Journal for the History of Blanchette on Frege on Analysis and Content Analytical Marcus Rossberg Volume 3, Number 7 All contributions included in the present issue were originally presented at an ‘Author Meets Critics’ session organised by Rich- Editor in Chief ard Zach at the Pacific Meeting of the American Philosophical As- Sandra Lapointe, McMaster University sociation in San Diego in the Spring of 2014.

Editorial Board Gary Ebbs, Indiana University Bloomington Greg Frost-Arnold, Hobart and William Smith Colleges Henry Jackman, York University Kevin C. Klement, University of Massachusetts Lydia Patton, Virginia Tech Marcus Rossberg, University of Connecticut Mark Textor, King’s College London Audrey Yap, University of Victoria Richard Zach, University of Calgary

Review Editors Juliet Floyd, Boston University Chris Pincock, Ohio State University

Assistant Review Editor Sean Morris, Metropolitan State University of Denver

Design Daniel Harris, Hunter College

©2015 Marcus Rossberg

Blanchette on Frege on Analysis and Content (EQ) F ≈ G As Frege puts it: “We carve up the content in a way different from Marcus Rossberg the original way, and this yields us a new concept.”1 Equivalence statements of cardinal numbers thus receive their content and truth conditions. 1. The Analysis of ‘Cardinal Number’ The Grundlagen analysis employs extensions. The cardinal With Frege’s Conception of , Patricia Blanchette presents a de- number of the concept F is defined as the extension of the second- tailed and comprehensive analysis of ’s notions of level concept equinumerous with F, ext(Φ ≈ F ), thus the target anal- logic, logical consequence, and conceptual analysis, and the role ysis, to be demonstrated to follow logically from (EQ) is these play in his logicist project. Blanchette’s investigation is of fundamental importance for Frege scholarship. It will prove to be (NGl) ext(Φ ≈ F) = ext(Ψ ≈ G) consequential not only exegetically, but also in providing contem- Blanchette makes the case that the logicist reduction proceeds by porary (not just Fregean) philosophy with a better understanding stepwise analysis. (N) is analyzed as (EQ). By purely logical inter- of distinctive contributions Frege made that have unjustly been derivability of (N ) and (EQ), these too have the same content. forgotten but provide enticing alternatives to current conceptions Gl Therefore, by transitivity, the sameness of content of (N) and (N ) of logic and analysis. Gl is established. There is little to complain about in Blanchette’s fabulous book. The mutual derivability (N ) and (EQ) is demonstrated using The friendly suggestion I want to make pertains to her account of Gl Frege’s notion of analysis in his mature logicist project, and the a principle that allows us to transform propositions of the form relation of this notion of analysis to her explanation of Frege’s (UG) ∀x(Fx ≡ Gx) conception of logical consequence. Blanchette investigates the question what exactly Frege’s logicist reduction of arithmetic into propositions of the form amounts to. Crucial here is the “recarving of content” that is said to be achieved by Hume’s Principle. Frege’s claim in Grundlagen is (EXT) ext(F) = ext(G) that statements for cardinal numbers of the form: and vice versa. In Grundlagen, Frege does not explicitly state this (N) NxFx = NxGx principle while clearly relying on it and its obviousness. By the time of Grundgesetze, Frege has this transformation formulated in have the same content as equinumerosity claims: that F is equi- full generality. Instead of extensions, he employs value-ranges. All numerous with G (i.e., that there exists a bijection between the functions, not just concepts, are now equipped with these corre- concepts): sponding objects. Frege gives the following explanation (Frege 2013, vol. I, §3, p. 7):

Journal for the History of Analytical Philosophy, vol. 3 no. 7 [2]

function (⇠)”

always as co-referential with the words

“the functions (⇠) and (⇠) always have the same functionfunction (⇠()”⇠)” function (⇠)” value for the same argument.” function (⇠)” alwaysalways as as co-referential co-referential with with the the words words always as co-referentialalways as co-referential with the with words the words Frege continuesfunction using (⇠)” the term “extension” for value-ranges of concepts, “the“the functions functions(⇠()⇠) and and (⇠()⇠) always always have have the the same same “theI use functions the“the words functions ( ⇠) and(⇠ )( and⇠) always(⇠) always have have the the same same ics to logic is carried out by the analysis of (N) as (EQ), which in valuesince for for the all samealways intents argument.” as co-referential and purposes with the words this is what they are. This “transforma- “thevaluevalue function for the for same theΦ(ξ same) argument.” has argument.”the same value-range as the function turn is logically equivalent to (N ). Blanchette’s central claim is value for the same argument.” Gg Ψ(ξ)” tion of the generality“the functions of(⇠ an) and equality (⇠) always into have a the value-range same that logical equality”, consequence as Frege is preserved over Frege’s analysis. Ac- FregeFregealwaysFrege continues continues continuesas co- usingreferential using using the term thewith the “extension”term the term words “extension” “extension” for value-ranges for for value-ranges value-ranges of concepts, of of concepts, concepts, value for the same argument.” cordingly, Frege not only shows that the laws of arithmetic can be Frege continues using“the the functionscalls term itΦ “extension”(ξ (Frege,) and Ψ(ξ) 2013, foralways value-ranges vol.haveI, the 9,same of p. concepts, value 14), for was the dubbed the “Initial Stipulation” sincesincesince for for all for all intents all intents intents and purposesand and purposes purposes this is whatthis this is they is what what are. they This they are. “transforma- are. This This “transforma- “transforma-derived from pure logic for his cardinal numbers, but also that the same argument.” § Frege continues using the term “extension” for value-ranges of concepts, since for alltion intents of the and generality purposes of an this equality is what into theya value-range are. This equality”, “transforma- as Frege same holds true for the cardinal numbers of our ordinary lives, tiontion of of the the generality generalityby Richard of of an an equality Heck equality (2011). into into a a value-range value-range It gives equality”, rise equality”, to the as as Frege ill-fated Frege Basic Law V in the Frege continues usingsince the for term all intents “extension” and purposes for this value is what-ranges they are. of This “transforma-those that mathematicians have proved theorems about all along tion of thecalls generality it (Frege, of 2013, an equality vol. I, 9, into p. 14), a wasvalue-range dubbed the equality”, “Initial Stipulation” as Frege concepts,callscalls it it (Frege, (Frege,since 2013,for 2013, all vol. § vol.intents I, I,9, 9,and p. p. 14), 14),purposes was was dubbed dubbed this is the thewhat “Initial “Initial they Stipulation” are. Stipulation” and continue to do so. Grundgesetzetion of the§ § generalitysystem of of anconcept-script: equality into a value-range equality”, as Frege calls it (Frege,byThis Richard 2013, “transformation vol.Heck I, (2011).9, p. 14), Itofgives the was generality rise dubbed to the ill-fatedthe of an “Initial equality Basic Stipulation” Law into V in athe value- I will not take issue with Blanchette’s characterization of the byby Richard Richard Heck Heck§ (2011). (2011).calls it It (Frege, It gives gives 2013, rise rise vol. to to I, the the9, ill-fatedp. ill-fated14), was Basic dubbed Basic Law the Law “Initial V V in in the Stipulation” the Grundgesetzerange equality”,system as of Frege concept-script: calls it (Frege, 2013,§ vol. I, §9, p. 14), was reduction involved in Frege’s project. I find it plausible that Frege by Richard Heck (2011). It givesby rise Richard to the Heck ill-fated (2011). ItBasic gives Law rise to V the in ill-fated the Basic Law V in the GrundgesetzeGrundgesetzesystemsystem of of concept-script: concept-script: must have had something very much like this in mind, albeit dubbed the “Initial Stipulation” by Richard– Heck– (2011). It givesa (V) Grundgesetze system of" concept-script:f(")=↵g(↵) = f(a)=probablyg(a) not in as much beautiful detail as Blanchette provides. Grundgesetze(V)risesystem to the of ill concept-script:-fated"–f (Basic")=↵– Lawg(↵) V= ina thef(a )=Grundgesetzeg(a) system of con- cept-script: What I disagree with, however, is the exact role Blanchette assigns – – – – a a (V)(V) "(V)f"(f"()=")=↵g↵(g↵()↵)="–=f(")=f(↵–fag()=(a↵)=) g=(ga(a)a)f(a)=g(a) to Basic Law V in her reconstruction of Frege’s project. (V) Note(V) that ‘"–"–’f (and(")= likewise,↵–g(↵) ‘↵–’)= denote–a f( thea)= second-levelg(a) function– that takes Note that ‘"’ (and likewise, ‘↵’) denote the second-level function that takes – first-level functions – – to theirNote value-range, that– ‘"–’ (and so thatlikewise,"f(" ‘↵)–’) is denote the value-range the second-level of function that takes NoteNoteNote that that ‘" ‘’" (and’(and (and likewise,likewise, likewise, ‘ ↵ ‘’)↵’) denote) denotedenote the thethe second-level second-levelsecond-level function function function that that takes takes–2. Sense and Reference and Basic Law V first-levela functions to their value-range, so that "f(") is the value-range of Note that ‘function"–’ (andf likewise,; that ‘ ‘↵’– is’) Frege’s denotefirst-level universal the functions second-level quantifier, to their functionvalue-range, and ‘a’ the thatso variable that takes"–f it(") is the value-range of thatfirst-levelfirst-level takes functionsfirst functions-level to functions to their their value-range, value-range, to their value so so that- thatrange,"–f"–(f "so()" is)that is the the value-range value-range is ofGrundgesetze of contains a second consequential change to Frege’s binds; and that Frege uses ‘=’ for both identityaa and biconditional. We will first-level functionsthe value to-range theirfunction value-range,of functionfunctionf; f sothat;; that that that ‘ "–f’ is(’"’ Frege’s)isis is Frege’s Frege’s the universal value-range universal universal quantifier, ofqua and quantifier,n- ‘a’ the variableproject. and it Between ‘a’ the Grundlagen variable it and Grundgesetze, Frege drew the functionfunctionf;f that; that ‘ a ‘ a’ is’ is Frege’s Frege’s universal universal quantifier, quantifier, and and ‘a’ ‘a the’ the variable variable it it gettifier, back and to the ‘�’ latter. the variablebinds; it and binds; that Frege and usesthat ‘=’ Frege for both uses identity ‘=’ for and both biconditional. distinction We will between sense and reference. As he puts it in the Fore- function f; that ‘ a ’ is Frege’sbinds; universal and that quantifier, Frege and uses ‘a’ ‘=’ the for variable both it identity and biconditional. We will identitybinds;binds;The analysans and and that biconditional. that of Frege (N) Frege inget usesGrundgesetze uses back ‘=’We to ‘=’ the forwill for latter. boththus get both becomesback identity identity to the and and latter. biconditional. biconditional. We We will willword to Grundgesetze: binds; and thatThe Frege analysans uses ‘=’ of for (N) both in Grundgesetze identity and biconditional.thus becomes We will getget back back to to the theget latter. latter. backThe to analysans the latter. of (N) in Grundgesetze thus becomes [C]ontent I called judgeable content. This now splits for me into what – – – – get back to(N theGg (N) latter.Gg) " H(" = ↵H(↵) H F ) = " H(" = ↵H(↵) H G) I call thought and what I call truth-value. This is a consequence of the TheThe analysans9 analysans of of (N) (N)^ in⇡ inGrundgesetzeGrundgesetze9 thusthus becomes becomes^ ⇡ The(N analysans) "– H(" = of↵–H (N)(↵) inH GrundgesetzeF ) = "– H(" = ↵–Hthus(↵) H becomesGdistinction) between the sense and the reference of a sign. In this in- Gg 9 ^ ⇡ 9 ^ ⇡ The analysansthat is,of the (N) cardinal in Grundgesetze number ofthus F is becomes now analyzed as the extension stance, the thought is the sense of a proposition and the truth-value is that is, the cardinal number of F is now analyzed as the extension of the first- of the first-–level– concept– – under which fall –all– and only– – those value- its reference. (Frege, 2013, vol. I, p. X) (N(NGgGg) ) " " HH("("==↵thatH↵H( is,↵()↵ the) H cardinalH FF) number)==" of" HF H(is" now(=" =↵ analyzedH↵H(↵()↵) asH theH extensionG)G) of the first- –levelrangesconcept of– functions under9 9 which that fall are all–^ andequinumerous^– only⇡⇡ those–– value-ranges 9with9 F. Again, of functions ^(N^) has⇡–⇡ – (NGg) " H(" = ↵H(↵(N) GgH )levelF )concept"= "H underH("("= which=↵↵HH fall(↵↵) all) andHH onlyG) thoseF ) value-ranges= " H( of" functionsSo= what↵H( ↵has) becomeH Gof) the sameness of content after content has the9 same content ^ as (N⇡ Gg) by 9the9 inter-derivability ^^ of⇡ the⇡ latter and 9 ^ ⇡ (EQ), this time explicitly mediated by Basic Law V. thus fallen apart into sense and reference? Is sense preserved be- thatthat is, is, the the cardinal cardinal number number of ofF4Fisis now now analyzed analyzed as as the the extension extension of of the thefirst-first- tween (N), (EQ), and (NGg), or is it merely reference? As explained that is, the cardinalBlanchette number argues of F is forcefully now analyzed and as in the detail extension that Frege’s4 of the first-project is bestlevellevel understoodconceptconcept underthat under in this is, which whichthe way. fall cardinal fallThe all all reduction and and number only only thoseof those ordinary of value-rangesF value-rangesis nowmathema analyzed of of functionst- functions asabove, the extension Blanchette of suggests the first- an intriguing alternative: logical en- level concept under which fall all and only those value-ranges of functions tailment is all that matters. level concept under which fall all and only those value-ranges of functions 4 4 Journal for the History of Analytical Philosophy, vol. 3 no. 7 [3]

4 4

Let us thus consider what Blanchette presents as Frege’s con- one and the same thought can be split up in different ways and so can ception of logical consequence. It is neither proof- nor model- be seen as put together out of parts in different ways. theoretic, but operates on thoughts, i.e., on the senses of (interpret- The example Frege cites here is however rather harmless. He ed) sentences. If logical consequence thus operates on sense, what considers splitting up the proposition ‘12 is greater than 2’, and of Basic Law V, which plays the all important role in the logicist accordingly the thought expressed by it, in two different ways: reduction? If by the Initial Stipulation we may infer an identity of value-ranges from a generality of an equivalence and vice versa, (1) ξ is greater than 2 then the thoughts expressed by these logically entail each other. 2 (2) ξ is greater than 2 This may suggest that Basic Law V states that its right- and left- 2 hand side have the same sense, or express the same thought. This is obviously much simpler than the recarving or multiple Indeed, Blanchette seems sympathetic to this view. She argues decomposition suggested for Basic Law V. Also Blanchette’s first that Frege held on to what is now usually called the “multiple example seems less radical. This example originates from “On decomposition” of content after the introduction of the sense– Concept and Object” (“Ueber Begriff und Gegenstand”). Blanchette reference distinction: “After the advent of the mature semantic quotes Frege (1892b, p. 200): theory, the entities multiply decomposable are thoughts” (Blanchette, 2012, p. 41). If this is so, Frege might have taken the It is thus not impossible that one way of analyzing a thought should left-hand side of Basic Law V, decomposed to involve value- make it appear as a singular judgment; another, as a particular ranges, to express the same thought as the right-hand side, de- judgement; and a third, as a universal judgement. composed not to involve value-ranges. The trouble is, nowhere in But the operative word here is “appear” [“erscheint”]. The sentence Grundgesetze does Frege say that the two sides of Basic Law V immediately preceding the one quote above is: have the same sense or express the same thought. Our questions thus are: To what extent are thoughts multiply decomposable in Language has means of presenting now one, now another, part of the Frege’s view? Does Frege take Basic Law V to express the same- thought as as the subject; one of the most familiar is the distinction of ness of sense for its two sides? If the answer to the latter is nega- active and passive forms. tive, does this pose a problem for Blanchette’s reconstruction of Frege’s examples are: Frege’s logicist project? Blanchette provides three examples in which Frege, after the (3) There is at least one square-root of 4. introduction of the sense–reference distinction, commits to multi- (4) The concept square-root of 4 is satisfied. ple decomposition. Her second example (p. 41), which we will (5) The number 4 has the that there is something of consider first, can be found in the short 1906 piece from Frege’s which it is the square. Nachlass, “A Brief Survey of my Logical Doctrines” (“Kurze Über- sicht meiner Logischen Lehren”, 1906, p. 218). Frege writes that

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While we could easily reject these examples as tangential to the question

of whether Basic Law V asserts the sameness of sense of its right- and left-

hand sides, this will not do as a response to Blanchette’s third example, the

passage from Function and Concept, where Frege claims that

x2 4x = x(x 4)

and

Frege’s point is this: “The thought itself does not yet determine "– "2 4" = ↵– ↵(↵ 4) what is to be regarded as the subject”, where the notion of subject in question is that of the grammatical subject. Note that “lan- express “the same sense,express but in a di“the↵erent same way” sense, (Frege but (1891), in a different p. 10; compare way” (Frege (1891), p. guage” in the latter quote above means natural language—for in- 10; compare Blanchette (2012), p. 42). stance, English or German—not the representationBlanchette of the thought (2012), p. 42). The 1891 lecture Function and Concept is the place where Frege in concept-script. (3)–(5) (arguably) are all of the logical form: The 1891 lecture Functionfirst introduces and Concept his isdistinction the place between where Frege sense first and reference. It could be pointed out that the relevant passage occurs before the (6) ∃x x2 = 4 introduces his distinctionpart between where sense Frege and draws reference. the distinction, It could be but pointed we should probably 4 albeit with different emphases. out that the relevant passagenot reject occurs taking before “sense” the part seriously where here Frege merely draws on the this ground. These examples are much more closely related to those Frege Had the phrase “the same sense, but in a different way” oc- distinction, but we should probably not reject taking “sense” seriously here discusses as early as Begriffsschrift, which Blanchette too describes curred in a Fregean writing before 1891, we would probably read as less radical than the recarvings encoded in Basicmerely Law on this V or ground. 4 it as a first, terminologically confused flickering of the sense– Hume’s Principle (Blanchette, 2012, p. 41). The former example reference distinction: “the same sense” should really be “the same Had the phrase “the same sense, but in a di↵erent way” occurred in a 5 can be seen as neglecting some structure that would be available reference”; “a different way” should be “a different sense”. But in the decomposition of the thought. In such a case,Fregean the decomp writingo- before 1891,Frege wedoes would say that probably the two read propositions it as a first, express termi- the same sense, sition is superimposed on the finest-grained structure (‘ξ is greater i.e., the same thought—a reference is not expressed. Nonetheless, nologically confused flickeringit just ofdoes the not sense–reference fit Frege’s attitude distinction: as we “the find same it in Grundgesetze or than ζ’ would be another option for the analysis). The latter exam- 4 any other writing after Function and Concept. The best, I think, we ples regard features of natural language, like activeKlement and passive (2002,p.87) suggest that “sense” here is a mere slip, that Frege may not have can say is: Frege should not have written this. voice, grammatical subject, etc., which are not seenhad“fully as logically mastered the distinction or the terminlogy” yet. Klement also (ibid., fn. 28) Frege is clear, in Grundgesetze and elsewhere, that for instance relevant and thus will not survive the analysis in concept-script; cites the fact that Frege’s claim‘22’ occursand ‘2 before + 2’ have the introduction the same ofreference the distinction but different (which senses (Frege, think of the notorious ‘Cato killed Cato’, ‘Cato was killed by Cato’, appears a few paragraphs later)2013, as possiblevol. I, evidencep. IX). thatIndeed, he might Frege not introduces have used “sense” the sense–reference ‘Cato killed himself’, etc. (Frege, 1879, §9).3 distinction precisely because he is seeking to make sense of in- While we could easily reject these examples asas a tangential technical term to in this passage. Simons (1992, p.765) finds it “out of character for formative identity statements (Frege, 1892a). If ‘‘22 = 2 + 2’ is in- the question of whether Basic Law V asserts the samenessFrege to have of sense made such a slip”. formative (and thus not a statement of sameness of sense, but of its right and left hand sides, this will not do as a response to merely of sameness of reference), surely, the sameness of reference Blanchette’s third example, the passage from Function and Concept, of the two formulae displayed above must count as informative where Frege claims that too, and thus10 neither they should have the same sense. The same x2 − 4x = x(x−4) goes for Basic Law V. Biconditionals in Grundgesetze are lumped together with identities, since the role of both is to express the and sameness of reference. (In the case of propositions, this reference is the True or the False.) So, Basic Law V states that its two sides

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have the Blanchettesame reference. observesBlanchette But (p. we 44) observes should that Frege(p. not 44)is conclude coy that about Frege that committing is coythe about to committingFrege Basic had tobeen Basic tempted to think that the sameness of sense ob- senseBlanchette too is observes the same. (p. 44) Basic that Law Frege V is is coy an about informative committing identity to Basic tained. In 1893, he had realized that his theory of sense and refer- Law V’s stating a sameness of sense. This may be understated. The im- Lawstatement V’sLaw stating if V’s ever a stating sameness there awas sameness of one. sense. of This sense. may This be understated. may be understated. The im- Theence im- does not yield this result. Blanchette observes (p. 44) that Frege isBlanchette coy about observes committingpression (p. to44) one Basic that get isFrege rather is coy that about he carefully committing avoids suggesting that the sense pression onepression get is one rather get thatis rather he carefully that he avoidscarefully suggesting avoids suggesting that the sense that the sense Law V’s stating a sameness of sense.to This Basic may Law be V’s understated. stating a sameness The im- of sense. This may be Blanchetteunder- observes (p. 44) that Frege is coy about committing to Basic is the same. In Grundgesetze, Frege repeatedly states that the two sides stated.is The the impr same.ession In Grundgesetze one get is rather, Frege that repeatedly Frege carefully states avoids that the two3. sides Definition and Explanation pression one get is rather that he carefullyis the same. avoids InsuggestingGrundgesetze that the, Frege sense repeatedly states thatLaw the twoV’s stating sides a sameness of sense. This may be understated. The im- suggestingof Basic that Law theof Vsense Basic are co-referential,is Law the Vsame. are co-referential,In but Grundgesetze never that but, theyFrege never express re- that they theThe same express principal the sameway for Frege to preserve sense, in Grundgesetze and is the same. In Grundgesetze, Fregeof repeatedly Basic Law states V are that co-referential, the two sides but never that they expresspression the one same get is rather that he carefully avoids suggesting that the sense peatedly states that the two sides of Basic Law V are co-referential, elsewhere, is by definition.6 But Basic Law V is of course not a def- sense. So too in the course of the infamous permutation argument6 in 10. sense. So too in the course of the infamous permutation argument6 in 10. of Basic Law V are co-referential,sense. butbut never Sonever too that that in they the they course express express of the the the sameinfamous same sense. permutation So too in argument theis course the insame. of10. In Grundgesetzeinition,§ but a, basic Frege§ law repeatedly of logic. states Neither that is thethe two“Initial sides Stipulation” 6 § sense. So too in the course of the infamousthe infamous permutationFrege starts permutation theargumentFrege section starts argument in by the10. describing6 section in §10. bythe describing problemFrege starts he the is the problem about sec- to he tackle: is aboutin “By§3 toof tackle:the first “By volume of Grundgesetze a definition, as Frege em- Frege starts the section by describing§ the problem he is aboutof to tackle:Basic Law “By V are co-referential, but never that they express the same tion by describing the problem he is about to tackle: “By– presen–t- phasizes in volume II in his discussion of the §3 stipulation: “This Frege starts the section by describing the problempresenting he is about thepresenting combination to tackle: the “By– of combination signs ‘–"–(" of)= signs↵– ( ‘↵")’( as")= co-referential↵ (↵)’ as co-referential with with presentinging the thecombination combination of of signs signs ‘"(")=↵ (↵)’ asas co-referentialco-sense.referential So with too in theconversion course of is the not infamous to be taken permutation as a definition” argument (Frege, in 10. 2013,6 vol. II, presenting the combination of signs ‘"–(")=↵– (↵)’ as co-referentiala with § ‘ a (a)= ‘(a)’,( wea)= have ( admittedlya)’, we have by admittedly no means by yet no completely means yet§146). fixed completely Also in fixedgeneral, Frege cautions his readers, “we should not ‘ a with(a)= (a)’, we have, admittedlywe have admittedly by no means by yet no completely meansFregestarts yet fixed the section by describing the problem he is about to tackle: “By ‘ a (a)= (a)’, we have admittedly by no means yet completely fixed regard the stipulations about the primitive signs in the first vol- the reference of a name– such as ‘"–(")’.” Only a few lines down, regarding completelythe reference fixed the of reference a name– such of a as name ‘"( "such)’.” Onlyas a few.” lines Only down, a regardingume as definitions.– Only what– is logically composite can be de- the reference of a name such as ‘"–(the")’.” reference Only a offew a lines name down, such as regarding ‘"(")’.” Only a few lines down,presenting regarding the combination of signs ‘"(")=↵ (↵)’ as co-referential with few lines down, regardingan assumed an assumed permutation permutation X, he writes– X, thathe writes “ ‘X(–"– (")) =fined; X(↵– what(↵))’ is too simple is can only be pointed to.” (Frege, 2013, vol. II, an assumed– permutation– X, he writes that– “ ‘X("(–")) = X(↵ (↵))’ too is an assumed permutation X, he writesan thatthat assumed ““ ‘X( permutation"(")) = X(↵ X, ( he↵))’ writes too is that co- “referential ‘X("(")) with = X( ↵‘ a (Φ↵())’(a�)=) too= is(a)’, we have admittedly by no means yet completely fixed a p. 148 fn. 1) So, the “stipulation” here is not a stipulative defini- co-referentiala with ‘ (a)= (a)’ ”, adding in a footnote: “Thereby it is co-referential with ‘ a (a)= (a)’co-referential ”, addingΨ(�)’”,co-referential inadding a with footnote: in ‘ a a withfootnote:( “Therebya)= ‘ ( “Therebya()’a it)= ”, is adding (a it)’ ”,is in addingnot a footnote: said in that a footnote:the “Thereby the reference sense “Thereby it is of ation, name it is and such indeed as ‘"– no(")’.” definition Only a at few all. lines down, regarding is the same.” Some notcommentators said that the have sense interpreted is the same.” this footnote Some commentators as haveAustin’s interpreted translation of Grundlagen (Frege, 1950, §64) calls the not said that the sense is the same.”not Some said commentators thatnot said the sensethat thehave is the sense interpreted same.” is the Some same.” commentators Some commentators have interpreted have interpreted – – evidence that Frege held that, in contrast, the samenessan of assumed sense permutationrecarving X, of he ‘straight writes that line “ a ‘X( is" parallel(")) = to X( straight↵ (↵))’ line too b is’ as ‘the di- this footnote as evidence that Frege held that, in contrast,this the footnote sameness7 as of evidence that Frege held that, in contrast, the sameness of thisdoes footnote obtainthis asfootnote for evidence Basic as Law evidencethat V. FregePerhaps that held Fregethat, it suggests held in contrast, that, that in Frege contrast,theco-referential sameness would the sameness of withrection ‘ a of( aof)= a =( athe)’ ”, direction adding in of a b footnote:’ a “definition”. “Thereby But it isthe German 7 sense does obtain for Basic Law V. Perhapshave liked it suggests the two that sensesides Frege doesof Basic would obtain Law for V Basic7 to have Law the V. 7samePerhaps sense. it suggestsword that here Frege is wouldnot “Definition ”; it is “Erklärung”: “explanation”. In sense doessense obtain does for obtain Basic for Law Basic V.7 LawPerhaps V. itPerhaps suggests it that suggestsnot Frege said that would that Frege the sensewould is the same.” Some commentators have interpreted have liked the two sides of Basic Law VBut to if have he had the thought same sense. that Buthe could if he claim this sameness of sense, it the passage regarding the equality of cardinal number in §62 of have liked the two sides of Basic Law V to have the same sense. But if he havewould likedhave be the hard liked two to sides the see two of what Basic sides prevented Law of Basic V to Lawhim have Vfrom the to havesamestating the sense. thisthis same footnoteclaim But sense. if he as But evidenceGrundlagen if he that, the Frege analogous held that, translation in contrast, of thethe samenesscognate verb of appears: had thought that he could claim this sameness of sense, it would be hard to explicitly, either inhad the thoughtfirst sentence that he of could the claimparagraph this sameness or in the of sense,Austin it would translates be hard to “erklären” as “define” instead of “explain”. It is had thoughthad that thought he could that claimhe could this claim sameness this sameness of sense, it of would sense,sense be it does would hard obtain to be hard for to Basic Law V.7 Perhaps it suggests that Frege would see what prevented him from stating thisfootnote. claimexplicitly, either in the first common and tempting to translate “Erklärung” in Frege in these see what prevented him from stating this claim explicitly, either in the first sentence of the paragraph or in thesee footnote. whatI mustsee prevented what submit: prevented him if from Frege him stating was from not this stating sloppy claim this explicitly, or claim confused, explicitly, eitherhave then in liked the either he first the in two thekinds sides first of Basiccontext Law as V “definition”; to have the sameand sense.there is But indeed if he a use of 8 changed his mind sentencebetween of 1891 the paragraphand 1893. orThis in the explanation footnote. is “Erklärung” in mathematical literature to mean just this. But Frege I must submit: if Frege was not sloppysentence orsentence of confused, the paragraph of then the heparagraph or changed in the orfootnote. his in the footnote. had thought that he could claim this sameness of sense, it would be hard to simpler than explaining why, in the ten years and probably more keeps the notions carefully separate, although he does not lose a mind between 1891 and 1893.8 This explanation is simpler thanI must explaining submit: if Frege was not sloppy or confused, then he changed his thatI must Frege submit:I mustheld ifon submit: Frege to value was if Frege- notranges, sloppy was he not ornever sloppy confused, again, or confused, then despite hesee changed what thenample he prevented his changedsingle him his fromword stating about thisthe claimdifference explicitly, in Grundgesetze either in the. He first might have 6Frege (2013), vol.I, p.16; see Heck (2012, ch.4), and Wehmeier and Schroeder-Heister 9 opportunity, statedmind that betweenan identity 18918 of andvalue 1893.-ranges8 This has explanation the same is simplertaken than it to explaining be too obvious to clarify. mind betweenmind 1891 between and 1891 1893. and8 This 1893. explanationThis explanation is simpler is than simpler explaining than explaining (2005) for discussion. sentence of the paragraph or in the footnote. sense as the corresponding6 generality of an equivalence. In 1891, 6 Frege (2013), vol.I, p.16; see Heck (2012, ch.4), and Wehmeier and Schroeder-Heister 6Frege (2013),Frege vol.I, (2013), p.16; vol.I, see Heck p.16; (2012, see Heck ch.4), (2012, and Wehmeierch.4), and and Wehmeier Schroeder-Heister and Schroeder-Heister 7See, e.g., Simons (1992, p.764); Schirn (2006) and Ebert (2014), for instance, disagree. I must submit: if Frege was not sloppy or confused, then he changed his (2005) for discussion. (2005) for discussion.(2005) for discussion. Journal for the History of Analytical Philosophy, vol. 3 no. 7 [6] 8This is also suggest by Ebert (2014, 2.2, fn. 26). mind between 1891 and 1893.8 This explanation is simpler than explaining § 7 7 7See, e.g., SimonsSee, (1992, e.g., p.764); Simons Schirn (1992, (2006) p.764); and Schirn Ebert (2006) (2014), and for Ebert instance, (2014), disagree. for instance, disagree. See, e.g., Simons (1992, p.764); Schirn (2006) and Ebert (2014), for instance,6Frege disagree. (2013), vol.I, p.16; see Heck (2012, ch.4), and Wehmeier and Schroeder-Heister 8 128 8 This is also suggest by Ebert (2014, 2.2, fn. 26). This is alsoThis suggest is also by suggest Ebert (2014, by Ebert2.2, (2014, fn. 26).2.2, fn. 26). § (2005) for discussion. § § 7See, e.g., Simons (1992, p.764); Schirn (2006) and Ebert (2014), for instance, disagree. 12 12 12 8This is also suggest by Ebert (2014, 2.2, fn. 26). §

12 Blanchette observes (p. 44) that Frege is coy about committing to Basic Blanchette observes (p. 44) that Frege is coy about committing to Basic Law V’s stating a sameness of sense. This may be understated. The im- Law V’s stating a sameness of sense. This may be understated. The im- pression one get is rather that he carefully avoids suggesting that the sense pression one get is rather that he carefully avoids suggesting that the sense is the same. In Grundgesetze, Frege repeatedly states that the two sides is the same. In Grundgesetze, Frege repeatedly states that the two sides of Basic Law V are co-referential, but never that they express the same of Basic Law V are co-referential, but never that they express the same sense. So too in the course of the infamous permutation argument in 10.6 § Explanation,sense. for So Frege, too in is the a wider course notion of the infamousthan definition. permutation Defin- argumentmost in laws10.6 in the realm of reference and relate only indirectly to ing is certainlyFrege startsan excellent the section wayby of describingexplaining, the given problem that hea defin is abouti- to tackle:sense. “By§ [...] Reference thus proves to be what is essential for sci- Frege starts the section by describing the problem he is about to tackle: “By tion stipulatespresenting the sameness the combination of sense of for signs explanans ‘"–(")= and↵– explana(↵)’ asn- co-referentialence.” with (Frege, 1892–1895, pp. 132–134), (Frege, 1979, pp. 122–123). dum. But not all notions can be defined. For instance,– –the “Erklä- Thus, what matters for logic and Frege’s logicism is that the two apresenting the combination of signs ‘"(")=↵ (↵)’ as co-referential with rung” of ‘ ’ in(a )=Grundgesetze (a)’, we (vol. have I,admittedly §8, p. 12) surely by no cannot means be yet a completely sides fixed of Basic Law V have the same reference. Since Basic Law V a definition:the ‘ reference’ is(a )=primitive. of ( aa)’, name Also we such have in vol. as admittedly ‘" –I (§4,(")’.” p.8), by Only noFrege a means few writes: lines yet down, completely regardingwas to fixed be a law of logic, the co-reference of its two sides is, of “The functions withof the two cognate arguments, verb ξ appears: = ζ and– Austinξ > ζ, always translates have “ erkl¨arena ”course, as “define” more in-than mere co-reference. It is sameness of reference as anthe assumed reference permutation of a name X, such he as writes ‘"(" that)’.” Only “ ‘X("– a few(")) lines = X( down,↵– (↵))’ regarding too is truth-value as value (at least if the signs ‘=’ and ‘>’ are explained a matter of logic. But this is not to say, sameness of sense. stead of “explain”. It is common and tempting– to translate– “Erkl¨arung”in in the appropriateco-referentialan assumed way) with permutation.” ‘=’ ‘ a is primitive(a)= X, he (a writesand)’ ”, addingthus that un “ in‘X(defined; a" footnote:(")) ‘>’ = X( “Thereby↵ (↵))’Blanchette’s too it is is insightful account of Fregean analysis and its role Frege in these kinds of context as “definition”; and there isin indeed the logicist a use ofproject should survive abandoning the thought that would end co-referentialup being defined. with ‘Fregea (a subsumed)= (a)’ ”, both, adding the in definition a footnote: “Thereby it is not said that the sense is the same.” Some commentators have interpretedBasic Law V expresses sameness of sense. The alteration proposed of ‘>’ and the explanation“Erkl¨arung ”of in ‘=’, mathematical under the literatureterm “explanation”. to mean just this. But Frege keeps Finally, Partthisnot 2 footnote of said vol. that I asof the evidenceGrundgesetze sense is that the, Fregeentitled same.” held Some“Definitions” that, commentators in contrast, (§26, the have sameness interpretedhere may of indeed be seen to be in keeping with her suggestion re- the notions carefully separate, although he does not lose a singlegarding word the about central importance of logical consequence for Fregean p. 43), startsthis in its footnote very first as evidence sentence that with: Frege “The7 held signs that, explained in contrast, so the sameness of sense does obtain for Basic Law V. Perhaps it suggests that Fregeanalysis. would Logical equivalence, and not sameness of sense, appears far will now be usedthe dito↵ erenceintroduce in Grundgesetze new names.”. HeIndeed, might as have the takentitle it to be too obvious to of the parthavesense correctly liked does the obtain twosuggests, sides for Basic ofnothing Basic Law Law V.has7 VPerhaps been to have defined it the suggests same in that sense. Fregeto But take would if he the role that the mysterious sameness of content played 9 Grundgesetze beforeclarify. this, but only primitive signs have been ex- before the sense–reference distinction in almost all the important hadhave thought liked the that two he couldsides of claim Basic this Law sameness V to have of sense, the same it would sense. betransitions, But hard if to he as befits a logicist project. The thought is that whatever plained. Explanation, for Frege, is a wider notion than definition. Defining is tive signs,Where he frequentlysee Fregehad what thought gives prevented mentions explanations, that he him reference,tive could from signs, claimor statingbut writes he this frequentlynever this samenessabout sense. claim explanations mentions explicitly, of In sense,Grundge- reference, it either would in bethe but the hard analysis never first to sense. of (N In) asGrundge- (EQ) guarantees, it is at least some strong, of primitive signs,certainly he frequently an excellent mentions way of reference, explaining, but given never that a definitionnot merely stipulates contingent, co-reference for all instances, even if it falls setze, 18, Fregesentencesee writes: what of “From prevented the paragraph thesetze himreference, from or18, in ofstating Fregethe the footnote. function-name writes: this claim “From explicitly,K the⇠ ( reference11) either inshort of the the of first function-name sameness of sense.K⇠ ( 11) So, (N) and (N ) too are guaranteed to sense.§ In Grundgesetzethe sameness, §18, Frege of sense writes:§ for explanans “From the and reference explanandum. §of But not all notions § Gg – – have the same reference, by the logical equivalence of (EQ) and a = K"(a = ") follows”,sentenceI must submit: ofwhereξ the paragraph11 if Frege containsa = K was" or(a thein not= the" explanation) sloppy follows”, footnote. or confused, where of the (primi-11 then contains he changed the explanation his of the (primi- the function-name \ (§11)§ follows”, where §11a can be defined. For instance, the “Erkl¨arung” of§ ‘ ’inGrundgesetze(NGg). If Blanchette’s(vol. I, arguments that logical consequence is pre- contains the explanation of the (primitive)8 concept-script substi- tive) concept-scriptmindI substitutebetween must submit: 1891 for theand iftive) Frege definite 1893. concept-script wasThis article, not explanation sloppy ‘K substitute⇠’. or The confused, explanationis simpler for the then definite than he changed explainingserved article, hisover ‘K⇠’. Fregean The explanation analysis can survive abandoning sense as a 8, p. 12) surely cannot be a definition: ‘ a ’ is primitive. Also in vol. I ( 4, tute for the6 definite article, ‘\ξ’. The explanation of ‘ ’ in §8, of ‘ a ’in 8, mentionedFrege§ (2013), above, vol.I, is allp.16; about seea Heck reference,8 (2012, ch.4), and and so tooWehmeier is the and ex- Schroeder-Heisterreplacement for§ content in the analysis of (N) as (EQ)—an investi- mentioned mindabove, between is all about 1891 ofreference, and ‘ 1893.’in 8,andThis mentioned so explanation too is above,the explan is simpler is alla- about than reference, explaining and so too is the ex- § p. 8), Frege writes: “The§ functions with two arguments, ⇠gation= ⇣ and as⇠ careful> ⇣, and detailed as Blanchette’s would have to (2005) for discussion.– planationtion of of ‘value ‘value-range’-6range’Frege (2013), and ‘ vol.I,"(")’ p.16; in §§3 see3 Heck and (2012, 9.9. Perha Perhaps ch.4),ps and most Wehmeier telling– and is Schroeder-Heister planation§§ of ‘value-range’ and ‘"(")’ in 3show and 9.— Perhapsthen (N) most and telling (NGg) isshould still be logically equivalent, 7 always have a truth-value as value (at least if the signs§§ ‘=’ and ‘>’ are is a passage inSee, §30 e.g., of Simons the first (1992, volume p.764); of Schirn Grundgesetze (2006) and,Ebert where (2014), Frege for instance, disagree. a passage in 30(2005) of the for first discussion. volume of Grundgesetze, where Frege uses the since (N)’s analysans (EQ) is logically equivalent to (NGg). Basic uses the §phrase “explanation ofa passagereference”. in 30 of the first volume of Grundgesetze, where Frege uses the 8 explained in the appropriate§ way).” ‘=’ is primitive andLaw thus V un-defined; sits in the middle of the derivation of (NGg) from (EQ), and This7See, is e.g., also Simons suggest (1992, by Ebert p.764); (2014, Schirn2.2, (2006) fn. 26). and Ebert (2014), for instance, disagree. phrase “explanationIn his “Comments of reference”. on Sensephrase and “explanationReference”,§ written of reference”. between of the latter from the former. The two sides of Basic Law V do not ‘>’ would end up being defined. Frege subsumed both, the definition of ‘>’ 1892 and 1895,8 Frege is clear that the focus of logic and mathemat- In his “CommentsThis on is alsoSense suggest and by Reference”, Ebert (2014, written2.2, fn. 26).between 1892 and have the same sense; nor do they have to. Sameness of reference as In his “Comments§ on Sense and Reference”, written between 1892 and ics is on reference:and “the the reference explanation and of ‘=’,not underthe12 sense the termof words “explanation”. [is] Finally,a matter Part of 2logic of suffices, and that is what Basic Law V would 1895, Frege is clear that the focus of logic and mathematics is on refer- 10 what is essential for logic [. . .1895, ] the laws Frege of is logic clear are that first the and focus fore- of logichave and mathematicsdelivered, had is it on not refer- been for its inconsistency. vol. I of Grundgesetze, entitled12 “Definitions” ( 26, p. 43), starts in its very ence: “the reference and not the sense of words [is] what is essential§ for ence: “the reference and not the sense of words [is] what is essential for first sentence with: “The signs explained so far will now be used to introduce Journal for the History of Analytical Philosophy, vol. 3 no. 7 [7] logic [. . . ] the laws of logic are first and foremost laws in the realm of refer- logic [. . . ] the laws of logic are first and foremost laws in the realm of refer- new names.” Indeed, as the title of the part correctly suggests, nothing has ence and relate only indirectly to sense. [...] Reference thus proves to be been defined inenceGrundgesetze and relatebefore only indirectly this, but toonly sense. primitive [...] signs Reference have been thus proves to be what is essential for science.” (Frege, 1892–1895, pp. 132–134), (Frege, 1979, explained. what is essential for science.” (Frege, 1892–1895, pp. 132–134), (Frege, 1979, pp. 122–123). Thus, what matters for logic and Frege’s logicism is that the Where Fregepp. 122–123). gives explanations, Thus, what or writes matters about for logic explanations and Frege’s of primi- logicism is that the two sides of Basic Law V have the same reference. Since Basic Law V was 9His discussiontwo of sides the terms of Basic in his Law debate V have with theHilbert same is a reference. di↵erent matter—and Since Basic Law V was to be a law of logic, the co-reference of its two sides is, of course, more than complicated byto the be fact a that law Frege of logic, picks the up onco-reference Hilbert’s use of of its “Erkl¨arung two sides”. is, of course, more than mere co-reference. It is sameness of reference as a matter of logic. But this mere co-reference. It is sameness of reference as a matter of logic. But this is not to say, sameness of sense. 14 is not to say, sameness of sense. Blanchette’s insightful account of Fregean analysis and its role in the Blanchette’s insightful account of Fregean analysis and its role in the logicist project should survive abandoning the thought that Basic Law V logicist project should survive abandoning the thought that Basic Law V expresses sameness of sense. The alteration proposed here may indeed be expresses sameness of sense. The alteration proposed here may indeed be seen to be in keeping with her suggestion regarding the central importance seen to be in keeping with her suggestion regarding the central importance of logical consequence for Fregean analysis. Logical equivalence, and not of logical consequence for Fregean analysis. Logical equivalence, and not sameness of sense, appears to take the role that the mysterious sameness of sameness of sense, appears to take the role that the mysterious sameness of 15 15

Notes

1 Marcus Rossberg Frege (1884)/(1950), p. 75. Ebert (2014) finds Austin’s translation of Frege’s word “zerspalten” as “carve up”, and the subsequent use University of Connecticut of “recarving” in the literature contentious. He suggests to follow [email protected] Dummett (1991, p. 168) in using the literal translation “split up”. 2 That Frege held that abstractions principles like Hume’s Principle or Basic Law V state the sameness of sense for the propositions on their two sides is argued, among others, by Beaney (2005), Burge (1990), Currie (1982), Milne (1989), and Simons (1992); see, e.g., (Dummett, 1991, ch. 14), Ebert (2014), or (Klement, 2002, ch. 3) for dissent. 3 Also note the striking similarity of these examples with the dif- ferent ways that Frege (1880–1881) lists of expressing ‘24 = 16’ in natural language, using, for example, the phrases ‘fourth root of 16‘ or ‘logarithm of 16 to the base 2’. He continues: “We may now also regard the 16 in x4 = 16 as replaceable in its turn, which we may represent, say, by x4 = y. In this way we arrive at the concept of a relation, namely of the relation of a number to its 4th power. And so instead of putting a judgement together out of an individ- ual as a subject and an already previously formed concept as pred- icate, we do the opposite and arrive at a concept by splitting up the content of possible judgement.” (Frege, 1979, p. 17) Compare also Picardi (1993, p. 76) for her categorization of five different classes of pairs of propositions Frege seems to consider as equipol- lent at some point or another, and her discussion of which of these may qualify for sameness of sense.

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4 Klement (2002, p. 87) suggest that “sense” here is a mere slip, that Frege may not have had “fully mastered the distinction or the terminology” yet. Klement also (ibid., fn. 28) cites the fact that Frege’s claim occurs before the introduction of the distinction (which appears a few paragraphs later) as possible evidence that he might not have used “sense” as a technical term in this passage. Simons (1992, p. 765) finds it “out of character for Frege to have made such a slip”. 5 Indeed, very similar phrases do occur before the distinction, for example in Begriffsschrift, §8: “the same content can be completely determined in different ways”, or Grundlagen, §62: “we must re- produce the content of the proposition in a different way”. (The proposition to which Frege refers is “the cardinal number which belongs to the concept F is the same as that which belongs to the concept G”.) Whether “content” here should be understood as “sense”, however, is our question. 6 Frege (2013), vol. I, p. 16; see Heck (2012, ch. 4), and Wehmeier and Schroeder-Heister (2005) for discussion. 7 See, e.g., Simons (1992, p. 764). Schirn (2006) and Ebert (2014), for instance, disagree. 8 This is also suggest by Ebert (2014, §2.2, fn. 26). 9 His discussion of the terms in his debate with Hilbert is a differ- ent matter—and complicated by the fact that Frege picks up on Hilbert’s use of “Erklärung”. 10 I am indebted to Patricia Blanchette, Roy Cook, Philip Ebert, Walter Pedriali, Kai Wehmeier, and Richard Zach for helpful comments and discussions.

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Breslau: Wilhelm Koebner. English translation: Frege References (1950). Beaney, Michael (2005). ‘Sinn, Bedeutung and the Paradox of ______. (1891). Function und Begriff. Jena: Pohle. Reprinted in Analysis’, in Michael Beaney and Erich H. Reck (eds.), Frege (1967), pp. 125–142; English translation by Peter Gottlob Frege: Critical Assessements of Leading Philosophers. 4 Geach in Frege (1984), pp. 137–156. Volumes, vol. 4. London: Routledge, pp. 288–310. ______. (1892a). ‘Über Sinn und Bedeutung’, Zeitschrift für Blanchette, Patricia (2012). Frege’s Conception of Logic. Oxford: Philosophie und philosophische Kritik 100:25–50. Reprinted in Oxford University Press. Frege (1967), pp. 143– 162; English translation by Max Black in Frege (1984), pp. 157–177. Burge, Tyler (1990). ‘Frege on Sense and Linguistic Meaning’, in David Bell and Neil Cooper (eds.), The Analytic Tradition. ______. (1892b). ‘Ueber Begriff und Gegenstand’, Oxford: Blackwell, pp. 30–60. Reprinted in Burge (2005), Vierteljahresschrift für wissenschaftliche Philosophie 16:192– pp. 242–269. 205. Reprinted in Frege (1967), pp. 167–178; English translation by Peter Geach in Frege (1984), pp. 182–194. _____. (2005). Truth, Thought, Reason: Essays on Frege. Oxford: Oxford University Press. ______. (1892–1895). ‘Ausführungen über Sinn und Bedeutung’, in Frege (1983), pp. 128–136. English translation by Peter Currie, Gregory (1982). ‘Frege, Sense and Mathematical Long and Roger White in Frege (1979), pp. 118–125. Knowledge’, Australasian Journal of Philosophy 60:5–19. ______. (1893). Grundgesetze der Arithmetik: Begriffsschriftlich Dummett, Michael (1991). Frege: Philosophy of Mathematics. abgeleitet. I. Band. Jena: Verlag H. Pohle. English translation London: Duckworth. in Frege (2013).

Ebert, Philip A. (2014). ‘Frege on Sense-Identity, Basic Law V, and ______. (1903). Grundgesetze der Arithmetik: Begriffsschriftlich Arbitrariness’. ms. abgeleitet. II. Band. Jena: Verlag H. Pohle. English translation in Frege (2013). Frege, Gottlob (1879). Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. d. (1906). ‘Kurze Übersicht meiner Logischen Lehren’, in Frege Saale: Verlag L. Nebert. English translation by S. Bauer- (1983), pp. 213–218. English translation by Peter Long and Mengelberg in van Heijenoort (1967), pp. 1–82. Roger White in Frege (1979), pp. 197–202.

______. (1880–1881). ‘Booles rechnende Logik und die ______. (1950) The Foundations of Arithmetic Translated by Begriffsschrift’, in Frege (1983), pp. 9–52. English ̇ translation by by Peter Long and Roger White in Frege J.L .Austin. Oxford: Blackwell. (1979), pp. 9–46. ______. (1967). Kleine Schriften. Edited by Ignacio Angelelli. ______. (1884). Die Grundlagen der Arithmetik. Eine logisch Hildesheim: Olms. mathematische Untersuchung über den Begriff der Zahl.

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______. (1979). Posthumous Writings. Translated by P. Long and R. van Heijenoort, Jean, ed. (1967). From Frege to Gödel: A Source White; edited by H. Hermes, F. Kambartel, F. Kaulbach, P. Book in Mathematical Logic, 1879–1931. Cambridge, MA: Long and R. White. Oxford: Blackwell. Harvard University Press.

______. (1983). Nachgelassene Schriften. Edited by Hans Hermes, Wehmeier, Kai F. and Peter Schroeder-Heister (2005). ‘Frege’s Friedrich Kambartel and Friedrich Kaulbach, 2nd, revised Permutation Argument Revisited’, Synthese 147:43–61. and expanded edition. Hamburg: Meiner. English translation of the first edition [1969]: Frege (1979).

______. (1984). Collected Papers on Mathematics, Logic, and Philosophy. Edited by Brian McGuinness. Oxford: Basil Blackwell.

______. (2013). Basic Laws of Arithmetic: derived using concept script. Volumes I & II, ed. and trans. P. A. Ebert and M. Rossberg. Oxford: Oxford University Press.

Heck, Jr, Richard G. (2011). Frege’s Theorem. Oxford: Oxford University Press.

______. (2012). Reading Frege’s Grundgesetze. Oxford: Oxford University Press.

Klement, Kevin C. (2002). Frege and the Logic of Sense and Reference. New York and London: Routledge.

Milne, Peter (1989). ‘Frege, Informative Identities, and Logicism’, British Journal for the Philosophy of Science 40:155–166.

Picardi, Eva (1993). ‘A Note on Dummett and Frege on Sense- Identity’, European Journal of Philosophy 1:69–70.

Schirn, Matthias (2006). ‘Hume’s Principle and Axiom V Reconsidered: Critical Reflections on Frege and His Interpreters’, Synthese 148:171–227.

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