the formula look like thHi:
~i~. L-..:iI'(') /1'(-)
belo~
Thi, (.) > - e .,b
b>O
pO
4
5
92
Summer Semester 1913
I I Frege's lectures on logic
93
I I I
A is identical to B if c¥erything that holds. for A also holds for B: and conversely.
I
--!rc'lM)a)
is. the same as
I-r- g (-&c Ira») L g(a = b) fIb)
u =' h; the tv.'o are identical. i.e., whateVcr holds of the one also holds of Ihe other:
(III
Ipp. 6-10 in CNaT'("cmp~'J This contains e....erything that can be said about identity.
E.g., as g we can take -';: 25. [H
Earlier we had:
~'(a)
L---.::: a=b ~ b)
(H,);- - - - - - - - - - - - --
In- fla) L:::: fIb)h (17
I »
ifM,(f(fJ))
~M,('(fJ))
(n.
Everything that's tnle for all 1·lle....el functions
is true for anyone of them. (ltta
«
We now apply this in the form 26.
, ~
I(a)
I(b)
,(,,) ~bl
11
~:.
"e.'Iier'~;:;" :: :J:i: ::-::rc
Six lin., from the bottom of the page. right column' Thee5 e I gpages lsee above), In terms of content, compare Crundg t!tz I, §
.
e
IIWhas the label "lIb." 26. Last line, right column: In Nc' the judgment suok.e in this formula is rnissing.
I
94
Summer Semester 1913
!-----------~
. Frege's lectlJres on Lolric •
I I
95
II
IlIa
I I I
I-n-r- Ila) L::= fib)
l1=:O
>
< We had Ig ~g(iJl-g(/,'l
ft g(<J11
t
= (--.- a)
1ft
Irl
t-r-
~ ,l.!(i1I-'!l(hl
I
a
(- Ill""
b
0>0 a>a
~
¢+b>(1
C.a>b C>O
(~b~ b) + b:> a
(a-b)+a>b
r
a>b tl-b>Q r+a>b
We also use: .... a>O
(H
0>0
l-r--a-b>O
(T
• La>b
a>b
a>a 1,>0 (j.> b r" 0
H
t -I
a-b>O
t+b>o r: t- a> b t:> 0
(lla
100
Summer Semester 191]
I I ~'s LecOJres on Logic
101
I I
I And we use:
~
32. (Ll): ---
_
~ -.,
a) 33.
(Id)::
a~b
t+h>a t+a>h
I
~;~
I
b>a
~~n,
I
(.1
I I
This is supposed to mean (~at
a and h arc real numbers, smce it is only for them that> is
'>0 b>a c>a d>b
(P
supposed to be defined. 31. (8)::
'+b>a
~ ----------
h-b>b • L(b-a)+a>b
t+u>h
0
-----------------------
~
b>a
b>b
IP) ------
(h -a) + h > a (h - a) + u::. h
(IIa)::
~ t
h>h
h
u>O
r+h>u r~u>h
roO
/8 32. Top of the page, first formula in the right column: In eN, the judgment stroke 15 . nu"ssing· f h
( )
33. Bottom 0 t e page: a Is by replacing ""a" by "b" and "'b" by "a. "Theinfer" Invoked . cnce involving (ld) in comes as~: fOllows: In the missing part (see above)., (Idl was apparently deduced the foabout em
a
This corresponds to In .the representation of lId) below , as well as in Grundgesetu I, § 49. (ld) is then invoked the Conn ~ (b'-a) +" > b (b- a )+ b > a (b-a)+a>b
thuHeplacing "b" by by" -rIb-a) sentence deduced invoking (Id).+ b) a." In eN, the jUdgment '''oke i. mi..ing in the
/9 I C'1I.1 t but-o ne forIl1u .. In J~. this ts nokein the las "d" by"! + a theJudgmen b "1 + b" an 34 Bottom of the page' I:r"t. C:!i;as been replac~ xi Also, when Invokmg (In erections in the e . replacement is effected by eor
d
102
Summer Semester 1913
103
I frege's Lectures on logic I
I I ,
, • b TI-:?,>a-f(b) r + a::> f(b)
f-r-I+b>a
b>.
~I+b>a
35.
l+a>h
0>0
(Ib
t>0
f-r-I+a>b ~I+b>a
l+a>b
tg f
{
(Id
l+b.>a l+a>b I> 0 r+h>a r+a>h '>0
This is the expression for the hmll . ' ofaIIJnction as the argument goes to infinity.
'li-;--,-------- a" b (IIa
b>.
.,>
L_-;-~===== b r > >0
Here we also have to
assume as known"
I-- I > 0
(E
b- feb)
,+ b > fIb)
.
b>b ~----b>O ___ _ _ _ _ '>0
(I
Thboth I . If q.e.d.
Thw;wc only nceded the sim . rk scntcnL·C.~ ,I H.
a
IS
u
and h arc r,m,t, '. a' th, ",gument go", to po"t;vc . ;nHnHy. then 0 and b cn;nc;de.
I;.'hal
w('
W(/'" I(J 1'/'0I '('
21 35. Top of the
36.
t>o -feb) ,+a>f(b)
20
!e~dap,
Ih, poge m \the pag" In CN. the jUdgmen"troke i, mi"ing in the<eeond fo,mula it on 36. Botto f becan" Ftege only 'ketehed the p,oofh ete withon' ,pelling e out. The whete lodgment 0 then have th"tatn' ofan "analy,i,." E\5e • hnwevet, Fteg u""he IUk, in G moke m ,uch ca"'"' well (fo> ""ntenc",")- N> no exception i' at i>Vue hete n ood p. 94), the judgmen,,(toke ha' hee added. Likewi" fm the "C·
tom""
fo,m'~nd8"'tzeI,
u a on the next page. "'ond f 37. Bottom of the page again, In CN, the tepte"ntarion of the lowe" loWe< letm in the on n) th, i only hinted at. Catnap ptovide> (with "fetenee to the lOWe< te<m ahove . nstructlo 'm,tead of a , b." In h" . 1atet h andwnung. . enth . I"dd,d' .. . n, ,. th e ..me. t he 'm,(tueU. tie tefe' and on,tead nfo, D." Again in hi' latet handwtiring lin par "'" with deiem"n ,enee tn hoth diteerion,). thi. temark folloW", "Note the diff"enee!" What he 5 IS the difference between free and bound variables: cf. the next note.
otmul~"
37.
104
Summer Semester 1913
105
Irejo\ lectures on logic
Earlier we already used:
Here we need the proposition:
t--(a-b)+b-u
(A
--.-IL
38.
..
I
L....!-.
l-
A
e+b>a
e+a>b
(1I1e):
• ~ el2 >a-l(~) e/2 + a >/(111) .>m d>m ~ ~ el2 > b-l(~) e/2 + b >l(.) bn d>n
•
(a
J-c
m)
j
m
(l
i.' •
(e
+ h) )-
LJ
e+(e+h»(a
m)+m
~
(Z) .----------------------
t-rT- e+(e+. b»a
L
39.
We also "",ant 10 use:
40.
(:+d>m+n
c>m
J:--n
(Z
e>(a-m)
(e+b»m
41.
(lib): - - ---- - - - -- -- -- - - -- - --
~
We also need an intermediate proposition:
e>a-m e+ b>m (1"+1')+ h=e+(e+b)
l- (p+
(H)::------t-rT-(ete)+h>a L('>a-m
1§(e+el+b>a (e+e)+u>h e -"a m t' ~ h "m (' .> h m (' Ie II
(e+e)+b>a
(' t ".~",
q)+ r=p+ (q+ r) (H
42.
(r
(Ic):
_ (Ie
~
-'"
t: :::~ : ~ ,~~ f'
m
'"
~--I'I" 'm
___.__ (0"
22 ~8. Top of ~e page: In eN, the followin r e . . . Carnap 5 later handwriting: "Dis . g ~arklsadded to the rust formula on thIS page. been used for the fraction or~ divided~h d and " .. For graphic reasons. "eI2" has In
I
d
I If . ,.
23 . ked by a line of dashes 39. Top ofthe page' In CN the inference involving gIlC)d;e:::e I. § 14. The formula (inste~ of a conti~uouslinel; CQ~poo:e. h.owev~:.+(;:nb»~. a; (a - ml + m, and b: a. (HIe lIS used after making the 6ubsUt.Uoons.jl ~ ). . . ns C • ~ d' e + b. aki the subsotuOO ••• 40. Top of the page, next line: (Z) is used after m ng i'IJ: a - m. and 11 : m. . ked by a continuous . . (IlIa) 15 mar u1 (III I ~1. Middle of the page: In CN. the inference Invot;1D~setz.:e I. p. 26f. The fo:m a a ~ne (instead of a line of dashes); but compare Cru ~e + e) + b. and b : e + Ie b). IS Used after making the substitutions jI,~ ): ~ > a. a. endy deduced i~ the 4.2 . th :-ht column was ap~ h substitUtions: .' ~ottom: The third formula (Ie) in e used after making t e 1Jl.1SS mg pan; Gru.fldg~setz.:e I. § 49. (
cr.
61:(e+r.)
1"
i:(i5
b>a. P: (e+e) +a> b.
_ _.. t ....··•. . .
106
43.
Summer Semester 1913
107
Frege's Lectures on Logic
(yle:
1§
~:::l:~;~
e>a m e! b>m e> b- m e-+a>m
(15'
---0I-
el2 +el2
=
e
I-
44.
n~:~~b
~(eI2 + e12) + b > a
45.
(o}
_____
(ei2 + el2) + a.> h
1§
46.
lIa:
d b>U e+a>b eI2>a-/( el2 + b t-/2 > h -f(d) l(d) e/2 > h-/(t/)
================ ('+o>h f'-'2 > a-f(d) ('/2 + a >,/.,( u - fed) e/2,'>a-f(J) el2 .- a >((d)
n
e:2
.>
('/2
+ h> I(dl
f'(d)
~ »
e+h>"
(Id
47.
~d2>(I-f,(.g)
..
"i2>u-/ld) el2 + a >j(d) d>m el2 > {/-f, (D) el2 .j 0 > (I)
13.
D>m
v
J
We suppose Ila to bc in this fonn (and similarly again with hand n).
---0--We suppose hill) ha ....e Iha IS used after makin gthesubstltunonsa:el2+el2,b: .. ';+a >b
45. Bottom of th epage: (oj is invoked a f ter the sub· . stltutlons e: e/2 and m:f(d).
~'."".'.,. .
~ ..
,
25 46 romWh the d~' I t"he 'form of. (ld) invoked here, the additional negation stroke comes r· olefact page' 47 T at b has heen replaced hy" --r- el2 > a - I(d)."
r.:
i~ .replaced oPofthepa b· thge.. I n the fIrst . transition involving (Ib) and (ld), the highest lower term terrn or lId) e lower term of (Ih) and the lowest lower term is replaced by the 4ft B . e [wo new lower terms are identical and are, thus, "fused."
h
~'b~'
low~r
. Ottorn "pieced b f h e page' In the ,econd tran'ition involving lIb) and (1d), "u" i, H," to be ""If, 'he I:we 'clat"e to the p,evioU> fMm of (Ib) 'nd lId). By mean' of ,he ,,'mition "" "pl"ed I" lowerte,m i, ,j(bl b>n
(l.) m>n:
/) f-c
m+ 1 >m m>O
K):
~
26
11>11
::. 0
m
n
27
49 Top orthe page; The form of (lIa) invoked here results from lila): Io---flu)
~f(.)
by r~placing "/( ~)" by the function (listed in the right column) ~ ell ." (/-.fl~)
L...= t.':'2 t
In.
The second time (llal is invoked, "a" is replaced by "b" and "m" by "n," 50. Bottom of the page: In eN, the two generalized lower terms of the last formu Isbondr.the to . ." 1nstea.d, arrows and the words "same as a b OV~. "refer f a tthe page are not gIven exp I·1C1uy. the corresponding lower terms in the previolls formula (top of the page). SImilarly 0 two genera.li..zed lower terms of the rUst formula on p_ 27.
____ 7
"'~
110 Summer Semester 1913
III
Frege's lectures on Logic
----------------
~
51.
m
.• I>m m+l>n 111>0 m"'>n
(I): ----------
~
Ie:
m .... ! ·>m
_
(Ira) .
~
~
m>n m>O
8
5;1.
mTI>m
m + 1 >n m+ I >m
I1a:
_
m>" b>m b>n m>O
~
That takes care 0 f Ihccase m>n
m+l>m
m+ J >n
.. () By transposition c oecomc5:
b>m
11>/1
-.---
-----------------m= n
b'm I lIL~ - Lb." ----m·(J
l-:::===~= m
~
b>m II >- n ) el2 > a -f(b ('/2 +0 >(t1) b>m ...) el2 > h-'J(~
a'b
a>h h>a (' " 0
d-Il
('/2I-h>f(b)
in the li'flll:
-:. m
I I
J
I .
55.
e+b>a e+a>b
'mII
II
n
d)
(II
---.---
tn-rl>n
.tt
b
a
h
-.---
::;>m
m + I >n m'>O
54.
in (he form:
m'l>m
m+ [>n
>/t
• 11/ 1/
-.--28 51. Top of the page, right column: In eN, the two lower terms of the form 0 hm a<e only hinted at. f (Ie) invoked c
52. Middle,a:right The form of (lIa) invoked here results from the followm stitutions; m + column: 1 and . g sUlr ~';;:.m
29
.... befo," the "has been r eplaced by n . . (I) twice here, m 54. Top of the page' In mvokiog d g,"phicolly fm
n -t 1 :>
11
n>m n>O
~ ~~i>m a
Ie:
n+I>11 n>O
_
30 56. Top of the J?age. right column: In eN . cd here are only hmted at. • the two Imver terms in the form af(Ie) mvok
31 . hoW to reconsttuct the cont~:concerning . Ior t he case m > n abo . 57. ~oncerning the empty pages: For gul.d a nee sODding deductlon llaUon of the deduction, compare the cor,re ~ Iso missing. The initial part of the subsequent deductIon 15 a
57.
115
Frege's Lectures on Logic
•
el2 > a-((iI) el2 + a >/(1I) 1I.:-o.
•
0.>0 e+b:> a e+a:> b ei2:> b -fib) ,/2 f b >(b) 8>n 11>0
el2 > b -f(b) el2 I- h >f(lI) 1l>11 n>O e+b>a e+a>b e/2 > a -f(D) el2 + a >((ll) 8>0.
0.>0
35
116
Summer Semester 1913
58.
117
Frege's Lectures on Logic
T1a: I-r-:----./(a)
~f(·)
• •
~ •
(IIa) ,-------
_ ell> b -fIb) e/2 + b>f('b) b>n n>O el-h>a
e+a>b e12::. 0 Pa-flb) t + a >f(lt)
b>. 0>0
pO
'T-Tr~·h"~TT'--
a
e+b>a e-la>b ..,.,c.rT1~",rrr e/2 > b . f(b) e/2 + b >I(b)
~>a-f.(b)
b
~+a>flbJ
b>.
b>. .>0
0>0 ¢> 0
lIa: 10-",·
L__-====== pO
•
•
.::=====
L__
:;:; ,j\:\} b>o
a- 0 t- ()
6C
€I> ()
0 e/2> >0 t>a- f(ib) t +o>/(b) b '----0>0 '------'>0
h-r-------,nro
>.
,
• L
•
Q>D
St).
e+a>f(b)
([Ia): - - - -- - - -- - - - - - - -- - - - -- - - - - -- - -- ---
el2> b - fib) el2 -to b >/(It) b>n e+b>a e + I.J:::' h e/2 ;.. ()
rrrr, e > a - !(It) b>.
e!2>a-f(b) e/2 + a >j(b) b>d
_---'====.
L
ell> 0
•
•
el2
This" has nothing
L----=_ _•
)
b>.
Q>O t >0
b is limit for function [with ar~~e~t] going to mhnlty.
e>a-~b)
o '--
to do with that R.
e+ b>a e+a>h el2 > () pb-f(b) .+h>/(b)
e+o>'[(b) b> (I .
aislimit
11>0
e>O
~
already lillishcl!
36 58. Top of the page, first line' Th
·erepreset' , conceptual Dotanon . .In eN " [m h ere on, more and more J'ust sk h n aOon In IS, ro · .. etc ed and conta" . . . h ' dgmen t strok e IS mIssIng in the fi fi Ins some mIstakes. In addinon, t e JU lI'st lye formulas ( [th d -. I
59 Bon f h o e eductIOn In the left column. ' . omo t epage.rightside:There 11." . .. cerntng the last formula means th rnar TIns Dhas nothing to do ......i th that 0 co nfrom the scope of the "0" in th I at the scope of the "0" in the upper term is different e Ower term Ie[ G .. " . rundgesetzeI, p.13); similarly for 6.
37 ·ouS page, right column), "a" "' (lIa) here (c [ . preYl . 6. O. Middle of the page: "When lllVO......ng lsrcplaced by "b." .. _ I" 't" concerning the lowest lower . Th emark a 15 Iml . . f the remark (';onfiI. Bottom of the page, right Side: e ~ Ie men ted along the hnes \) . term of the second formula has to be s pp f:l;rning the lower term above it.
6.
IIa
119
Summer Semester 1913
IA)"
------------- -----------------
I-r- e/2 > 0 . L e.>O
'Tr-------,-,n-- e + h > a
frege's lectures on Logic
(A
e I- a> b e>O ./Torr- t> b -fib) b > fIb)
11), --- --- - - - - - -- - - - - - - - - - - - - -- - - - -- - --
'+ b>11
L_-=--.:::=~=== d
IL,
. C
>0 >0
• ~b
\
'"'IT'--'T,,:rrTT-r , > a -fib) r+£1>/(b) 11>0
tab
'""' ~. \
.
L__~===== 1£>0 > 0
G==h
~
I' L
I.-
t> b -fib)
,+ b >f(b) b>1l
.>0
r>O
I' L
L--
t>o-J(II)
,+a>IIO) b>Q
.>0
0
, ,
•
r+b>a t+a>h
This is the proposition we wanted to prove.
r>o t> b -fib) + h:>l(b)
t
'ntcnce that occurs in it alway~ contains The deuuction is so complicateu because every sc 'Th" .' impOrlant for ., ., ' . • assumptions in mmd, IS IS all of Its condltlOns; one does not Just keep ,
b>.
,
•
0>0
r>O
t>a -((b) r ta 'Itb)
II .' II
Q.> ()
t ' ()
We proved carlkr'
the rigor of proof.
I)~"b It It ! II ,(( It I
(f
>
II
r ·0
38
39
121
120 Summer Semester 1913
Frege's Lectures on Logic
Mathcm I and
I." Thus we have 3 sentences; thus 3 suh-concepts that can be regarded as characteristics of the concept prime number. (It does not matter that u is not in subject position in the 1sl sentence.)
can a so occur in other s
something saturated the 2"" ,
IS
Analogously for the definition (~la relation; the only difference is that we ne-ed 2
entences; they have a sense in themselves. The I >I
unsaturated
letters [Buchstahen]. A relation is contained in a sentence in which 2 proper names
We can say several things about . " . one and the same object, also put together in one S IS a .," , POSItive number and a yd root."
uccur, e.g., J > 2. A relation name is doubly in need of supplementation
sentence:
A rC{luin.:rnent for any dcllnition (whether of a concept or a relation): It has to hold in
gent'ral withollt qualilil:atinns; or if qualified, then 2 definitions ha\'c to hold; the
. the case . . the staled for III wtllch
tid
I'· . . ', .. ,.. r. d Otherwise it can happen thaI qUcd ooly for pusitive whole numhers,
\l1lC
chooses a diflcrenl sign for it. e.g·. ---.. .
21
., back to
In any
CO". thm alwuy' hos to he a comptete .,yst,,,, at hand that;' lugicolly unproblematic.
20 At
(thi~ is an empty
function.g _ ( also for objects other than numbers, e.g,: If both arguments are not
lt~n
t
.g>';
concept nothing falls under it). Functions have to be defined everywhere from the beginning; for instance, the
r
eN. an arrow points from the word "h ..
Functions of two arguments that always have a truth value as
> 0 (the concept of positive number). Or we form the concept
One more thing concerning the eonfiuSlon . of object . . and fi . Its value: The value of th" _ unctIOn, or of a fum:tion and e [unci IOn I !.; .,~ r.or a II arguments (at I 1 H • . ut on..:: cannot say th I" ' , east lor numbers! is at IS the funcllOn. Since e ' . the point 2. we h'lve to I ' .g.. In order to differentiate say at , rep ace the argument position wi h • (2+4:)) (I +2 2) _ I 2+k etc., so (I +(2+/i)Hut In "["there .IS no argument po 'it" b h y 2 + k. Thus even the f ," s I at could be replacetl . uncllons which arc called "CI .• ..' contused with their valu' h'h" msldnl 111 Analysis arc not to he c. W II.: IS an object.
90. Six lines from th e top: In 2-:\}:.!," 91. Two 1" f
(~_ ';).
value are relations. Therefore we can transform the relation';:> ; into a concept, e.g., .;
. F. A_,function of 2 argu menls .IS fundamentall d·j·j· I erent from a Cu ,.g.• It I saturate one a r g ' .. nc IOn 0 I argument. ument positIOn in "c; - (" I et a f . only a second saturdtion I d ." g unction of one argument, and ca s to an object.
y
S. can be transformed into a function of one
argument in two ditTerent ways: either by a saturation (.; - 2) or by identifying the two
But (I -I 2tl isn't th c fu nctlOn . of a function R h a particular argument" e g (I 2 . at er, it is the value of a function for , .., + '3) can be th . If I designate th .. c argument jm the function ¢'c (I 2·3)'. e posItIon of the argument by J; the I h that is the vI
t
95.9 97.
If a is not:> b, and if a is not = h, then b:> a.
To prove: If L B > L A. then AC > BC
A sentence that is supposed to be an axiom has to be true; a false axiom is selfcontradictory.
94.
160
Summer Semester 1914
Therefore:
If not AC
:>
Be. then not L B > L A
161
Frege's Lect.ures on Logic
(5
If L B > L A, then AC> Be.
-.-
[',5'
If L B > L. A, then not Be > AC
(X
A, J'
. not Be ~Ac., . ' then AC>BC. If L. B > L A and If
(p
2',4'
If L B > L A. then not Be
(F
f.l, v
If L B ;. L. A. then AC > BC
=:
AC
This is what was to be proved,
(Here we have repeatedly used "transposition": antecedent [Bedingungssalzl and consequent [FolgesafZ] are interchanged, but both in negated form. (The English
. The difference between an indirect and a direct proo f'IS therefore not as big as
logicians call this contraposition.)
usually assumed.
Thus here we have used the false Sentence "not AC
:>
He," too; but not in itself, only
as a condition in a bigger sentence so that nothing has been said about whether this
.
. d not draw any conclusions from a false As we have seen, in an indIrect proof we 0 . as assumption; sinee we do not actually have the falsehood as a premise, but always only
condition is satisfied or not.
the condition in a conditional judgment.
The same proof can also be arranged in such a tonn that it looks like a direct proof.
Ncvertheless. in mathematics people have
.
. de cd tn.ed to draw conclUSIOns from
III
We make use of the following sentences: false premises. e.g., in non-Euclidean geometry. . ". . 'ne the other is intersectcd by It, too, "'If one of 2 parallel lines is Intersected by a Ii , . lIel . t" I f one does not use thiS para or ' "'There is only one parallel line through a pOIn . . . . ~tisn~ . , , cl what can be proved without It. t . aXIOm for the moment and asks mer y . " It is 'd without lIS111g I . i ich ~cntenccs can h c prove nhjeclilmanlc. One learns. Ihen, w 1 ~ , . ' that 1 Illstead an aXIOm if one assuITIC.~ mlOl hcr ;IXIOll . , quile" different Imiller. IHlwevl'r. t
]') I[not LA >/.8, thennol Be >AC'. 2")
!fllm LAc-LB, then not Be "-AC.
3') Ifoo! Bc" >AC andifnol 8e -A(', then AC >BC 4') If LIJ >L.A. thcnnol LA :'1')
LIJ.
Ir / H ., L A, then not / A ~ / H.
. .. rl' we have tn {lojcct· One can only lakc a me Clllllradil.:ts it Against Ifllll pltllcdll I w.e it as IhL" , "rt"ltl,.·c 11IIllufalscl.lI1c"!hulony . Ihllll!J;hl hI Ill' Ihl' prt'lllisl' ill Ull mIL:. . ' . 11 dc-rivc-d sclllcnces. " {;\'Ildilion is thell CIIlTlcd llillll~ III a Clluditioli llllll:lllilliliull
"8. Bottom ofpage; This pIcture IS presuma Ysuppose. l' d me that a is axiom; "Assume that A B Care three points that do not lie on a mean assu AB'In a . ' , C If h I' a meets the segment ahneintheplaneABCthatdoesnotmeetA,B, t e segment me S·C"m a pOI·n,"IDavid . AC = the . pumt, then it also meets either the seb'1Tlent or. . h h ,"on~ .. ~,·tjon of thl' 114 Startlng"'lt t f' Cl. "r." .I . A_' der Geometrlf~, . A-ionllI 4 • Fn~g" H 1 bert, . (;Fundlagen . . n.J\.10m I" d under the same num her, J'Ul tex.t thIS axIOm replaces another axIom 15 e . . ) USes the latter axiom in his Nachge/a.Hene Schriften j 1983). p. 246.
.
.
'
.......
_,
t
Summer Semester 1914
164
165 Frege's Lectures on Logic
.------------_.-
These arc pseudo-axioms; since the expression "between" is nut dctcnninatc yet. It is analogous to presenting a number of sentences as the definition of a number, as
.
If par stoud for [hezeichnet] a re 1atmn, we cou
follo .....s: il 2
-
"If a parb
4 II is still doubtful here whether a numher has been defined at all:
and a par c,
(/ < 6 whether there really is such a number, or perhaps several.
then b
etc.
Thcsc "axioms" arc, therefore, really definitiuns, and not even uniquely determining.
c. for any a. b, and c." . . . her true or false. E.g.. it IS tme if we lfwe replace par by a relation, thiS sentence IS ell . . h' '":I"d level concept. namely Ihl:: replace par by .'=." Thus here we are also deahng Wit a ~
concept of relations
This is further nbscun:d by the facllhal for us the word "between" is not new,
thus:
".-I pat
Compare:
U tar (''';
nd
U
IUO
h I f between three object\ (In In "B pat A tar C· we arc dealing wit a rc a IOn . . . , . " , a fundion of three arg.unll:nls who..c . [traditional] logic: "relatIOn With 3 bases ), I.e, , lation as thaI stand.. to iI concepl I value is a truth value. (It thus stands to an ordlOary re
lcvcll:onccpL
"If a is a
"in:' so as tu . If 1'1 level b t also different from th c casc ( bUI
concepts.
"lfHrat A lar(",then Bpat ("!lull."
I len: we arc dealing. with a 2
h· h e,g., the identity relation falls.
10 W IC ,
. _ . indicate that thiS case IS analogous to,
"A li...' s hl'lwccn Hand C" say
.
0:=
.. d r" With respect to 2 nd level concepts we don't want to say un e
('llIlsCqUClltly it is hettcr to replace it by new words:
\Ill.'
Id fonnulate the sentence .
v~
andhisalf', then u -,-- h, for any a and h." Here we have the 2nd h:vcl concept "conct=pt of concepts under which only one object
e
More predscly. Hilbert's 1'1 Axiom says: . . . the line detennincd . I If 8 IS a pOInt on "If A is a point. if C is a pom . .. tar C. then 8 pat C tar.--l. is different from A. and if B pat A
h~ A and C.
If
falls,"
30
31 L
99. Eight hnes hom the bottom: In eN. Carnap has WTitten in his lat~r handwriting "this i!> right (.w richti/{I:' rt>pt>atcd rhl" word "pat." and drawn an arrow to an occurr("n~(" nfir furtht>r dl')WT1.
eN this sentenCI' i .. mar ..#' 100. Ten lines from the top: In "in" in the previous l'if,:nten~e,
'
J
a~
a
('onllllt'nt (,n
th ...... urd
166
Summer Semester 1914
Literature Cited
And this is su pposed ' to b e satisfied whatever A, B, and C may be. The whole thing has a sense only if pat and tar have a sense. Later Hilbert not only uses .. lh e word ..hetween" with a
d'l~ I teTcnt
. meanmg, he also
often I " ... differently . . uses "point" ' "I'me,."" pane, from Euclid. What is unclear, then, is this: • • makes clear how else he understands them. , he never says so ex.plicitly and h·c never Often he
lISCS
the ex.pressions as indicating indefinitely , J' us'1 as we usc Ietters.
Awodey, S. and Carus, A. 2001. "Carnap, Completeness, and Categoriciry: The Gabelbarkeitssatz ofl928" Erkenntnis 54, pp. 145-72. Awodey, S. and Reck, E, 2002, "Compleleness and Catcgoricity, Part I: Nineteenth Century Axiomatics to 'IWentieth Century Metalogic" History and Philosophy ofLogic 23, pp, 1-30, Awodey, S. and Klein, C" eds. 2004, Carnap Brought Home: The Viewfrom lena, Chicago: Open Court, Beaney, M., ed, 1997, The Frege Reader. Oxford: Blackwell. Bynum, T, W, 1972, "On the Life and Work of Gottloh Frege " in Frege, Conceptual Notation and related articles. Oxford: Oxford University Press, pp. 1-54. __, 1976, "The Evolution of Frege's Logicism" in (Schiro. M.. ed, 1976), pp.279-9 9 , Carnap. R. 1922. flu Raam: Ein Beitrag zur Wis.senschaft.slehre, KantStudien. Erganzungsheft 56, Berlin: Reuther & Reichard. __. 1927, "Eigenlliche und Uneigendiche Begriffe" Sympo"ion 1, pp.
32
355-74, __, 1929, Abril'S der Logistik. Vienna: Springer, __, 1930a, "Die ahe und die neue Logik" Erkenntnis 1, pp- 12-26, _ _. 1930b. "Bericht tiber Untersuchungen ZUT allgemeinen Axiomatik" Erkenntnis 1, pp, 303-10. _ _. 1932. "Die physikalische Sprache als Universalsprachc dec Wissenschaft" Erkenntm:., 2. pp. 4:~2-65. _ _. 1947. MeaniflK (Jfl(i Nt.It's,'lit)'. Chit'IIp:0: tJnivt'rsit)' ofChi','ap;o !J~«.~ o __. 1956. Meaning and Nf'C('8Sity. 2nd ('tl.. Chinlp:o: llnl\if'rsuy 01 ChH'ag
..
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