Frege's Lectures on Logic: Carnap's Jena Notes, 1910-1914 (Full Circle)

-&'1' . . . .IS, t~Vt.J., Ytl 111' Ig to tht~ right of The scope of. the quantlftl~r

-& .. . ' the usual sense. ' . bl to express all of the other which is a variable-bmding operatlon III Combining these basic clements, F~ege ~s a die.. ction and existential ., h s conJunctIon, sJun. Q) , . ". famihar lOgIcal operatIons sue a .' . P & Q' written as -,( P ---4' -, In IS quantification. For examp Ie, t h e conJunctlon the form:

ifl~

then Q

~;~~~mpll.re again (Kreiser 2001), pp. 280-84. for some of the details in the last para-

~Q P . . d' ted by several dl'I'feren t forms of The application of rules of inference IS III . lca. I I I d with the names of the horizontal lines between formu Ias, someumes a)e e

28 Erich H. Reck and Steve Awodq

other formulas being invoked F . . . rule (see below): . ormstance, an apphcatIon ofthe transposition

Frege's lectures on logic and Their Influence

29

Universal generalization is indicated by a "sagging" line:

Finally, a vertical line (the "judgment stroke"):

Iis indicated by writing a "flattened X" . In

--

place of the horizontal line:

is added to the left-most end of a formula to indicate that its content is asserted as true, for instance when it has been derived from the axioms. Frege applies his inference rules only to such derived formulas; there are no conditional derivatious involving formulas without the judgment stroke.

P .... ~R .... ~Q

2. SwnmaryofFrcge's logical system

When convenient as in the £ di ' , ' oregOIng w hall tIonal SIgn -7 associates to th e . h' e S use the convention that the cong ten more simply as P .... Q Rfl t , so the formula P .... (Q .... R) ean be writremi '. - 7 . T h e rule of Detach ' , .P ses, IS wrItten with a sin Ie h ' . ment, WhICh reqmres two v IIl oked. Thus if we have alrea~ la::l~~~tall1ne. labeled by the formula being (II - I ' W,'

"un int,'r Q from p .... Q b D Y etachment, by writing:

TQ (I):: _ _P_

-Q

Propositional letters: P, Q. ... ludividllalletters: a, b, ... Predieate letters: f~ G, . High(~r-ord('J' pn'dieafe letters: a. FUllctionleuers:'/: K· . Variahlt~s: .f·, y, ...• X, Y. ... ,

If We have already labeled. (2)

TQ P

then the inference from P to rather than a double colon.

The system of logic presented in the lectures is a hybrid fragment of the systems presented in Frege's published work; like Crundgesetze der Arithmetik it uses several ditIercnt rules of inference and fewer axioms, but like Begriffsschrift it makes no use of the theory of extensions of concepts or the description operator. Thus it is essentially what we would now call a system of higher-order, predicate logic, similar in many respects to the system of Whitehead and Russell's Principia Mathematica (1910/1913), bmwith "simple" rather than "ramified" types. We begin by summarizing (in modern notation) the system presented in the lectures, before relating it more precisely to Fregc's other logical systems, 14 The logical language is much like the usual one of predicate logic; it consists of the following kinds ofsymbols:

Qf rom (2) by Detachment is

writt~n with a ' I . SInge

-P

p....

We shall follow Frege's custom of using speciallencrs .'T, y, X; Y, etc. for variables involved in quantification. Atomic formulas are the following:

(2):

I' a=h

-Q Applications of the other n I li1Ilo....'jngstvlt>s

. . " ()

t',·

JrWs;

F(a, .. , h)

It'S (Sf'l' bf'low) are ff'('orde.rl, II

I

ana ogously, using the

a (F. _.. ,G)

for a propositional letter P for individual letters a, h for Fa predicate letter and a, ... , h individual letters for a a higher predicate letter and F, ...• G predicate letters

(Cut) - - - -- -- (Negation)

14. In doing so, we disregard some specific features of Frege's system, particul?-~Iy as relating to the use of distinct letters in quantification and the theory of propositIOnal equality.

,

----------------------1" 30

Erich H. Reck and Steve Awodey

For our purposes. it is sufficient to use different sorts of letters in this way to indicate different syntactic types. A more elaborate system of notation, say, involving numerical subscripts as in Alonzo Church's "A Formulation of the Simple Theory ofTypes" (1940), would be required for a more rigorous presentation. Frege's own convention was lOuse different styles ofletters, labeled with appropriate styles ofvariables indicating their argument types. No types higher than those indicated here OCcur in the notes, but the further extension is obviously intended. Note that the equality sign = occurs only between individual letlers. Function symbols with appropriate arguments are permitted in place of individuals as arguments, as in/(a) ~ band F(g(a), b). Occasionally, functions of propositions are also considered, as iuf(a::::: b). A special case ofthis is the "horizontaf' function: -'I'

. ' ·1·k an d relanons 1 e l+a'anda<3are SpeCI'f'IC, us'ually arl'thmetical , [unctlons also often used in atomic formulas. _ f th t given in the The system of deduction is essentially a fragment 0 ~ 11 : tbree I f ' f enee and the.o owmg 1 Grundgesctze. It consists of severa. ~u e~ 0 10 e~ . duality respectively. axioms for propositional and quanuf1canonallogtc an eq , Axiom I:

P-4P

Axiom II:

';Ix F(.r)

-4

';IX a(X)

Axiom III:

sometimes called the ·'content stroke," which represents a function that can be applied to any arb7llment cp to yield the value true if and only if the argument is true, andjalse otherwise. Thus, e.g., the expression:

3I

Frege's Lectures on Logic and Their Influence

gla" b)

F(a)

-4

-4

a(E)

g(';IX(X(a)

-4

X(b)))

Aspecial case of (III) is the more familiar: a~b

--> ';IX(X(a)

-4

X(b))

-(/=(1

and another case is:

elt'nottOs true, and the expression: ~a~b

-->

~';IX(X(a) -4

X(b»

d . f 'ndiscernibles." The funcThus (Ill) yields Leibniz's Law of t~e. 1 enn,ly ~ I ,'thin auycontext. . . . I' It' Its applIcatIon w . tlOnal formulation lOVO Vlllgg perm s . kinds: those involving a The rules of inference arc broken down lOto n;o mulas to draw a couclu' .formula as premise, . amI those " that use . , twO lor smgle !jion. The rult~s of the first sort lire liS follows: W

denotesfalse. Formulas are built from the atomic formulas by means of (the horizonti.ll and) the propositional operations: negation conditional and the (universal) quantifiers: ';I.r 'I' (x) ';IX'1'(Xj

for an individual variable x for a predicate variable X

In principle: higher predicate letters could also be quantified, but this does nm happt'n In tht'. notes. In addition to this primitive notation, Frege often t~st's rh.(' ('om pit-x ~Ylllb()ls corresponding to the other familiar logical operations. hkt,: '1'&", 3xq>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'.~.:


in

qJ

(a)""

'.

'.

, simIlarly for functIOn

and predicate let-

q>(F) VXq>(X)

where X m
There arc also three "forms of inference" involving t wo prcmlst:"s: . P

(Detachment)

P-4 Q _

Q
Q

on

ill

This rule, like the others, can also be applied in the case where there are several other conditions P present. This completes the description of the logical system presented in the notes. We now briefly analyze it, comparing it mth Frege's published systems. The system of the notes, consisting of three (groups of) axioms and seven rules of inference (together with rules of substitution and change of bound variables) can be shown to be a complete system of deduction for this fragment of predi: :a~e logic, which is of course sufficient for all of predicate logic by suitahle defInItions. It differs from the system of Bt·griffsschrift by the addition of several new rules of inference (beyond DetS ofaxi0111 III. hut the rClJlilining fiv(~ axioms of Ht,fJ,r{!I.\·...("!Jr[/i can now he (krjVt~d. ,. Tbt, system is esS('ntially tlw "logic.,I'· fragment of the sysu~TT1 ~rJvell 111 ~rund~t',\'f'f::;t" Spe(;ifinllly. it omits from that system the three axioms gc~ve,rn~ mg propositional etluality (IV). extensions of COIH'epts (V). and the descnpuon operator (VI), which we now briefly describe.

.. COndItIOns, e.g.:

P-4Q-4R P--.:. R

A ft'l:lt.t'd rule is tht> 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
36

Erich H. Reck and Steve Awodl!f

Grundgesetze ("Absorption"), which states that the horizontal function -

({I is

idempotent: -(-'1')=-'1'

He concludes the lectures with a discussion of the hierarchy of objects, concepts, relations, and second-level concepts. The universal quantifier is cited as an example of a second-level concept, as is the property of heing satisfied by exactly one object, which is expressed in conceptual notation. Between Parts I and II, two notcs have heen included which, from their content, seem to belong ronghly there. The first of these notes labeled (by us) Appendix A is an analysis ofthe ontological argument for the existence of God. It uses conceptual notation to clearly make the point that existence is not a characteristic of the concept "'God." but rather a feature of it- to u.se Frcge's terminology. By analogy, a house may be made of stones and mortar (its characteristics), but not of spaciousness and comfort (its features). The second note is also of considerable interest. as it deals Vlith numbers. It hrieny indicates how the number ofa concept is a feature ofit, and thus a seC(mel-level concept. Fregc considers the example "'two tall towers" in which the towers are both t.all, hut are not hoth two. He also considers some other types of second-level concepts. Pan II is occupied with several different topics, including some further l~x.amples of the ex.pression of mathematical concepts in conceptual notation; th(,~ development of a system of formal deduction; and several extended ex.ampies of it.s use, There is also an extended discussion of rigor in mathematical arguments and definitions, the use of variahles, and the nature of functions. There are two gaps in Part II of the notes. one of five pages and one off(H1r. The contents of these gaps is, however, not difficult to infer. Part II hegins with a brief summary of the "'semantics" of the coneepw al notation. tht~ rdt~rt~nccs and senses of sentences and their parts, i.e. names and functional expressions. The bask language of horizontal, negation, conditi~nal, identi,ty, and quantification is recalled, Two examples ofthe expression 01 mathematIcal concepts are then given: continuity of an analytic function at a point, and the limit of a function for positive arguments increasing to infinity, ~he statement is continuous at the point x" is expressed in conceptual notatIOn as:

':r

'1e>O 3d>0 '1a (x- d < a <.r+ d -> - e
'1e>O 3d>0 '1'1 >d (-e < A -f(a) < e)

I:0.. .

>

.. 1 1. ' .

37

Frege's Lectures on Logic and Their Influence

. the note b 00 k ' At. this point the lectures There follow five hlank pages 10 d may have included the two examples of the °hnto~olgllcal ahrog~:v:~t t~:y w .at wows, ' ) Ju d gt.ng from . numerical concepts (see a b ove, f 'tion of . I ' I d d the hegmmngs 0 an exposl must In any case have a so mc u e . . III f llowed hy deduction. For the notes resum~Vlit~ the sta~~ml~~lti~~:~I::rks:h:tentirely formal deductions of the sort ?lve~ ill Frege s ~ axioms I and II are invoked absent from the notes up to thiS pOlnt. M?reove,' marks like '"earlier we in the course of the suhsequent deductlons with re had. , .." , . . from axiom III (notably, After deriving some propertIes of Identity Ii ' of - . ) F e turns to twO app cations Leibniz's Law, re~exIVlty, and. symmetry, re~ The first is a detailed proof of "how one can denve proofs usmg our symbol~" h ' ter than the other ' , that two num h ers are . equal If eac IS grea the proposltlOn when increased hy an arbitrarily small amount: (Vx>O (b+x>a & a+x>b)) -> a =b

from the premises:

(a- b)+b=a ,(a> a)

a>h--....7a-b>O ,(b> a ) & ~(a > b) -> a = b 1>0

This occupies five pages. ' d ' I 'h()ws that limits are , h (.0 II ows. lTl10le late y. s Th~ second example, whlc . . .. (,... I'the formula: uUlclue. Spt~ufll'aUy. a proo IS glvell 0 V,'>U3d>OVa>d(- ,'<;1-f(a)<,,) & V" > 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.<.t. Particularly noteworthy in that respect is the r,cvI~cd lOgIcal system [)f(.:'sented in the Icdures, in which, as pointed out in eellon fI °h11" I'" , , . 1I' the apparatus . 0 <.a III ercnc(' IS retamed hut the ('onstructive mac h mery resp hI" h h • ht'ell disnlrt.ll"d, The , on~1 e or t e t cory oflobric:al objeNs has resu Ihug system, hke,modern 't . r I I ' , s y s ems 0 (t'( lwtIVC logiC, has no domain of its own, b ut can he' apphed to reasoni ng"1 I ' a )out ot Iler (Omalns, As Carll a}) himself.' h aps recallIng Frege's lectu per £ I " later: res. ormu
S

The

' ,

,

proposulOns oflogIc and mathematics [ ] are l' " T cance for science since the .d' h "" .0 great slgm ITO ., '. Y31 10 t e transformanon of [scientific I ~ po.slUofns,.[.,,] LOgIC and mathematics are not sciences with a omam 0 ohJects of thei [1 Th e assumptIon ' of·'formal" or "·d al" b' rown. ... l e o Jeets, as opposed to the .... r al" b' f t:'nces. is dropped (Carna l e o Jecrs 0 the empirical sci. p 932. p, 433. ollrtranslation)

2. Frf."gt"s innuencf" on Carnap It. might rt:>asonably be asked wh at, I'f any mfluence ' F 'd . ('Ially the later ont:>s c()uld h h'd' rege s octrmes. espe~ , , a v e a on the subs' l t d I ' t-.11V l'n then l-'Vt'n his earlier . e( uen eve opment of'I oglc. WrItten works were not well~known or widely appre-

Frege's Lectures on Logic and Their Influence

39

ciated until much later. Indeed, many of his contributions seem to have been rediscovered "independently," and to have entered the discipline through other sources, snch as Russell, only to be recognized much later as having also been contained in Frege. Afurther important aspect of these lectures is that we know through them that Carnap, in particular. was already acquainted with many of these topics directly through Frege. One specific case of Frege's indirect influence on modern logic through Carnap appears to be the general adoption of higher-order logic with s~m~le types, as opposed to ramified types, in the late 1920s and early 30s. As thIS ~m­ eage ofthe theory has not heen generally acknowledged (even by Carnap hImself), we take this opportunity to briefly make the case. IS . The first thing to note in this connection is that Carnap's textbook Abrts.~ der Logistik, pnblished in 1929, hut completed and circulated as early ~s 1927, seems to have heen the first systematic treatment ofhigher-orde~lo~c with simple types. If! The other frequently cited sources, Leon ChWlstek,s "The Theory of Constructive Types I" (1924) and Frank Ramsey s "Foundations of Mathematics" (1925), actually contain much less detaIl tha~ was already in Frege; and Hilbert and Ackermann's Grundzilge de~ lheoretlschen Logik (1928), occasionally also mentioned in this conn~cuon, used ramified types until the second edition of 1938. Perhaps more Importantly, Carnap's Abriss and related works were familiar to Godcl (see below), whose "Uber formal unentscheidbare Siitze dec Principia Mathematica und verwandter Systeme I" (1931) is generally reCob'llized as one ofIhc m~in sources of the modern theory, Finally, given his familiarity with Fn~ge's hIerarchy of objects, concepts, higher-level concepts, etc., especially m the form, contained in the 1910-1914 lectures it is reasonable to assume that Carnap s use of simple types; in Abriss, A and elsewhcr~. was derive~ from that Sourr.e,17 Not~ also that, un(ler Frt'gl-~'S influl-~nce. simply-typed hIgher-ord~r lngit' was neVl'r just a dt'vic'(' lor avoiding contradi<'tion for Cllrnap. hut was III t~e very mIltm: oflogic, It-ndillg it 1111 inht,rt'nt plmlsihility that other eom~eplions la('k(~d. " . . tl 'I Frcgt>'s lectures are hiS Anot h er rangt: OllSSUCS ot'eurrtng prOlllll1Cn y n . . semantic doctrines of sense and reference; intension and extenSIOn: names e and objects; individuals and descriptions~ concepts and extensions; et~. Thes , • 'fl t'a! book Meanmg and lSSues also form the backbone of Carnap s m uen 1

lybau,

ecent very different 15 , Compare (Kamareddine. Laan. and Nederpelt 200 21 f or a r , aCCOUnt. , . ik com are (Reck 2004), P Hi, For more on the relevant history and role ofAbnss der Logtst, th modern sources ~7. To be sure. in the Abriss Carnap cites only Russell and ~~':l~?:.~it7ngs that, despite f .type theory, relegating Frege's works to the catei?ory 0 , 1 ble insights for the hlJlng superceded by more recent logical theory. snll conta~~ va u; be explained by the presem" (Carnap 1929. p, 107, our translation)., Howe~~r~::e~~reg~'ssystem of type fact that one's own parents always seem old-fashlOned. '. menu (The n . I· t d type-theore nc argu . otaUon did not lend itself well to more c~mp lca. e r; h positional calculus.) same can be said, mutatis mutandis, about hiS notanOn or t e pro

___4111

40

Erich H. Reck and Steve Awodey

Nf·(,f'.~·silY

(1947), in which he "developed a semantic method int1uenced by Fregc''i distinction between the nominatum ("Bedeutung," i.e., the named entity) and the sense ("Sinn") of an expression" (Carnap 1963, p, 63), Here,

however, we probably have less a case of the specific influence of Frege's lectures than of his more general influence on Car nap, since these topics were also discus~ed pr~minently in Frege's published writings. . Fmally, ,It seem.s that Frege's influence on Carnap extended beyond technwal and phIlosophical logic to also include what may be called the ""scientific temperament" that he passed on to analytic philosophy: From Freg: I le.ar~ed caref~lncss and clarity in the analysis of con-

cept~ and h~gulstlc expresslOn.[".] Furthermore, the following con-

ecptJOn, which derives essentially from Frege, seemed to me of paramount importance: It is the task of logic and of mathematics ~thin the total ~ystem of knowledge to supply thc forms of concepts. statt:ments, and Inferences, forms which are then applicahle everywlwr,c, he~ee also to non-logical knowledge. It follows from these ('onsHlerations that the nature oflogic and mathematics can be ~'Jt.arly UIH~t'rst~,)()donly if close altention is given to their application Ifilloll-iogwal hdds, especially in empirical science, (Ibid" p. 12)

,

,

3. From Frt'gc to (;(j,dd, via Carnap WI' {'onclude this.essay t" , . , ' hy callI' ng ,a t entlOn to a particularly noteworthy case ot' Frege S InnuCnCt' which Ie-Ids trl K C"d . .., , , . un 0 e I an d some of the most imj)ortallt early results In modt:r I . IH A -I. .', . n ogIe: s recent scholarship has established, during t IIe ate 19205.Car nap was , d'In IogIcal ' research on axiomatie sys, aettv e Iyengage

[ems adndl9v2a8rulUs 1,lotlOns of complctt·I1eSs. Ht~ finished a hook manllscript aroun , which, ht, 'I e Untasudwll/-ff'n zur All . A' " circulated un( Ier t h e tit gkememen xwmallk, hut never published. 1'1 A.... Wt~ will nnw indh"lt(' (his wor apparently had some inn >, h ' • 'a1 . uenCe on t t: young Kurt C{.del. WhO.. . I~ subs(>' t Ioglt" results In turn contrihlll 'd quen (' • , , ' , t. to .,aenap s dt'('ISlOtI to ahandon hIS project around 1930. . .. >

Much of the background for C ' ' th 'd ' . arnap s A,nomatik consists of Fregean emes an assumptIOns HIS goal h approach to th r d.' • owever, was to combine a Fregean logicist . . h H' . e loun anons of mathe developing a "theo o f ' , manes WIt a dberuan formalist one by 303) in which ry laxlOmauc systems of arhitrary form" (Carnap 1930. p, , variOUS ogIcal and m a l ' al ' d "h' et oglc questIons could be addresse systematically As he . . put It In t e Introduction to the Axiomalik: W.lnthiss{'ctinnwt"drawheavily (A ' (R\'~'k 2004)' ('ompa" al' l thO~ wdodeyandCarus 2001), (Awodl"Vand Reck2002), .' . at elntro U't' Ie ., 19, Carn'lp did puhlish 1 C IOn to arnap 2000), " some re ated . t' l ' , l Jnt>lgt>nt!lcht> BegrifffO" (1927' "B . ar .~c es, In partIcular "Eigentliche und , 'k" ( 19'.,\Ob), and "Vb "er ' Ax IOnl/ltl E t c h . . uber U n I ersuc h ungen zur allgemelnen ' er xuemalaxt " Ie ' IJOO k mllnuscnptwBsonlyrecentl redis orne. arnap and Bachmann 1936). }iIS Y covered. edued, and publisheda6 (Carnap 2000). and

41

Frege's lectures on logic and Their Influence

In l.,.] recent investigations into general properties ofaxiom~tic~~sterns such as completeness. monomorphy (categorieity), decldabl~Ity. consistencj', etc,. [.,.] it has become increasingly clear that the mam difficulty [...] lies in the insufficient precision of the concepts used. The mostimportaut requirement [.,,) is, on the one hand. to establish explicitly the logical basis to be used in each case, [, .. 1; and on the other hand, to giveprecise definitionsfor the concepts used on .that basis, In what follows, my aim will be to satisfy those two re~Ulre- , rnems and to [, .. 1derive a number oftheorems ofgeneral axlOmall CS . (Carnap 2000, p, 59. our translation, original emphases) At the time, Carnap was one of a few logicians engaged in research of this sort, and he discussed his results wi th Fraenkel, Godcl and Tarski, among others. Despite some interaction, however, Carnap's basic viewpoint was decidedly Fregean, compared to that of those influenced by the Hilbcrt schooL TI?,us" fO,r instance, no distinction was made hetween what we would now call the obJe~~ language" of the axiomatic systems heing investigated and the '"mctalangu~g~ d , I'k F ge he worked ' used to conduct the investlgatlons; , ' ' bemg mstea I e re, . ' W1thm Olle "universal" language. Yet again. Carnap wauted to r~co~cl~e such an approach with tbe use of the axiomatic method, rather than dismlssl~g the latteras "theft." as Russell had done. Here one can perhaps discern the mnuen~e ' d h " ntial role oflogl c , orthe later Frege ofthe lectures, who emphaSIze t c mlere . "f h d b t - wilh HIlbert (as preCarnap also adopted Fregc's posluon rom t e e a e . , • ., M h ' ' ' ) that an axIOm system sented III the lecture notes for" LOgiC 111 at ematlCS , ' d h' . I .cal research project deterrnllles a highcr.ordcr concept. lndee , IS enure °6r:L can he seen as an altcmpt to reconcile the two sides of that dehate. 'd , I' ' . matic systems cons! ThE~ s.pecific logit'al nntions and properties 0 aXlO , . . . I" ,. lomcal consequence. ered by Carnap indud(' sueh modern-sotHH mg topH.S as r;. ' , '<1 1'1' d 10,rH~al cOInplettness. satlsfa(~tion, ('onSi,'ilt>IHj', catq.{oridty. deel n II Ity, an, t"' ~ , h' , , . ' h h t wei to l)roVI' an 1 ,It uny . Amon~ tht. "tlwon'ms of W'IH·r:.,llIXHHIJUW'S t at t' r . ' II ' ,.' . I I ' 1 syswm IS IOhr:LCU Y cOnl(:fHISlstt'nt aXiom ~ystt~m IHIS II mudd. lUll t tat UlIllXIO n . h _' '... . ' I' "fn I)oint ofvlt~w, owcver. p1etc Just lilt IS calq!;Ol"lt'al. As IS dt'ur rom u mo( I . , II .' I k d rt'ventcd him from rea Y " t he mathemaucal tl"anH.~work JIl whH~h \e wor e P . .. h' If " ' h 'dequately. Carnap Imse prOVing these conjectures, or even staung t em a "with . .' h through diSCUSSIOns heCame aware of the deficienCies In hIS approac Tarski in 1930 and , somewhat earlier, with Godel.1 dent at the U' mvers!'ty of In the late 1920s. Kurt Code! was a doctora stu f h Vienna Circle. V ' B h e memhers 0 t e lenna. where Carnap was teachlllg, ot w rc d' of mathematics. I' d foun allons ' an d they often met privately to dISCUSS oglc an '"M talooik .. and on e t:I • • Moreover, Codel attended Carnap •s 1928 I'ect u~e course I d h ~ manuscripl of his he was one of the few people to whom Carn~p ClfCU ~le d ~ ~o important tht,A,riomatik. Cbdcl's work in logic from that time r~~uhlte . III laled to Carnap's I re ogulla e as re orems. the contents of which are c Icar y c, '
, "

___all

42

Erich H. Reck and Steve Awodey

and the incompleteness of axiomatic systems of arithmetic (Gode1 ]931). In addition to the well known and less direct influence of the Hilbert school on G6del's work, Carnap's very direct influence thus seems unmistakable. And indeed, Codel himself refers to Carnap explicitly in the former work, and irnplicitlyin the early public statements ofthe results from the latter.:w Finally, in later years, he identifies "Carnap's lectures on Metalogic" alongside HilhertAckermann as his two main early inOuences. 21 Thus we have found a clear historical and conceptual path from the texts presented in this book to the celebrated logical results of Godel-who 3.':1 a student in Vienna attended the logic lectures of Carnap. who in turn took down these lectures on logic by Frege in lena.

20. See fGi.idel 1986) p 62 f 3. 2(03). •. • n ..• compare also (Awodey and Caru5 2001) and (Goldfarb 21. See (Awodt'y and Cams 2001).

____.......\MIIi

~~----------

Carnap's Student Notes

What follows are notes, taken by Rudolf Carnap, in Frege's 1910-1914 courses ~n lo~c and the foundations of mathematics at the University of lena, trans"ated .mto English by the editors. The notes consist of three parts: Begnffsschrift I," "Begriffsschrift II," and "Logic in Mathematics." Their ~ont~nt ~eriv~s from transcriptions, supervised and edited by Gottfried abnel, from material contained in Carnap 's Nachlass. The first two parts are

~upplementedby editorial tootnotes from Gabriel'5 German version, also trans~ ted into English. Apart from our treatment of footnotes, we have adhered to the typog "" .see " " by . rap h Ie conventIons ()f the German verSIOn; the Ultro d uctlon Gabne~ (above) for details. However, we follow CaTnap's original page breaks. ~he tturd part, "U)bric in Mathematics," is published here for the first time, either in ('...erman or III " E-,nghsh " (although a related " 0 f Frege ' s text. ' conSISting OWn lecture notes, was published in his Nllrhp,t'/assenen Srhriften). We adopt the same tv<) h" " " h" I" h I" " " .J t ograp Ie conVl~nllol1s tor t IS text as or t e orcgOlng ones.

45

-_.

, R.Carnap University ofJena

Winter Semester 1910-1911 Frege Begriffsschrift I

10 (

2+J~'

~

cont;nt judgment stroke

~trokc

-,3>5

negation

~

the same as

3<5

Only th~ thought. not yet as a fact.

1.

- 3 < 5.

condition stroke

L3~>2 ] >2

L2~>2 2>2

II IV

}

This may at first seem counter-intuitive. since we know the individual sentences, or their falsehood. But as far as logic is concerned. it dot:sn't mallcr whether we can survey the sentences.

r:.~.lhis makes mote scnsc:

I

~~s~::e~: ~~"Begriffsschrift I" and "Begriffsschrift II," editorial footnotes will be margins.' ~:n the te~t by. brie! descriptio~5, as well as Teferenc~ numbe.rs in th.e outsi~e e flotation "11 coununghnes 10 the deSCrIptions, each formula m Freg 5 Begriffsschrift Nates. W~ be counted as one line. eN will be used as an abbreviation for Carnap 's

GottfTiedlGa~~e

exception of footno'te 99, all notes are from the German version by neL l.Topoftbepa~ ° h t 51°d e. 10mes 2-3: "Only the thought, not yet as a f act, "ThoJS IS ° to b e Underst d. l:>~' rig thus 0. f In the ,sense "not yet asserted as a fact"; since 2+3 -= 5 is 0. true thought and pende to Der Gedanke (1916) p. 74: "A fact is a thought that is true"-indent y of whether it has been recognized as true and asserted by us,

o~ a~t-accord.ing

49

~

't16

50

Winter Semester /910-1911

T ~

Frege's Lectures on Logic

51

not {A false, B true}

4 cases are possible: III: A false, B true. not-A and B

A is true, B is true.

I)

II) A is true, B is false. III) A is false, B is true.

IV) A is false, B is false. 1: A true, B true

«

A andB

Case III cannot OCCur.

}

IV: A false and B false neither A nor B

Thus, only means that case III is ruled OUl; doesn't say anything about the truth of individual sentences.

«

II: A true and B false

A andnot-B III not lA false, B true}

A ornot-B

«

L

I

not (A lmc, B truel

not-A

ill"

not: A false, B true;

no!

"

lA true, B lalscl

2.

nol-11'

IV not ! A f
--c;:::: .,;

.1 "r !lIme <",('h~' 1I<1I1-l"u:llISJV~'

~ and;

~ neith.:r-nor.

"ul'"')

not-A or IJ

2

3

~------~~ d s "or"and "and" . I n eN,n) otonlythewor .are-doubly . 2,Bott~mofthepage,lasttwoltnes: but also the correspondmg SIgns'In underhned (here represented by boldface. lJegriffsschrift notation.

.-

,

52

Winter Semes[er 19/0-1911

not (not-Maud r)

Ican also regard Let us replace Mby:

S3

Frt!ge's Lectures on logic

L

A

as upper tenn to rand cJ. or

S

--rcA L

~

as upper tenn to .1 ,

not ( not-A and B) since:

not ( not~A and 8 and T)

,~ L

B and rare interchangeable.

LJ

not (not-N and .1)

Wewrire:

We call A the "upper term"; Band rthe lower terms.

»The lower terms arc interchangeable.

( oot-A and Band T)

not-N

« not ( not-A and B and rand,1)

TeAl L:: r

not (not-Maud rand,j)

"

Let us replace M by

L ~

For not-M Wl' get: not-A and B.

nnl (nol-A ,md

n anu rimd .1): H,

is the upper term; rand ,1 are the lower !trolS. I can', rc~ard B as a lower term

lll' L: r,I

I: ,I 1lIIl'n:h;1l1~l'ahle

r Ilol ( not-I) lind ["ami, 1 )

nut \ nul (not-A lind H)und "und .11·

lower Icrm[ s I

) mllY not he br\lkcn up.

4

5

--,-,-.'1 LH

not-A and B

54

Winter Semester 1910-1911

lS ~

I

not ( not-A "'d not-B and not-F)

55

Frege's Leewres on Logic

AorBorT nol (not-A and nol-8 and not-n

3.

A or B or r (non-exclusive "or")

I nltlrht'r A

Instead of I, [put

1Ior

B nor

r

I

LandT

---r-c= BA

not-B or A

I I

A andB and r

shall mean: ··'05ults by transposition."

A andBand r . h kind of transition. In ordinary language we simp Iy say "thus" without specifymg t c Here we arc more preClse. (-) and ,1

In~t('ad of (~).

let us pill

'E;;

I-r'-r- ,1r

I

><:

L(~n.,

4.

5.

For.l

Tht'!W\l .1ft' i",cn-JlllfI~t,llhlc:

we CUll wnte: Rho Iranl'pll",Uinn

S{'

1-.-.1

, L, /"

7

6

no~auon

. for "'uans-Frege's Slgn endieuetze I, p. 27)two lines are perp 4. Second half of the page; Elsewhere (c f · Grundges is smaller., and the .. .. I the fiattene d X , d POSlnon" (contrapOSH.lOn , . the premise aD 1ar to each other. We follow CN. missing both 1D . dgment ar~. .. 5. Bottom of the page: In CN, the JU f suokes "transpmiloo n. in the conclusion of the second example 0 a

3. Top of the page, second line: In eN, above the word "or" in shorthand . (inUlTI" the phrase "Ilon-€xclusive 'or"') the same word occurs again in ordinary n01:aUon, pres ably to highlight this case and make it easier to recognize.

...

fIIIIIi

56

57 Wimer Semester 1910-1911

Frege's Lectures on logic

Can of COUrse
lIere, 100. we use the flattened X:

The main point is: »In a transposition the upper term negated takes the place urthe lower ham, and the lower term negated, the place of the upper [enn. «

The intemlcdiate

~tcp

A

~r ,J

B

><

Another application of transposition (according lu the same principle)'

Rille:

The case or severallo,.'er tenns:

t J. form into the upper in nega e the rest' 0f the lower We can :'Ilw'ays tum anyone of. the lower . . !cnllSlower teml: . neg ated tom) tOlo a term and the upper term In terms remain unchanged. E.g.:

We know thai the lower terms are interchangeable. therefore:

1n:::~ 1rc:::~

C

~;

r "

1

Ct "

.

,

E -

><

~~:

t

A II

A

"

8 6. Bottom of the page last Ii 'C'''h of th e Iower terms . ' formula ne.furthest H as, to erroneously, In the the right. ".1" instead of"r" in the lowest

9

is simply lett out.

"

58 Winter Semester 1910 191!

59

Frege's Leewres on Logic

Besides transposition there are also less simple inferences.

We can regard 2 terms together as the upper term:

(H

(I-c~ not ( not-A and B )

case

1[1

B is true II and IV arc ruled out.

is ruled out.

(I-A

From these two propositions it obviously follows:

A is true case I

We write (the propositions arc labeled for later use): If we only Want 10 h.ave one lower Ie nn, we proceed as follows:

(a

I-cAB (fJ)::--

I-A

1'=~r

Now we ICllhc I I I \"

. (p mes be Inc upper Il~rm:

---

Irl :

I-c~

Thus we nOw hI11/C onl Y WIt' 1uwer tefm: II amll't1",1 ,'1.

IIi( :- - -

I-A

~-----

~/O~i

'--.-'_'

I-B

I-r

(fJ

Ir

Or we write:

1-8 (a):---

I-A

Or we write briefly'

m-~

L

r

(fJ.r):-

I-A

----------~

60

W'Inter Semester 1910-1911

61

Frege's lectures on Logic

More Intricate Farms oj -Inference:

(0

\\!lIat can

Possible cases:

>

(0

IY

B true <

7,

At

B false

M,

Bt)

Af

Bf

Bt

rt

Bt

rr

(B f

rt)

Bf

rr

r

t:

Bt:

At.

If A

l

Br

rr

Ruled out:

AC

n.

If

"

A true

(A f

>

be In . t'erred from th"

<

Thus it follows:

!

f-r:; We-write

f-r:AH (PI'

f-r:A

(a):

;,t., / II .\ II

Fusion of iJcl1ticallowcr tcnns:

r

/2

/3 oomb;ned i page, ,econdhne, left ,ide, The b.-cke" indicate rbat rhree term' have been 7.To pofthe . terms "r" nto one upper term (in relation to the remaining lower term "T·'). The lower fl, S and".d"h ave been exchanged without comment. . arne line asI i n 7" nght . . . '. '. C'undge",ze "de, (E) ,e,ult' f,om (0) hy applymg tran,po""on, ","c, ("f, , p. 28.1ert column, bottom halO.

63

62

Winter Semester 1910-1911

Frege's Lectures on Logic

Instead of alilhat we write briefly:

»

~~

~f

or:

~

(0)'

(y),---------

~j

~i

z

if we have two premises [with the same upper term] such that a lower term of one occurs negated in the other, then we draw the inference: leave the upper term alone and take as lower terms the lower tenus of the premises except for that term which occurs both affmncd and denied. «

z

We write: Yel a third fonn of' f

1Tc ~r

m",nc, - 2~p"m;"r; with the ""m, upper term

L.2= .1

><

(¢

E

('7

~~

What follows from this'?

(~)o--

(a),

d -.-- -

(a

14 9 80 resents the inference indica~ed b' " ttom urthe page, last line but one: Frege usually rep . ~ eriod at the beginOlng y -, _ . _ . _ . _ . _" typographically in such a way that there 1~D'~ eN. and at the end (cf. Grundgesetze I, p. 30. right column). We fa

9.

65 Winter Semester 1910-1911

frege's lectures on logic

Inferences Involring "e v nera /.uy.

always true:

( 21"

4

(-I)

4 4

0

~ 1 >- 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

II>

---" 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


Contrast:

2

can be decomposed in different ways into a saturated and an unsaturated

part:

(-2) is a square rool of 4

~

2'

;' = 8.

3;

2;'::: 8.

8;

2'

something unsaturated

= p

"

i= I 12,

But:

is obviously false; it would say: Everything is a square root of I.

(-2JfaJls under~the concept"::;quare rool of 4 .. :

,

'~•

'

I

would not be true either: Nothing is square root of 1.

~--)

something y self-sufficient

ThIs bnngs about the connection.

Thus in order to negate generality, let us make use of German leners:

) falls under (

is false: Everything is square root of I.

'-,---J

is true: Not cverything when squured is I.

even requires a douhlc completion

is flllsl,:: Nolhin~ whcn sqUlIrcd is 1

~.

Thus I need

h-tr-r/8 ~2. ~~ddJ.e of the page: In eN t . the. WIthin the bracket • he thLrd. cUdy bracket d •. means of additional und tJ:te expressIOn "square oesn t Include the definite article er lnmg, root of 4" is highlighted further by

i-

/9

a. - 1:

1(1 IIC~IIIC

it:

It is false, that nothing is a square rool of 1, or existt:ntiul propo.\'ition:

Th~re is a square root of l.

13. Top of the page, first two lines: In eN, the quotation marks are missing.

13.

69

68

Winter Semester 1910-1911

Earlier

\\If;:

Frege's Lectures on logic

had:

is not true, thus:

We can also write: n

.4

f 1 is a}'<1 root of I." "At least one square root (= at least one) (particular affirmative) some

,?

== 1

""Ta'~ 1

Ilcrc the judgment stroke IS . not neccssat)'; we can also leave It. as a mere t h DUg hi , which is not possihle with letters of generality.

Vi

is an

is false, thus:

(J~it'c/ (not a concept), is not in need of completion. hy contrast. is unsaturated.

. 3Td root of 1," "At least one square roo t of 1 IS"not a . ) "some not (particular negative

is a nonsensical notation. \h' havc allowl'd ourselves III he misled hy ordinary usc of language: If SOlTll'lhing j"" square Toot of 1, it is + I or 1:

-&,- <1>1.\

L'I'la\

~I') 'I'lal

~(al If something is a square rool of I. it is not a 3"1 mol Or: No :>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-
. . '-'- 'f'(o)

some _

Some:

~<1l(0)

says therefore: There is at least one object, that is both tP and If'.

.'

we can also write:

~
The 4 judgments: all, some, no, mme not are oot that IrnpO .

~

'C':::

~X(O)

t

So the particular affirmative judgment contains an existentIal Judgmen , affirmative does not.

'1'(0)

ronn to make a connection with ordinary IOgIC.

the general

[taot for us We give their ~e basic elements.

But we have even rno


16.

tufe of things.

There is at Jeast one X.

· d es violence to the oa The distinction between subject and predIcate 0

Instead of X( ~) we have:

E,g" the four concepts in

~

E~~:~ x(o)

flhH are aII .Interchangeable· The'e , ,·s "' least one object thai e.g.:

'1:':
says: A is a 'P and A is n m.

22 15. Top of the page, first line; In eN thewo" . of the two diagonals in the logical 5 • uare;..d contradictory" takes (as usual) the pla~e reasons. q - e present form was chosen for typographIC

__

!,,~

.~.M.··

. t/Jand If'and Xand il. IS

. . __

23

:th

. "object" instead of e bottom: In eN, 1£ says sub'eet_predicate m :6. Se~ond half of the page. third line {r.o the replacement of the J Pl'edicate"; but in the text we are dealing f rion strUcture. st~ctureof the sentences by their argument- unc

73 72

Winter Semester 1910-1911

Frege's Lecwres on Logic

There is a ditlerence between 3:> 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
Om: first ddines the concept in the detinition.

··Goo. ' but includes existence as a charactenstll . "[Ht'rkmal] Therefore in the onr
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, -one ,elation on p. 82 (eompart> also. Grundlff'snu I. p. We are probably dealing with a mistake in Ca.rnap hott'S. hpn'. thiS representation ?Ccurs nOWhere else in Frege. It is. however. logieally
sln(~t'

40~.

~

w points from the statement

24.

Summer Semester 1913 Frege Begriffsschrift II The meanings of the parts of a sentence are not parts of the meaning of the sentence.

However: The sense of a part of the sentence is part of the sense ofthe sentence.

The meaning of a sentence is its truth-value. lfa sentence has no meaning, but only sense, then we arc not in the realm of science, but

of fiction.

A sign expresses its sense, refers to its meaning.

(E.g., the meaning of a proper name is the object it names,)

I) . . n('('d o!I'omplcfioll (e.g. . Sorne signs arc m

0=

).

lfsunnJ. . . . f" . ' 't"I'/lin the case of two , .... ~mcllta'lllfI YH~lds a ~cntcncc. then the Slit{l n.~ crs to a (on< arguments, a rt'!,jfjO/r).

More generally: .Iimctivn. 2 J Th

.

. f I entation are called proper osc parts of a sentence that arc not 10 need 0 supp em

name.~;

they refer to objects.

~---------------' 87

I

88

Summer Semester 1913

COlh.:ept of the true:

l=:, ; == ;,.

concepl ofth...

I

}

non~(rue:

I I I,

1;1 [eve-! function

(the arguments are objects)

identity rdation

I frqe'$lectures on logic

2r.J lewl function lthc argumcnts an: 1-, le....el functionsl

"

We wan( to express the cool'inuit} of an analytic funl:tion al a particular point.

for every t' there

.h thatlhl" fclatwll hold:,-

eXIsts
y' ThlL~: In the interval from x - d

fo x -t- dlhe difference berween the ....alue of the function and the value at x can be made arbitrarily small.

." ,

·,

~~ ·.,, ,

L__~_~

I'his is supposed to hilid ti,r

SUllie I'

II.

2

TIT

d'\_1

(/'(\)

if'{l\I)

(/'1'1'

·0

• ·0

"

,

, fonllnll\l)'.

\

90

Summer Semester 1913

91 ""', Lectures on Logic

Example. The function ~2 is continuous at 1: Let the number A be the limit of a function
toWHn!s infurity (A ~ lim


I'T' A -
FOI" every a > dthe difference A - fj)(a) must be < e in absolute v!ilue; and for every positive e there has to be such a number d.

';=' 00

•

a>d

y

~A-(.) > - 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

><

I-rr fib) L fla )

><

27.

u~b

(lIIe

la

-0--

1Sf(~t fib) (lIId

I-r: flu) f(a)

usually

'Laa

Now we usc lhe introduction of German letters; we can ~Iways do so jfwc let the cavity immediately follow the

'----'

Judgment stroke .

~fI")

In III. instead of _ .,t= for g we can take (11)"

---,!~('::')

I-

12

l/ -- {/

Imtc:ld of h I Cllliid also have written a everYWhere

{ in III and a

.

(Ilk

/3 ." . led line fro mth e topofthepQue' On the use of the Slgn 27reekl Tl, G a els). cOlnpaTe Grundgeset:;e 1. p. 66f. ettel"s( as Ih --,:l"

-~- "and

0

f

0_·"

"".1lJ,J.

97 Summer Semester 1913

, lnjo'l1.e
I' ,

I In IIIc we replace I'I.¢. ) by .; '" a:

I

lfil- - - - - - - - - - -

I ,

( --r-

"'1-111

<J

( 1I1~

l

I

-0I

28

Want to

Ilk

29

replace a by -

11.

and b by -.-

J.:(u)

gtl'tl

>< We had Ig ~g(iJl-g(/,'l

ft g(<J11

t

= (--.- a)

1ft

Irl

t-r-

~ ,l.!(i1I-'!l(hl

I

a

(- Ill""


a)

I I 14

I '----

28. SeVen lines from the top of the page: WIth per formed .'_;: in ," . . respect to the subsutuuons .. (IIIe), the funcrionletter "f" has, in additioll, been replaced by the fUDcnon name The horizontals have been fused (cr. Crundgesetze I, p. 67). In eN. the judgment Ii in (IIIe) i .. missing.

rr~k.e

29. Eight lines from the top of the page: In the ttansition u!>iog (Ig) the formula

L~ is considered to be a lower te~m in fIg) that coineideswith the upper term in (IIle). The 67) .

pr~sent derivation of (lUg) dIffers from the onf: in Grundsest!tze 1 fp.

••••••'..v

~ "g'"" t'Xlh}l

hglhll a)

-0-

~"

III

_

~;-a)=(--.-a)

ilia

I I

U"

~Lal~(-.-al

(lg), --- - - - - - - - -

~(:-

--0--

I

/5

/'

tlllh

99 98

Summer Semester 1913

!

:i"I~e~"~le:cta:",,~o:n~L:o~gi:C

_

We now give some applicatiuns: show how one can cunduct fJroof~ with our notation. 30.

E.g., we want to prove the propusition

~ t

We had IIIe

Here we want to make use of the proposition: (a-b) H~a (lIIc),,---_ _

I--

(A

~ a>a

(a-b)+b>a

31.

a~b

r+b>a r+a>b t> 0

~

oj

h-- u>a . L(a_b)+~~_~ (Ib)" -------------

!(b)

~

Instead of a" a

We use it here by replacing j(¢) by ¢> a a by (a-b)+h b by a

~ ~ e

(a

~

(n

a>a (0 b)+h>a

a>a

a-b>O t+h>a

r

(a-b)+b=a

We had Ib:

~ •

We now write il in th~ fonn: (a h)+h>a (a -- h) + h ~ a

(0

~r(·)

1(0

I(a) a=b

lIIe then assumes the form

~

h-:;--!(a)

lIa: _

b)t-a>b

16 ~o. ":o~ of the page, second line: In eN, the jUdgment sO'oke in the sentence to be proved

IS mISSing.

31. Middl: o~·the page: In eN, the brackets in the expression "(a _ b) + b > a" in (IIId ao. d (a) aTe nllSSIn.g~ compare. however, the form in which (a) is used. In eN, the label (IX) IS duuhly underhned.

17

t+o>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) <'12 + a >/(d)

>f(~

e12+e12 =-- e

---0--~~b ---0--~~u ---0--We

~uppose

I-r-~

(fJ

(Ib

(Ib, Id)e:

~ ~

0.0 ••••• 0.

e+h>"

L - ,,12+0>((;) ;>m

e+a>h ,,;2 >a-f(d)

ei2 + a >f'(t/) e2+h>l(d) e/2 > h-/(t/)

================ ('+o>h f'-'2 > a-f(d) ('/2 + a >,/.,(<1)

Ib to have the Conn:

el2 > 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 Ih<: llmn" l'il I (/ , ((d) /'/2 . ,/ ((iI) 1,;2

j

/I

- I(d)

24 43. Top of the p a ge. fust . line' (y). . 44. Middleo f t h epage: (HIe). . IS Invoked afte rrep I ' aClng"b"by"a" and "a" by '"b." e, 8 I1 d f(¢"I: ~ ¢tb>a 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, ,<,placed hy 'he lowe, teem of (1bl and the lowe' teem ahove thus, "t'used:/y the lower term of (Id). The twO new lower terms are identical and are,

108

49.

Summer Semester 1913

(II., II.)::

109

Frege's Lectures on Logic

==========================

We use the sentence: h-m+l>m : Lm>O

b> m b>n

e+h>a e+a>b d>m eI2>a-f(b) el2 +a > fib) b>m d>n el2 > b- feb) el2 + b >f(b)

~-Tr

e+h>a e+a>b el2 > a -feD) el2 + a >f(D} b> m e/2 > h -feb) el2 I b >/('o)

b>n

---.---

b>n

(l

r-

I have lo addth a.'t since h m can b e somethmg ot er than a number. We also use: (£

l-n- ~ ~ ~ L: b>c

(K

m+I>1I

~ _.-

in the form:

m + 1 .> m

m>n

b

50.

b

d>m d>n e+b>a e+a>b el2 > a -fib) el2 + a >f(b) b> m el2 > b-j"(b) el2 + h >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


If; 1

;: > 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

><:

52.

~

a

b

_.b>/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 f
separa~eat comes bpfore.

second time. r ula here is last lorman a dd e ndum to w .'is. Bottom of the page: In eN, thenstitute ofthe proof, thus appearing to co

1.-.-;>/1

~

53. Bottom. nght column, In the 'entence dedueed by mean, of Icli. 'he I owe r ternl m . ,. m >twice n" still appears. The transition is thus meant in the sense that this lower tet was used in (.1).

rt

112 Summer Semester 1913

!

II J

rrege's Lectures on Logic

(2.) n >m

f-rrn+l>m

L

56.

Ie):

I) .

f1

>m

n>O

~ ~

Ila) : -

_

~::~: n>m n::>O

~

n+l>m

n-r-l>n n + 1 >m n+l>n

---0---

n + 1:> n_

[PI'. 31-31 in CVareempryl

n+l>m

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
not exhibited clearly; we just say "thus" or "therefore." the corresponding rule of

inference is not indicated. Consequently it is easy to use "therefore" in cases where we are not dealing wilh a gap-free proof. For ex.ample, in geometry we somelimes rely on I) the pure inferences

intuition; One is then only aware of the fact that the result is evident. says "therefore," but does not know the reason for the evidence.

2) the commentary on them.

Our conceptual notatiun avoids this logical deficiency of langu
mathematical rigor. In conceptual .a1 to negatively influence This mixture has the potentt upcrfluo . us s f" ordsare . a completc understanding 0 It, W notation. assummg

( )

(),,----

( i ( I" ( 1 Imnsposillon: introduction of a (icnnan Ictlcr:

40

41

J

",.I,~

/22 Summer Semester 1913

123

frege's lectures on LogiC

In our proof we used the assumptions: J-a-h+b==a

(E

(4

I-r-a>a

(E

~

'T=a-b>O a>h

~f;~

E~~b}\1

C-+-d-.m.j.-f1

l'

'nl

(2

d .n

1-(11 1 In

t

r

"

1

(II

(q + r)

(/1

62.

These lower terms don't seem to be necessary; note. however, that in conceptual

l-r•( L

notation the [etters do nor juS! refer to numbers. but to objects in generaL these lower

e: 1 = e

~'; () :.j could have adde '"l I'

d")

terms are supposed to replace the assumption that we are dealing with numbers. For the function'; > (can only take the value True if both [arguments] are numbers. Our

l-r-m+l::>m

(I

. Lm>O

conceptual notation is thus different from arithmetic and analysis insofar as it does not

ne~'er determined exaer(l" (in mathematics) where the limit ,.ea/~v is, what can be indicated by a letter (the realm ofnurnbers

just deal with numbers. 11 is

fiesh,. What u numher

is constantly cxtendclI: negative, rationaL complex). One does not know how far the

(K

realm of numbers may he further cXlcnllcd, As a cllllscquenl:c the Sl,.'flSe of all propositions is inllcterminatc. l'tIT as soon as the realm Ofll1l11lhers i.'< oh:nded. the Propositions that Were already proved nel'" no Illn,Lter hold. This is provelltltc

.I('1He,

not Ihe form. Anll thcI('1/sl' has 10 he proved

Sll

anl'\~

nlT;lIlSC I ha .... e

1'11- 0 I'

III onln III

,·0

preserve thc validity IIfp rnp nsiIiIlIlS.l1\)uplt. havl'rlm.. held 1I11llllin' :J1lI1 lllnrl'tHI!ll'il !'wm. lakinll: it to he whm i.. essl'Iltiul

42

43

62. Bottom of the page: In eN, the whole paUg?aph inserted here is marked as a parenthetical remark.

d

124 Summer Semester 19 t]

125

Frege's lectures on Logic

Here functions are essentially the same as in analysis. but with an extension: not only numbers, but any object at all can be argument and value of a function; especially truth.

We can add thcse rwo numbers:

(I + I') + (2·1)

similarly:

(l + 2') + (z.ll

values (concept and relation). "In general it is not wcll understood what a function is really SUpposed

(0

be." This shows itself in various things people say. They talk, for

ex.ample, about the SUm of two functions (f(x)"!"- g(x). But really we cannot talk about is, and numbers are, after all, objects. In the"!"- sign we have the sign for a function ofm'o

. the sym bol for a function from the . now distingUish But in this composite symbol we can . n in the two cases and what symbol for its argument. W e can distinguish whalls comma

arguments. It is, in fact, a function we can call a 1'I level function of two arguments.

is different.

sums of functions. Only in the case of numbers is it defined what the sum of the numbers

That is to say: in each ofthc argument places only names for objects can be put, not functions; indeed, functions are fundamentally different from objects, including numbers. Thus it is impossible to have the argument place for a proper name occupied by a function name.

2

lx.

We combine them to: 2

I +x +2x.

63. How should we understand this'! We cannot add timctions.

I+

Different:

1, 2, ...

.

h . rwo functIOns.. '. we have built this function out oft o~e .. t We can, in a certam sense, say that ..' d We can conSIder dlfferen . the way Just mdlcate . . flhe But that is only to be understood m d tennine the values 0 . two functions. We can e n . h 0 different cases, we ca values for the samc argument In the Th by companng t e rw . n functions and add them together. en, t which is in need of samratlO , . his common componen , . ' . f th n >ay: This functIOn IS identi " a common component. r 'J r one can e . . I abbreviated lorm . we caB the new function. n . two functIOns. . of addition, out of those compuscd, by the operation

Let's look at a special case, say:

1+ X

.;2 + 2';

Common:

W~' alwHys h:IVC tn

distinguish the function from ils value for a cer1ain argument Fur instancc, Iherc is thL' value of the function for the argument I.

44

, function of 2 arguments. E.g.. . addition. but for any The !i:lmc holds nol only tor

45

63. Three lines from the bottom of t:h. I uestion mark. e page: 11 eN, a period takes the place of the q

_ _1 ,.

J

.#!

.,.j. .

_

126

Summer Semester 1913

127

Fr!ge's Lectures on Logic

thus, e.g.

F(]~2'::. :;.::,

FfI-·J 2 ,2·J}

We have thus obtained a new function. but it isn 'f the function

I~( of the funclion

2,';'

since the first tllnction is a 1-, level function.

lien: I can again think 01. this as composcJ f case!i and one that \arit;:s I, 2. 3 . II a componl,.'n! that is the same in all three Then again. it is possible to have a function of a function:

\\'hat Slays the same' We can say that I is the limit of the function F(l--~2. 2'';1

Strictly speaki ng H. is ....TO fun ction of a function - ng to say: the sum or th e produci of functions. Also wrong: a E.g., our functions:

~ 2-1- 4'

as the arguments go to oc.

Similarly, I is the limit of the function

Similarly, I is limit of the function

as the argumto=nts go to infinity.

3( - s+1 as the arguments go to 00. 3s~ + 5

Here we have a common unsaturated component. \Vhat varies, however, is lhe fw:Jction .

As an argument for the argument " first function w e can take . e.g.. t he value of the 2 nd for the

It is a de· . were an CllClency of en'l:rJay lanRl/lIge that we have to tal k as I·1· the fiunctIOn

object. Or the valu!;: of the 2nd ~

unction for lh c argument

I

j

Wl'. havl'. thus. II

~

III such a

case Wl' ran talk ahl1ut a lilll\;ti(lll of 1I function.

12'2)'

Or Ihl" vlIllle Ol'lhl' .,n,1 f ... 1Ifidiun lilt th (' lu",Ulllclll

lhl' nWitakl' is illwl1YS lil

I I (2.])'

Thl"cum mun component is:

..__.....

2'~1 levd lilOdion who!'lc llrguml"nts arc 1'I1evc1 functions.

46

47

l'lll1rU~l" the Vllille of the fUlll'liol1 with the functiun itSl'lf.

128

129

Summer Semester 1913

refer in the same way to variable numbers as these symbols do to Constant numhers? Do

to the value of a , . b ut functions. Genera lIy one re Iers . This is all connected to what I Said a 0 . . ,ha'y is a functlOo, . 1 d' to thUlkmg . then mls e 10 function of .. with}": }' =1(x), and one IS a variable number; a . . all neither a constant nor . , Th~ letter v docs not refer to anythlOg at , . in itself. Rather, It IS '. , ,= I + ..2 does not refer to anythmg x combination of S1gnS such as. J '0 which the leners with a sense 1 er g Mit· to be understood as part of a b'I g , a sentence I .thmeticallanguage mucb

We have ditferent variable numbers x, y, z, just as we have different constant numbers 1,

andy arc used to confer generality to t

One speaks, for instance, of variable quantities, variables. What is that? With the 64.

symbol I we refer to a dctenninate or, as One says, ··constant" number. Likewise for Lhe sign 2. But how do We refer to

65.

Frege's lectures on Logic

H

variable number? For that we have letters; but do letters

2,3? No, we don't know how they are different; we cannot say anything about that

AbDUl the COnstant numbers [we] can say: There arc integers, real numbers, prime

he whole. In our usua an

. us ryday language, th . Much is left to eve is not expressed that really Sb ou Id be exprcssed. . d all by itself, some thought . . own as though It express e , eraHty lhe fonnula y 1 T Y! 150 wnnen d The letterS of gen . I ' componellt. In itself it does, however, not d a that . It IS on y a only confcr genera lily on the 7"C

numbers. etc., classes which have certain properties in common. Do we have for variable Ones something like, e,g., the

di~tinetion between prime numbers and non-pnme

, hin in themselves, they used to occurring in it do not refer to anyt g _ b + ac the letters are (b+c)-a , whole. For example, if we have the sentence a dd' if a h, c are numbers.) Id have to a . , (A tually one wou make thc sentence general. e e can see what t he whole t his case w . 'e, we have . an y thing. In But c g nalone doesn't express h Ie thing; for Instant,; ' . " do not have the w 0 s~ntenec b. Ordinarily, however, we

nUlnbers? No, we don', really have that. We don't know how they are dirrerent. We know what the sum of two x

I-

eoo~lant numbers is; but not what the sum of two variables

y is. Nowhere is illaid down how we arc supposed 10 add 2 variable numbers.

in words. , conditinns that arc only cxp rcsscd

48

49 eN an

64. Top of the page, line three: In CN, a period takes the plaee of the Question mark.

lin.~

65, .Sixth from the. topabout of the page: In CN, a question mark takes the place of the penod after !lay an.yth~n.g that."

h page: In • bottom of t e .. above. 66. Four hnes from .the "a(b + c)" ab + oc brack.et to the equatlon

arTOW

f the f'nd of the POints rom

66.

130

Summer Semester 1913

Letters also have this task of expressing genemlity in analysis. But that is not always

131 Frege's Lectures on Logic

aUoget her. . the cxpresslOn . "ariablc" As I said, it is better to aVOId v

easily recognizable.

We have here the

imegral lncidcntally, it also happens that letters arc used to exprcss what we did by "there exists": h· h we can write as follows: as a particular example of the general case, w IC For example, it happens that in an algehraic proposition aile wants to express existencc.

If

Yet no clear distinction is made hetween these two ways of using letters. It is only through our conceptual notation that we are made aware how completcly different the Iwo usages arc.

The point where things are usually unclear is that one doesn't distinguish between the expression "function" and "value ofa function." Similarly. one writcs/instead ofl{.t).

From a logical point of view that is to be rejected. What is unsaturated appears then to be complete.

(a)da.

b 2 Thus we (lre dealing . functions· above, e.g., Y a . We can then replacef(a) by v a n o u s , W have to . , 1'1 level function as argument. e nd with a 2 level function, WhICh takes a . t variable to fill besides usmg an apparen .. W d 't know w h a I' 0 do indIcate thiS clearly. e on . bl One also writes . uld not be reCOb'1l1Za c. . the argument place; withOut 1·t the function wo

dx' dx

-~2x.

And then there is a usage of letters in Analysis that Pcano h
r' , J:)a~da. 67,

This refers to a completely dctennined numher, namdy or that expressiun tnakci'i on difference. This

(J

li.~

Wh1..'lhl'f I writc down 1/.1

Ls thus a pSClJdU.V'lfi,thk

50

nly differentiate functions, not . . h a function, since one ean 0 . Here, too. we arc dealmg WIt .. the letter.t, ObVIOusly . ·nvolved IS we use what the function I numhers And tn make clear . For here we can infer a . t- h",· '" de I- (1(. . ,. " ,·\·lettcrthanlll(d this IS a dJifercnt us\,: () t Ie I_ •. ith numbers. lacing the etters w particu1
51

67. Two lines from the bottom ofth , h pre-vious integral. epage; In eN, an arrow points from "This' to t e

.i

JI..

_

132

Summer Semester 1913

133

rrege's Lectures on Logic

Above , how c\er, . we cannot write:

dl'

dl 68.

~21

because we are d eatmg . wilh a 2nd I . I " . The I,\ level function has to be . . e"e ,unctIOn. mdlcated cl ear Iy, which . it isn't in this case.

We are confronted wilh peculiar questions. The solution will always consist in leaming

These are , th en, a couple of cases I wante . Roman letters. d to mentIOn. in connection with the US
to distinguish between the/unction itselfand its value; as well as in always distinguishing

between the symbols and what is referred to by the symbols. Some people think that the symbols are what arithmetic is about. But that doesn't work in the end. One contradicts

v .

69.

anous other things in Arithme . tIc depend on this issue as welL Indeed, O[]e should always cianI'" 1 ':J, a ways ask the foIl . symbol 0 . . OWing question anew: Is what I am confronted with a , r IS It lhe meaning of the s y m b o l ' . fonned by llsin " e.g., an Integral: Is the mtegml a symbol, g the stroke J. or IS it the meani . power series. ng of a combmation of symbols? Is the a groups of symhols. or is it Iha . Regardless of wh' h t to whIch such a group of symbols rcfers'!

oneself continually. Instead, the symbols are just tools for inquiry, not what the inquiry is about just as the microscope is a tool for botanical inquiry, not what that inquiry is about.

I have now suggested various questions, which I recommend to you for further reflection.

IC alternative one
52

53

68, Fourth line f = 2'1 " rom the top: In eN an . ' arrow points fro <.<.' • • 69. Lines 3 6 • d 7 rn In thIS case" to "[he expressIon •• n fromth b question mark. e ottorn: In eN• a penod . tak . each case, the plac.e ofthe es. In

"d12/dl

____t

J

Summer Semester 1914 Frege Logic in Mathematics b Logir . role ..0 olher sci,nces "' it doC' in mathemarics. At does not play th ' ,arne '

cst In law (definiti ons ), but very differcnt subject matter. matlcs purely logical? Or are therc specifically Are the infcrences in mathe· .

in£ mathematical inferences tb at are not govemed by the general laws of logic" (F..g., the 00serence from n to n + I.) B ut here, too, there" Ihe. mferenec - a general law on wh"h . ." as a ccrtam property tp and ,f It holds genendly tor this ed: "If the number one h . . ..

propcny

, POSitive whole number [n] has it, its successor (n-\-I) also has it, then that _if a ...

({J

each [positive) woe h I number has the property rp." 70.

E.:w.mple. (a+ b)

+m

~

a _ (b + m) is to be proved by Bernoulli's induction. We conceive of the --l-

ProPOsition as. a p ro pert)' of the number m, llssuming a and h to be given.

We have to prove that

M -'" n ~'(I • (n ~ I) iJ

(0 I

(tl

+ h)

+ th I n) + (I' + (n

1-

1))

\'~ ",,",",," (0 I

(h ! n)) t

hoh.!s, also holds; acc\lrding 10 the proposition: c t (1II-1ll (c I 11I).1: We apply the proposition: "h, any C~lllle?\t onc can repln ec a nurnher

by \Jnc identical with iI"·

I~

:~~~~~~~ ~he ~~g~~"humo~,

ve m ut of pag" The following example (Ihe "a"oeiati law"l i' "p.,ated by '''pondi e" 'Om the mam text in eN, As it i' mentioned, but no"pelledo in thce explain text, F"ge probahlydevelOped it freelyTC in hi' lecture, Thi' would of "te y e followmg outline of a proofi"omewhat oh,cu in the note'· It con'"'' of ,u,tJ:t"e tramfmmation of"(a' bl .(n' 1)" ioto "a' (b·(n· I) )." Fo' a "iriei,m p compare L, Wittgeo"ein, Ph;lo,"phi,al R,"wck>, 119751, p, 194[. 11 .' a proof,"

"ght ar.. : ,om t h e h ottom, ught "d" In eN, the pa"nthe'" . 00 the .xpr""on .. Flvelinesfr .. . on t h e

~

',,, b);' m,mng, The propo,ition "c • 1m' 11 ' I' • m) • 1" i' u"rl loy replacing "c" hy I" re' 1m
w su tsfrom "(a'" b) + (n + 1)" ""Thee, Ii " , , ,1n, ""'1'1' n" from the bottom, ngh",d" In" (n • b) • n) d . rhe. expee,,'on • b)" • n " • Ih the expee"ion "n • (b • n I" in a"oroanee with the a"umpt'on "I n • b 1 • n " ad

:'::'~

;~;,t;:t

line, left,id" In eN, the ",oond parenth"i' behind "n" i' ,"i"ing and. in,te . on Car '. an addinonal parenm"i, behind "1." The longe' arrnw on the left, wh"h ide ""pnap' nnt" abo begin. at me level of the lower line, i' ,uppo"d to indicate thar the of ,;,""wn:'( a • (b • n I) • I" ",ul" by ,uh,riturion (cf. note 721 from theright hand. e pteV10us line.

137 136

74.

Summer Semester 1914

a+«b+n)+ 1)

(a+(h+fl»+];

(a+h)+(n+l)

al«b+n)+1);

~~

lectures on Logic

. . ore and more, endlessly. But we can The manifold of mathemattcal truths gro..... s m JI d it ber of truths gets smaller and sma eT an

We get again applying the proposition:

also trace the inferences back; then the Dum

h+{n+l)=(b+n)+l;

. .

(a+ b) I (n+ I) = a+(b+(n + 1));

h ps also definltlons. finally has to come to an end. Axioms and postulates; per a

Thus what holds for n also holds for n + 1. This is the first premise. The second

look at that in more delilillater.

We will

. . I es those . ths in a chalO of tn erenc , "Theorems" we only caU the most Importanl tN

premise is the general law above. We conclude: If the numher 1 has this property, then every positive [whole] number

that arc used as premises in several directions:

has the property. From this proposition and the proposition that the nomber 1 has the property (as above: (a + b) + 1 =

75.

Q

+ (b + i) it follows that every positive whole number

has this property.

o

Thus every mathematical inference is analyzed into a general mathematical theorem

t 'onvince (in the in Euclid) less the purpose 0 c . , . but to establish a logical Sometimes a proof has (also already , ld beheve anyway). case of simple propositions whtch wc WOll . 't ro of truths . . ' 1 of mathcmatlcs a sys C , connection, Euclid envu>loned as the Idea " ake the number of bastc. . . , The aspiratIOn IS to m . mterconnected hy logical mferences. fwhich all ofrnathematlcS .' . to find the kernel out 0 . ' unprovable truths as small as pOSSible, I urety logical mferences. . . necessary to draW on y P can be developed. For that purpose It IS th logical laws, on the one Iy between e -) separute Sh arp One must (sec the example K hOVI,; h ther . . " d theorel11s used, on teo ' hand, lind the m.lthenmtleal aXlOl11S an If ' i1l10 procricl? lia.~ n t 1I"/1i<'h ht' did 1I0t 1'11/, /I .l TillS ,dcill, III \\'hiC"JI ":lIdid fl.Win'd II

or axiom and a purely logical inference. We distinguish: inferences frum 2 premises and inferences from] premise.

From 2 p"mis," a new truth followse

A

From Ih" and a 3'" a new one follow, ag"n

~

We ean al", d
I

c.g., if II rrupositilltl holds for C'very positive whuk numher (Ull! we

r

SlIY

tlwl it

1111.S

.. ldfllt'tlli//it'st he,'lIlllmo.I'/ .'/lIil'dl' /0.1'/ ill "WI'(' rt/t ("t "

alst)

holds for the number 2.

2 74. Second line, right side- In eN an srrow . f h . n "c + (m + 1) '" (c + m) + 1" abo~e.' pOInts rom ere back to the equauo 75. Ten lines from the top: End of the examplo . " ( c f . note 7 0).

3 . diagram illuscrates a PO . . . The followmg 76. SlXhnes from the top. 'nts the theorem to which the arrow pOl '

SSI'ble fork originating in

76.

138

Summer Semester 1914

Every mathematician works in his own area without attending to how to integrate it

into the whole system. Thus the disorder.

It is not just a matter of extending the chains of inference further and further and increasing the manifold of Imths more and more. but <'Ilso of going backwards and discovering the basic truths so as to develop out of them the system o/marhematic!i,

139 frege's lectures on Logic

~ d "'J" "derived sentence "'Axiom," "theorem," not: "basic sentence [(mm saL, . . b tht: sensually perceptible sign [Lehrsatzl" Since we want to understand a sentence to e . f I Th ught is not meant 1tI a whose sense is a thought. Only this thought is tlUt: or a se. 0 . is independent psychological scnse h..:re! Th "... sense of the Pythagorean Theorem, e.g.,

convillcing, but their logical interconneUion is also import<'lnt How one thing follows

from the various subjective thoughts. . ... . antee that an object with ccrtam Postulates are really also aXIOms, they guar . Is· not meant as !I. properties exists. (E.g., in Euehu:, a I,· ne through any two pom . ,

from another. Today too much weight is put on showing that a claim is evident; nol

construction. )

[n mathematics we do not, like in the other sciences. just have to make claims

enough on which web of inferences that supports it.

.. f si s can he replaced by a simple sign. Definitions are supulatlOnS that a group 0 go

Thcf(ml1da(irms consist of axioms, postulates. and perhaps definitions.

Thus we distinguish:

Axioms are truths that do not need proof and that within the system, ure also not

I) tht: "defining group of signs"

proved. (But that is not enough: sinee Euclid already proved various things that did not

2) the ·'defined sign" which is new.

seem to need prooL, Thus: I)

trulhs; an untrue system is a contradiction in it~c1f.

2)

belonging to a CI'I"'(/i/l.\T.~/em: it is possihlc for there to he scvcf:ll systems of lIlathl'matics. E.g., A H ('/)HF; possibly A could he proved from Hand ( ... F. or alsll IJ from A and ( ... f·; then We have lhl' choin' or whdher nnd IJ as a Iheorem Of "ict'

In

lakl'

,.j

as

an ,1.l(lOltI

\.I'I".\"(f.

. the parts orthe hI· to the particular sIgns. (To the sentence corresponds a thoug , -l Id have to adu In hI) Tht:rcforc we wou scntcm:e, correspond parts 0 r th e thoug . . . . . , . as the roup of signs has a sense. _ .' conncction with dclltlltlOns. wsof"r g b h nCW simple . • .. I re lace thc group of signs Y t e . It". in accmdann.' With the delltlltlon, p . tly a definition 'Inn l'S nnl the thought- (tltlsequcn sign, thcn it is only the sentence thaI c , g., _ I ,,~]. only for . . [I ,·k Jes b,stmmll'rl Uln.,.,. . 1 1 " f clmncctlons -(l~ r is mIl rl'<.,l1y necessary lor I lC ll~pe \J lhc case Ill" c'Iprcssioll

4

5 . . . n" is underlined three time5. DO 'l" the word "deClnl 77. Nine lines from the (Op: In C~~,

_ _ _.1~

140 Summer Semester 1914

A definition is introduced by means 0 I a sentence th t' h . identisch ist] I th a lS t en Identical lderdunn . n c subsequent Construction f til . as a premise for' fi 0 e system that sentence is used fonnally III ercnces, although not contentu Ii . the whole . , a y. from the sentence that contains group of sIgns the fonnally sImp Ier sentence can b d . d b law of identity. c enve y means of the

141

frege's Lectures on Logic

Objection: How can it really be doubtful whether the sense of a complex sign

coincides with the sense of a sign that has already been in use for a while and whose

,ensc has long been fixed. Well, suppose it is the case, but we only "sec it as if through a Frequently something is smuggled into a . recognized h mathematical definition that should he

fog"!

as a I corem or axiom first.

Some definitions in mathematical textbooks are never cited or used later on, not even

Definitions are logically superfluous,

implicitly; they arc pure ornament ("ornamental definitions"). Already in Euclid: e.g., u

but psychologically valuable.

line is length without breadth, etc.

Dejinitions don',·Just. he1p to construct but als . order to reduce th , 0 to ana~vze what IS complex, e.g., in e number of axioms. Such an an I . only feci that on ' h,," a YS1S cannot be proved right; one can c as hit the naIl on the head . M . ,and It can prove itself fruitful ore preCIsely'. we construct ' . the system again b . Y USlOg the result of the analvsis Occasion"lly hI' .. walsfi d' . xc In a definition is. the .se nse 0 Ia ' use fbr a h'l sign that has already been in W Ie . Oncc'annut . prove that, thou h· j h· . . ' .', . Stlpulation bu,' . g , t as to be eVident; It lsn t an arbitrary an aXIOm. , . LctAbcthcoldsi . gn,'1' et s assume that a c '. . ertam complex sign coincides in sense with A, If we don't k nuw that fur SUre w that B is t h . , e proceed as follows: We stipulate arbitrarily o ave the 'sense . u l'lh c complex si If' , sense of A h ' gn. the I' definition was correct then the as to coinCide with that olB . WcavOld ' th . system by using B If e sign A and rcconstmet the whole . that reeo t ' , ns ruetlOn succeeds . . . . . IOtroduee the sign If " " we can. tor pragmatic reasons, also agam; we Just have to re '. . . ., gard II as newly IOtrnduced. as if it hadn't had 1Io sense before th d e efimtlon.

Given the way in which mathematical texts are written today one can never see whether a defmition is (implicitly) used or not. Each proposition should really carryall its premises explicitly with it. Sometimes things are called "definitions" which really aren't. If in algebra the 3 numbers x, .1', z occur in 3 equations, thcy are not always detennine(] completely thereby: It is possible that several such systems of) numbers satisfy the equations Thus: If the expressions "point," "line." "plane" occur in several sentences. their sense is not necessarily detennined

u"ique~r.

It is not clear whether se\-'cral solutions are

possible, or none at all. That is. then. no definition. When an "empty sign" in a sentence has no sense, the whole sentence also has no

~nse; it docs not express a thought, only a task: to find a sense for the empty sign in such a way that the whole sentence acquires a sense and becomes troe.

i

6

J

7

______ 1

,,1-. ,ill

143

142

Summer Semester 1914

frege'!i lectures on Logic

If a concept is given to us, we don't know yet ifan object falls under it. It is

possible that an object falls under the concept, but also that no object f~lls under it. It is false that a proper name is a concept [word] with only one object falling under it. In the

sentence: All humans are mortal, its sense does not include; Cato is mortal; for that a further premiM: and some inferences are necessary. More precisely, the sentence would fead: Ifsomething is human, then it is mortal. lfwe move from the general to the particular, we get: IfCato is human. he is mortal. To this we add a 2 nd premise: Calo is human. Only from both of them we get: Cato is mortal. As this is not directly contained in the sentence: All humans are mortal, the concept docs not denote [hezeichnet nichtj the object falling under it We have to distinguish: Whether we replace an old group of signs with a new SIgn or whether We assen something about something.

An example for the

r

J

pcrceivable by the sen!ies. ry different system!i of .. , nee tions. From them three ve Thus three very dItferent (,,:0 p . not present so far. ber noW temsIS, thus, arise nile s y ' I d be a num be r, too'. this num arithmetic would havc t o · u railroad tram wo According to Weierstrass a

kind ufdefinition: We give 2 sentences

i!i a prime number. Increased by 2, x is l1ivisible by 4.

X

}

Hereby an object. namely 2.

is determined.

Here we have a l:oncepl with two characteristics; thereby I have not uclcnnillcl1 an

. Herlin.". , tic', as if the differences cnmcs radng along !rom '. 'usl one onthnte .. h whole there \S J • One always ads as 1\ 01\ t c

object. but a concept; evcn if in this casc only 1 tlhject fulls under it. An object falling under a concept is. lIncr all. nol the mcaning of the sign concept.

, ]"eg • ,. 1/5 [WorrdefinitlOnen, ." some . , b " mi1Jaj de/ml tlU . ly . , . expreSSIon can on One talks reprovmgly a out no . . But thIS reprovmg . . h F gean definitions. sense people in connectIOn WIt re but must also have a .. I consist of words. mean: What gets defined must not on y . d 't give proofs for groups ' .. Since we on . ltd in mathem aUt.: s . of that This however is oftcn v\o a e 'bl to get a clear grasp ." . and it must be POSSI e . ell of signs but for the sensc thcy have, . t giving definitlOnS as w . • . . more careful attention a sense. MathematiCians should pay b' gs f similar t m . A number is a group 0 Weierstra~'s: , ,"3" A number is the stgn. . . ed by the senses; I 2"d (Thomae?): . that cannot be perceIv . 3 in them is not A number is somethmg 3'd Wrege?): . S' so what IS can. e.g., , aIk about 3 axtOm ,

f(H

lhi,~

resided only in

8

'11l;ant • l1etails.

insl~lIl

9

Carnap adds as a . thIS sentence. I ., . I connection WlthlHeiterknt Jm Zelltrum ' he bottom. n the center 78. Three lines from t ' . "laughter 10 comment in the left marg:tn.

d

78,

145 144

Summer Semester 191-4

~'s

lectures on logic

Mathematicians are not concerned about that in any further way, The formulations of propositions in the different systems of arithmetic
d d as a unit and " de can itselfbe regar e U f '.h ] the concept a ' to (bcelc ne j ·bbbb'" Let b r e e r " · h neept be posited repeatedly: t = . g . , . . h r the sign: h b b b, t e co . t repeatedly. or rat e .] magnitude Ifl now posit thlS concep .50 instt=ad, to [posit a . . ew results. probably one I , . that ., d h to stand for the express tram, doesn't change at all, and nothmg n , d But even It I regar the determinate single magmtu e Or erhapS, contrary to ositing it repeatedly. p led into the [rain does not get changed by P . ' t In that case we are .' . of the train that IS mean. different formulation above, It IS my Idea . . ent people would have , d b'ective; then dlfter . bout the realm of the psychologICal an su j tly not be talking a ld , const=quen , s and they wou . . ·d numbers, smce different I ea., .

philosopher to analyzlo: their sense. A self-respecting mathematician will nol bother to do that. "At most he will occasionally, in an unguarded moment, drop a definition in 79,

passing, or at least something that looks like a definition." At bonom then: is the realization, I assume, that the content of the thought is

80,

the main thing. But perhaps the different conceptions ofnwnber do, implicitly, coincide at bottom? While everyone misses the mark slightly in his or her definition. Then, however, the proper sense has to become manifest in proofs. Yet in them no use is made of the definitions when inft=rring corresponding laws. It is for that reason, too, that the different definitions do not stand in conflict with each other; Lhey lay heside each other.

81.

E.g., according to Weierstrass: A row of books in a bookcase has to be a number, the express train at 5: 14 as welt. If I now multiply those numbers with each other. I have to get another number. But how, then, do I find it From Weier.\'trass's lectures: "According to our definition a numerical magnitude

[Zuhlengrr)jJe] results from thl': repeated positing of similar clements." Thc definition OLI

row of similar things, Things arc to be counted as

similar if they cuincidt= in a certain wmplclI. of characteristics. Such a row is, theil, what we mean by a numerical magnitude." Does the sentence above rt'ally Itlliow from Ihis definition? Accurding to the delinitioll it is nolthe idea 1lt"t11i: row, but the row itscJfthat is the numerical

ma~nilude.

copt ofmag m

same thing.

like the animals in paradise [sie liegen neheneinander wie die Tiere im Paradiese].

says Ihis: "We can form the idea

Further it says: "Now the can

. ration? IC, b" HoW do we know that? Is it an How do we get to mulllp S d that contam all these . the roW that thuS obtalOlllg ' There IS now a magmtu e resident Wilson, ] (hiS P a:uom? Ex-ample W, POSit repeatedly d appeUatiVUm lApp elatlvum ), m S Wilson (concept wor , h we mean the sum conlams all the presuient peatedly "By a )( t ng • name) appears re . b an a_times POSl I row president Wilson (proper .' merical magnItude y . .. "We obta lO thiS nu b b_limes • if h Islhe consisling of , 1 terms of h. .. " But whal is meant y .. suddenly , " sltmg ofa . eHexerE-'I]. of fl,
Su, c.g., a railroad lrain is supposed Itl resull from lhe

repeated positing of a railroad car?

10

"

tis "'b"·

79. Five lines from the top; Here Carnap adds in the left ' . '"I hter on the left [Heiterkeit /inks]." margm. aug 80. Seven lines from the top: What is meant is "the din d d i . f the cOllf:t:"ptofnumher." e.rentun erstan ngsa . 81. TWdve.lines from the. top; H~re Carnap adds the marginal com ent: '"The old mall waxes poetic! [DerAlte ll.!lrdpoetuchfJ" m

___ djh

)

'~,

02,

, 1-46

Summer Semester 1914

147

hIp.'s lectUres on logic

\

I

.

I This sleight of hand is cleverer than the usual: the performer doesn't just deceivc the audience, but also himself -the pinnacle of art. In other scicnces such sleights of hand 83.

Occasionally Weierstrass also talks about

<'I

"set," as well as about the "value" of <'I

numerical magnitude. Here hc probably means what is usually meant by number. "A numerical magnitude is determined if it is declared which clements it contains and of

the number has indeed been defincd. Before that the concept of number is extended in order to make division always possible. Now a number is no longer, as before, supposed to consist of similar, but of dissimilar clements. The definition, however, was: "Each ofthc repeating clements of the row are called the unit ofthc numerical quantity." Out ··the unit" is a proper name: so

Thus the actual number is, contrary 10 the initial definition, brought in through the back door as a "sct," "value," or as a "how often."

12 ~3. Three lines ~rom the top: In connection with this whole paragraph Carnap comments

..

0= -

I I I I

it can't mean each one of them.

the left rnargm: "laughter in the whole house {Heiterkeit im gan.zen. Hause]."

I

-j

each of them how oRen they are included." If one knows what ';how often" means, then

In

.

0=

arc not popular at all; hut in mathematics they are, so to speak, presentable.

Plv~h~

to this questlOn, " thin" of va \ue . , t ibuted some '" h difficult Weicrstrass would ha\e l:on r d.dn't even sec t e 't~~1 'allv thought about \ . intelleclual powers. had he rc , " as a system. I ~ .d 'a1 of mathcmatll,;S .dent ica1, 1 roblem in it. He was lal,;king t 11.: I C t things that afe nonP . conncc 5 f -quality a1so \ Wcierstrass thinks the sign 0 e . \ ,. I + h '" h + lJ. therthey arc and h~ gives as hiS ex-amp \;. , . .ome respect, ra would . t similar III S But: l 3 .., =:- 'i hoth sides arc not JUs . . ( .. ee for that task we n -r - - ' tas\<. of addlllg sm . the , .. we don't mean the 1 f oIl1 completlllg identical: since by ·'3 er that reSll Is r . + ') l- 4)_ but the numb . res ect to a new not have to add a number. (3 ~ fi dO\ltlater\lllth p . S Just as we may, e.g·. III Iready kneW task and that is the same as - . . . .d ntical With one we a d n10untatn that it IS I C mmet or a newly dlseo\,ere different before and called something else. \It the sentences have "') is the same, b has cognlll\ e S nd'iO=]-+-d entence The meaning of 5 a - Thc secon S . haitI different senses heliocentriC VieW of \ thought ('Im/ntt [Gt'Jdllkt'l/In ' ., d .·thc founder of the h I say: "Copernicus ail . ren ee between w en t .. till a dirte content [Erkcnntnisillhtl/ ]. ,.' 1e man: but thcre: IS S ..... steml" and "mcan tht.: s.Ul t·the solar s, tcsnarsystcm h 1 ,·.vicwo ~ lrIC . • 11' the hc!llll,;Cn . 'cS. "C\lpcrnicu:- is thc hmndcr {I . h'wc different "ens Tho:-": scntences ' ·'Copernicus is l 'npcrllJl,;IIS.

/3

148

Summer Semester 1914

Frege's Lectures on Logic

149

If we consider this to be one sentence, I ean negate it as a whole; then there arc the following possibilities: the I
1) a meaning: the thing abollt which something is said;

is a positive number and

0

is a 3,d root." We see: If we want to define a concept, we

have to indicate in general the place for the object (e.g., by means of a), since the concept

2) a sense that is part of the thought.

(predicate) is supposed to have general validity.

The name Scylla has no mean', I " lng, on y a sense. Mount Aetna is higher than Vesuvius." Here it is not the m .. . . oUntam Itself, with all Its masses ofmck that is part of the thought; rather somethin invi "bl ' g 51 e has to be part of the invisible thought: the sense of the name Aetna. Similarly 5 and 3 + 2 have the same meaning, but ditferent senses.

Ifboth concepts have sharp boundaries, i.e., for each object it is determined whether it falls under the concept or not, then the composite concept also has sharp boundaries. Another example: Suppose "(I is a prime number" is the expression to be defined: "There is no whole number> I and <

0

such that a is a multiple of it." But, e.g., the

number 3/2 also falls under this concept; thus we have to add:

We define concepts r I t' , e a IOns, and (what is most difficult) ohjects. A concept is in ~ecd of supplementation al . . E.g. ' ways has predlcallvc character.

64.

~~ I

The two pan

art . P

IS

S

"0

is a positi'o'c whole

H5.

number"; "a is > 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 qU
lhe sense of a group of signs is indctcmlinate, E"Vcry sentence, i.e., every complex of signs (according to their rules), hilS to be either true or false.

/4 84 , Four ij nes from the b

understandin

they still d

?

f ottorn: This remark' g 0 the Conte:x.t princi 1 . If 15 revealing with respect to Frege's (later) o so as a Contribution to thP e, the parts have senses "in themselves," then e senSes ofwhole sentences.

/5 "." ositive" is containpd in "'0 i.~ 85. Middle of page: The emphasis is on "whole num b er • p . ' ). 1," thus superfluous. . f h in tf'xt hy flngl/' Ilrack,-to;,. 66. Last line: In eN, this paragraph IS separated rom t f> rna -

P.6.

,

Summer Semester 1914

150

151

frege's Lectures on Logic

E.g., if we apply the general definition

Isaa~~ltIPleof7

b)

Arithmetic signs, too, have to be defined in such a way that, no matter which object

a

IS

)

a is congruent to h modulo 7

a whole number

and

(not only numbers) we put in the place that needs to bl: supp!cmentt:d, we get a detioitt:

b IS a ...... holc number meanmg. we gd: Ifwe claim that the sentence "Aetna is higher than Vesuvius" is true, then the two

(16 - 2) ; " multiple 7

proper names do not just have a sense (as even names in fiction do), but also a meaning:

and

J

16 is a whole number

the real, external things that are designated lbezeichnet]. Now tht: whole sentence also

and

(

hus a me,ming, besides its sense. This meaning has to remain the same if we replace

16 is congruent to 2 modulo 7

2 is a whole number

some parts oflhe sentence with others with the same meaning, even if they have different But if we write:

senses. E.g.,

and

"17 - J is a multiple of7." The 'nllll vul/le remains unchanged. It is,

)

(16~3) is a mulliplcof7

"\0 - 2 is a multiple of7"

16 is a whole numht=r

therefur~, to be recognized as Ihe mt=aning of

and

(

3 is a whole numher

16 is congruent to 3 modulo 7,

the sentcm::c. then the whole thing is still right the sense of the individual sentence is independent of

In ordinary usage: 'The sentence (or thought) is Irue," it seems as ira property is ascrihed to the sentence. We du nut seem 10 be dealing wilh a relation bdween the

whether it is true or hllse. We can, in addition, give a sentence assertoric force or not

sentence and its meaning (a thingl, hut with a relation hctwl'l'll an Oh.il'l"t

(c.g .. in tidion). We nUl grasp a thought without asserting it as tn/C, without judging.

111\d

this

property. However: n\llhing is added to lhe senSl' \.r"'i . r' hy sayil1~ 111<11 it is trill'. ('he scnll'nn' "(II P\ISSlllLI ('111/('1'/)'

16 87: Two lines from the bottom: The object would here be the sentence seen pnnted sentential sign,

8S

a WTitten or

17

2) is a lIlullipk ofT' is in ill'ed nrslIpplc\llentation: it fomts a

nSSl'ftillll wlnrh .. :Ill,

" "\",,,.. ",' I (,', if Ilto so, I subs.um..: 16 under this.

,I, l'.~'"

...

Summer Semester 1914

152

,

Frege's Lectures on Logic

153

The equality sign = does not stand for [hedeurt:t nich/] the copula, but for identity Consequently Analogously to how (3 -- 2) is the value of the function (a - 2) for the argument 3, we can say that

apparently says that

"(16 - 2) is a multiple ofT'

2 is a square root of 4;

and

(-2)'(-2) - 4

"(17 - 2) is a multiple ofT' arc values of the function

says that

-2 is a square root of 4;

"(a - 2) is a multiple of 7'"

bot

for the arguments 16 amI 17.

docs not say that

What ajimclion is cannot be defined, it cannot be reduced logically to something

2 is a square root of 4,

more simple; one c<Jn only hint at it. elucidate it.

bot What lor us is an argument is called "subject" in [traditionalJ logic, our concept is

2 is a positivc square rool of 4

l'allcd "predicate."

The signs indicating generaliry are used to confer generality to the sentence. They III\.

rhe usc of limdillll in mathcmatil,:s is quite obscure. One talks ubout variables as if

have to be rcplaee
general to the particular.

2 by

Therefore a letter call not stand for [hcJclIleII] a "variable number·'; since there isn't

2

H9.

4

2

anything at all the lettcr is supposed to stand fnr---thcre arc no variable numbers. In pure

5

2.

arithmcti..: time

doc~ nol play ,1

rnle. lim, conscquenlly. docs variation. One talks about

it is not as. ifol1c thin"ev<Jrics -..my Vllflallon -. . ' " Sm" u 111 t hc thing that ....arics would Iwvl'to

vari.1hk "qll
ilf

r~mal1l

/illl l .t;O/1 whl'!"e f indil·all's till' (l1'!l:lll11l'nt pllsilill11,

BUI now

the saine: (like a monarch Whll
il'On that

chal1ge~ over time i~ a

we havc moved over to the

uPlilinl/ioli ot"llrithllll·tic Till· v:lriatioll dlll·S nllt bl.'hl1lg hl anlhll11.'lic proper.

Dnt: ntkn confusl's tht: valuc l)f the fUlll-tilll1 with the I"undion itself; the value is an

18 88. Eight lines fTOm the botton· Wh at F rege means, more precisely, is "the use of the expression 'fu~ction'." 1, 89. Lastp.line: in F rege ' 5 Nachgclassene Sr.hriften (19fl3j, 236.Comparc the corresponding passage _. <

object and it isn't in lIceJ ofSUpp!clllcntatioll any more, There is a diHiculty here: The expressions ';conccpf' and "function" arc logically objectionable. Since one c;'ln say ·"the function," which is then a proper name and has to stand for an object_ The value of the function is. indeed, an object, which is thus easily confused with the function itself.

19

155 154

Summer Semester 1914

x is the argume not t f h c function 1 + 2'(. N ow, it looks as if this func!,·on

2,/' .' , so In the functIOn .g-.

w'-'~

the

A function of2 arguments, e.g., .; -

"'.,

argument in (l + .

argument positions

91.

r

numbers. it is to be "the False." If a function is first defined narrowly and later expanded, then this expansion changes the function; and if one still keeps using the same sign. this ambiguity can easily lead to confusion. In the development of mathematics one does, however, reach certain points where one wants to expand the system. But then one has to begin from

.

Ines rom the bOt[om' Fo

der Anthmetik. .p. 72 • t'ootnote.-

~cratch again.

E.g., une would h",e to proceed" folluwS' ,slong as the plus sigo + i, u>cd 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 v
~

90.

Frege's lectures on Logic

(1 +

s• Grulldgcsetze r the expressio n ' .. constant," cf. F rege •

-_.,.,,~.

156

Summer Semester 1914

A sentence, e.g" 2 + 3 "" 5, has

Let's assume for the . One cannot develop non-Euclidean geometry b y saymg:

1) a sense, which is its thought content [sein Gedanke]. By combining the words 92.

157

Frege's Lectures on Logic

moment that the parallel axiom is not true, so drop it.

which stand for [hezeichnen] parts (lfthe sense one can form a variety of

But is it possible to draw a conclusion from a false sentence?

sentences. If I have the sentence:

If A is true, then B is true,

then by means of the axiom:

A is true,

proper name docs not just have a sense, but also stands for [bezeichnef] an

we can conclude:

B is true.

object. The predicative parts of the sentence stand for [bezeichnen] a concept.

On the other hand, if A is not true we cannot infer anything.

2) a truth value, which is its meaning. The closed part ufthe sentence, a proper name, stands for (bezeichne/l an object; in science it is necessary that this

Object and concept are quite different.

If A is true, then B is true,

Then again, suppose I have:

If B is true, then

r

is true.

Conclusion:

If A is true, then

r

is true

Continuing further:

If r is true, then

Conclusion:

If A is true, then t1 is true.

Among the sentences we distinguish between axioms [AxiomeJ and theorems

(Lehrsii/ze]. Axioms are true and unprovable; they have not been derived themselves, instead

.:::1

is true.

everything is derived from them. An aKiom has to be true; and for that it is necessary that no part of it is still indeterminate. (EKcept for general signs, which confer generality uf content to the sentence.) ()e/inition.\· arc stipulations that a familiar complex. sign is ID be rerlaced hya simple,

new one. This new sign hus thereby acquired a definite me.ming, and tile st'n1cncc hecome.s tautologieul. This talltological.selltenn' cun then he Ilsed;ls

t in all the sentences. But usually one docs be d assume '., . sentence, althoug h they have to not write down all the premises 01 a . . . A is tme I can still dra...... . ' . 'fl Jon't prt:suppose the aXiom. ' implicitly <11' conditlOl1s, 1 hus I . , , t ee . - . If 4 is true, III ewry sen en . condusiollS. alhcil alwilYs with the condItIOn. ' .

t

The condition "If A is true" remainS cons an

il

prt'misc,

lOll

22

23

92. Three lines from the top: According to Frege's al . 1 it should say .. exp re ss [ ausd" usu t:ermlno ogy, rue k en )" h ere'Instead 0 f "stand for [ber.eichnen]."

t

158

Summer Semester 1914

Frege's Lectures on Logic

159

Ewmpie

93.

We presuppose: If2 sides ofa triangle arc equal, then the opposite angles a.re also equal. [To prove:] In ......... ·V·~ J tn·,ngle the larger angle lies opposite to the larger Side.

~C

Similarly for indirect proofs. E.g., suppose we have to prove that E is true.

A

Assume I have the sentence: If E is not truc, then Z is not true; ifI then know that Z is Thus:

true, I can conclude that E is true, since the sentence abo ....e can be transposed to:

II

If Z is true, then E is true.

III

A sentence that is an axiom in one system can he a theorem, thus something to be proved, in another.

IfAe - BC, then L B -= LA.

IV Ifa=b,thennoth>a.

IfAC L B.

V If a> h, then not h > a.

lfnot AC >BC, then AC~BC or BC >AC If AC =BC, then LB =LA. If BC > AC then LA> L B.

Ifan axiom is only posited hypothetically, It should really be added as a condition to

(a I

II

each sentence, a1 least implicitly. If we don't do that, the theorems derived from il

If LB =LA,thcnnot LB >LA.

IV

(ilppcaf to) have more general validity than they really do.

If AC - BC then not L H

:>

L A.


If / A

:>

L A,

---.--> /

B, Ihen not L B

If BC' ','1(', then rwl L H' / A.

V

(y (a

11'11111 .'Ie' . JI(',

(I')

24

iflltlt .-/('

IU',thclIl1ol.!B>LA.

''/.-f, then 11111 (I(' /Ie. [I' /1/ ·/A.ifnol AC ·>B(', then nol L B > LA,

If. /I

25 . . of a roof, several arrows are a ddedin 93. First line; In the followmg outhne p

eN

he third line after ( (1 I to the lim'! (fj)~ II 'g(a)andfro mt - from the line immediately fa OWln • after (fJ) to the line (r). . a f ter ( a ) and from the line and b !Itand for the ang les, L_ A and L R. - from the second fme

-from (a) back to (II II ;

. h' line the letters a L B " 94. Fifth line, right SIde: In t I S , 1 . "If AC> Be, then LA) - . B . 95. Sixth line, left side: In C'II.T'tsayswrongy. J~, 1 db stand for the englesLAendL . his line, the letters a an . ents AC and BC. . " 96. Sixth line, nght Side. In t b and for the line segm ., he letters a and !'it 97, Seventh line: In thIS hne, t

.'>I

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