aa r X i v : . [ m a t h . HO ] J un Frege’s theory of types
Bruno Bentzen
Abstract
There is a widespread assumption in type theory thatthe discipline begins with Russell’s efforts to resolve paradoxesconcerning the naive notion of a class. My aim in this paperis to argue that Frege’s sharp distinction between terms denotingobjects and terms denoting functions on the basis of their saturationanticipate a simple type theory, although Frege vacillates betweenregarding functions as closed terms of a function type and openterms formed under a hypothetical judgment. Frege fails to expresshis logical views consistently due to his logicist ambitions, whichrequire him to endorse the view that value-ranges are objects.
There is a widespread assumption in type theory that thediscipline begins with Russell’s efforts to resolve paradoxesconcerning the naive notion of a class since 1903. However, if bya theory of types we understand a logical formalism where everyterm must have a type and every operation must be restrictedto terms of a certain type, then it can be argued that Fregeanticipated much of a theory of types in his two-volume book
Grundgesetze der Arithmetik (1893/1903), which introduces themature version of the ideography that he intended to use asa formal logical system to vindicate his longstanding logicistclaim that arithmetic is reducible to logic.1 Of course, never once in his career did Frege advocate afull-fledged doctrine of types, but his sharp distinction betweennames of objects and names of functions leaves no doubt thathis ideography is always implicitly operating on typed terms,for objects and functions form two disjoint categories of things.For Frege, objects are saturated and complete, while functionsare unsaturated and in need of completion (
Grundgesetze I § ι and a type formerfor functions σ → τ , where σ and τ stand for types. Therequirement that every term definable in the theory must havea unique type thus incorporates the principle that every termeither stands for an object or a function. Given that numbers,truth values, and value-ranges do not exhibit saturation, theycannot be viewed as functions, so, according to this dichotomy,the only alternative left is to regard them as objects Objects stand opposed to functions. Accordingly,I count as an object everything that is not afunction, e.g., numbers, truth-values and the value-ranges introduced below. Thus, names of objects,the proper names , do not in themselves carryargument places; like the objects themselves, theyare saturated. (
Grundgesetze I § • Roman object letter: a–e , i–l , n–z ; • Greek object letter: α – ε , Γ – Ρ ; • Gothic object letter: a – e ; • first-level Roman function letter: f–h , F–H ; • first-level Greek function letter: Φ – Ψ ; • first-level Gothic function letter: f{h ; • second-level Roman function letter: M ; • second-level Greek function letter: Ω .The distinction between Roman, Greek, and Gothic marks isequally essential to Frege’s theory of types since, as I willdescribe in more detail in the next section, each kind of letterhas their own purpose in the ideography. Not only typeannotations are implicitly present in the ideography in theform of metavariables ranging over types, but also operationalconstraints that depend on the type assigned to a term.Frege famously takes the notion of function as primitive,viewing predicates and relations as functions that assign objectsto reified truth values and distinguishing between functions offirst and higher level depending on whether they only admitobject or functions of lower levels as arguments. That Fregetakes those constraints very seriously can be most readily seenfrom his initial response to the discovery of Russell’s paradox,where, in his first letter to Russell, he explains that thecircular expression “a predicate is predicated of itself” is notan acceptable term in the ideography because a predicate is afunction of first level. More generally, if we adopt the usualnotation a : σ to express that a is a term of type σ , then it is easyto see that for every term f : σ → τ , the function application f ( f ) will always be ill-typed. Unfortunately, as value-rangesare taken to be objects, self-application is completely able toenter the ideography through the back door, compromising thecoherence of the whole type system of the ideography.Even though there can be no doubt that Frege failed toarrive at a consistent conception of type constraint, the factremains that many of his logical insights are still acceptedtoday since the development of Church’s simple type theory. See Frege (1980, p.131–133).
This point has been alluded to by some authors. In particular,Quine (1940) has pointed out that Church revived Frege’sconception of function abstraction and predicates with hislambda abstraction and his treatment of predicates as functionsto the type of booleans. Later, Quine (1955) comes torecognize Frege’s hierarchy of objects, functions of first level,functions of second level and so forth as an anticipation, tosome degree, of the theory of types. Potts (1979) explores somesimilarities and differences between the ideography and thelambda calculus, and Klement (2003) adds detail and precisionto the comparison of the role played by value-range and lambdaterms in their respective systems, although major emphasis isplaced on the influence of Frege’s conception of value-ranges inRussell’s early work. Simons (2019) examines Frege’s use ofdouble value-ranges to provide extensions for binary functions,a technique known as currying in type theory. Despite that,however, I believe Frege’s theory of types still has not receivedthe attention it deserves from type theorists, and it is far fromobvious how Frege’s logical views should be understood from amodern type-theoretic perspective.Indeed, what I hope to show in the present paper is thatmore sense can be made of Frege’s theory of types than isgenerally assumed in the type theory literature. In Section 2,I claim that Frege’s conventional usage of Roman and Greekmarks allows him to distinguish between open and closed termsin a robust way that resembles the type-theoretic distinctionbetween hypothetical and categorical judgments. In Section 3, Iinvestigate Frege’s conception of value-ranges as objects and hisanticipation of the lambda calculus, and I conclude that Frege’sunderstanding of functions is based on a confusion betweenopen terms and closed terms of the function type. In Section 4,I examine how the view that value-ranges are functions wouldbe in conflict with Frege’s conception of identity and his logicistambition of establishing numbers as logical objects. After the discovery by Howard (1980) of a correspondence betweenpropositions and types in the context of constructive logic, this approachhas been generalized, giving birth to type families, indexed families of typesthat associate every term to a type.
The two forms of judgment that have become standard inmodern logic can be traced back to Frege’s ideography. First,we have the usual judgment form that states that a propositionis true, which he famously writes in a turnstile notation A and, second, we have the accompanying judgment form thatstates that a sentence expresses a proposition, which isinitially proposed in the early version of the ideography in Begriffsschrift through the notion of the content stroke A which characterizes a proposition as the content of a turnstilejudgment, that is, a “judgeable content”. The notion of contentwas originally the main semantic unit of the ideography, and itwas not until after the distinction between sense and referencethat a proposition came to be explained as the sense or thoughtof a sentence. That is, a sentence expresses a proposition andrefers to a truth value. When the theory of sense and referenceis formally incorporated into the ideography in Grundgesetze ,the turnstile judgment gets to be explained as the assertionthat a sentence refers to the true and the content stroke endsup being treated as a function that refers to the true for thetrue as argument and the false otherwise (the horizontal).As a result, it is no longer possible to assert the fact thata sentence expresses a proposition, which is to say that a termexpresses a thought, since the ideography does not have acounterpart for the content stroke anymore. In practice, thatis achieved through the manipulation of functions into truthvalues such as the horizontal, negation, identity, conditional,and universal quantifier, which, when fully saturated, areassumed to have a truth value as reference, therefore expressinga thought. Since sentences are handled as terms that referto objects, and a sentence that refers to exactly one truthvalue cannot be both true and false, a consistency proof forthe ideography can be given by showing that every well-formedterm in the system has a unique reference. That is preciselywhat Frege sets out to do in
Grundgesetze I §§ The assertion of a turnstile judgment is commonly supportedby one or more implicit typing annotations, because the factthat the proposition expressed by a particular sentence is trueoften depends on assumptions that a letter occurring in thesentence refers to an object or function. In order to ensure fulltransparency with respect to the assumptions needed in theassertion of a turnstile judgment such as A Ą ( B Ą A )I shall write all typing assumptions as explicit hypotheses onthe left-hand side of the turnstile in a modern notation, where a true indicates that a refers to the true A : ι, B : ι ⊢ A Ą ( B ⊃ A ) true . In fact, I shall be treating the relation a : σ and predicate a true as forms of judgments as well, meaning that they may haveassertive force and can be subject to rules of inference. Thus,to give an example, I may write x : ι, f : ι → ι ⊢ f ( x ) : ι to express that f ( x ) is an object under the assumption that f is a function and x is an object. In type theory, it is commonto refer to judgments made under typing assumptions such asthe last two illustrated above as hypothetical judgments andjudgments made under no typing assumptions such as the firstone above as categorical judgments.Although it would not be wrong to say that Frege makesextensive use of categorical judgments, it would seem thatwhen all the typing annotations of a claim are made explicitin the way I just suggested very few categorical judgmentscan be found in the ideography. This indicates that mostjudgments in the ideography are in effect hypothetical ones inour sense, and, as a matter of fact, the notion of hypotheticaljudgment is implied in the ideography through the adoption ofRoman letters since § Grundgesetze
I, where Frege speaksrepeatedly of the unsaturated nature of functions. In particular,(2 + 3 x ) x cannot refer to an object, since, when we substitute thenumerals 0, 1, 2, and 3 for the argument place x , we obtainterms referring to different objects, namely, the numbers 0,5, 28, and 87, respectively. That is because (2 + 3 x ) x isan open rather than a closed term, that is, it is a term withfree occurrences of variables. I shall express this fact with thehypothetical judgment x : ι ⊢ (2 + 3 x ) x : ι that asserts that (2 + 3 x ) x refers to an object provided that x also refers to an object, or, what amounts to the same thing,since every closed term is supposed to have a reference, that(2 + 3 x ) x is a closed object term for a closed object term x .Every function term is unsaturated in the ideography, butsurprisingly not every unsaturated expression is accepted asa function term. Perhaps the most straightforward way ofseeing this is by observing that every theorem in the ideographyis actually a schema formed by open sentences with freeoccurrences of sentential variables. Curiously, Frege does notview open terms as function terms because he does not acceptthem as referential terms:I shall call names only those signs or combinationsof signs that refer to something. Roman letters,and combinations of signs in which those occur,are thus not names as they merely indicate . Acombination of signs which contains Roman letters,and which always results in a proper name whenevery Roman letter is replaced by a name, I will call a Roman object-marker . In addition, acombination of signs which contains Roman lettersand which always results in a function-name whenevery Roman letter is replaced by a name, I willcall a
Roman function-marker or Roman marker ofa function. (
Grundgesetze I § ξ or ζ while in an open term we alwayshave Roman letters playing the role. For Frege, however, openterms only serve as indicators of objects. They are what hecalls “Roman marks”, unsaturated combination of signs thatresult in a closed term when every Roman letter occurring in itis replaced with a closed term. For an open term, the best wecan hope for is that a referential expression is achieved when alltheir variables are instantiated with closed terms. On the otherhand, although Function terms are unsaturated, we can showthat they are referential if they always result in a referentialexpression when properly applied to closed terms.By arguing that function terms are unsaturated, Frege notonly creates an unnecessary distinction between two classesof unsaturated expressions, but also commits himself to thepuzzling thesis that function terms behave like open terms x : ι ⊢ f ( x ) : ι despite actually being closed terms of a functiontype f : ι → ι . By “unsaturated expression” Frege seems tomean an open term. I will come back to this discussion inSection 3.3 after examining Frege’s conception of value-rangesand his view of abstracted functions as saturated entities. Although not a part of the referential terms of the language,uppercase Greek letters are in sharp contrast with Romanmarks in that they are used as a stand-in for referential terms.Heck (1997) views them as auxiliary names that are added tothe language subject to the condition that they must refer tosome object in the domain. Instead of relying on an assignmentof values to free variables, as it common in modern logic sincethe seminal work of Tarski, Frege makes use of auxiliary namesthat are assumed only to refer to some object.The supporting role of auxiliary names is most evident inFrege’s proof of referentiality in §§ Grundgesetze
I,where he intends to show that every closed term of theideography is referential by arguing by induction on theirstructure that f ( ξ ) is referential if f (∆) always has a referencefor every object term ∆ and that a is referential if Φ( a ) alwayshas a reference for every function term Φ( ξ ). Thus, Frege doesthink of uppercase Greek letters as terms that are assumed tohave a reference, although not only in the domain of objectsbut that of functions as well. In fact, since every closed termis supposed to be referential, I believe the claim that ∆ is anauxiliary name amounts to the hypothesis that it is closed term.Therefore, in a turnstile judgment∆ Ą (Γ Ą ∆)consisting entirely of uppercase Greek letters such as ∆ andΓ, what we actually have is a categorical judgment where everyauxiliary name is assumed to be closed, which is to say, in otherwords, that, by hypothesis, they are assumed to be derivableunder no typing assumptions. The presuppositions involved inthe reasoning above can be expressed more accurately as aninference where the notion of a closed term is made explicit ⊢ A : ι ⊢ B : ι ⊢ A Ą ( B Ą A ) true . In words, the inference above states that we are able to showthat the sentence A Ą ( B Ą A ) is true provided that we canassert that A and B are closed object terms. When judgmentsare represented in this way, as I shall do, we no longer have aneed to write auxiliary names as uppercase Greek letters.0 Considering that Frege is determined not to call into doubthis distinction between objects and functions, to questionthe premise that value-ranges are objects is to embrace theapparently superfluous thesis that the resulting value-rangeof a function is again a function. In the ideography, theconception of value-ranges as objects can be expressed as therequirement that that the value-range term ` ǫf ( ǫ ) refers to anobject for a function f , which, in our notation, can be moreeasily articulated via the hypothetical judgment f : ι → ι ⊢ ` ǫf ( ǫ ) : ι. (1)More precisely, value-ranges are regarded as first-class objectsobtained by function abstraction, which, in turn, areincorporated as a function of second level which has the value-range of a first-level unary function as value. In modernterminology, we can think of the value-range ` ǫf ( ǫ ) as as thegraph of the function f ( x ) for an argument x . The notion of value-range is governed by Basic Law V, theinfamous axiom that states that the value-range of the function f is the same as the value-range of the function g just in case f and g have the same values for the same arguments` ǫf ( ǫ ) = ` αg ( α ) ↔ ∀ ( a ) f ( a ) = g ( a ) . As I noted in the previous section, this is in fact an axiomschema since the function letters f and g are open terms, so, tobegin with, we observe that we have a hypothetical judgment f, g : ι → ι ⊢ ` ǫf ( ǫ ) = ` αg ( α ) ↔ ∀ ( a ) f ( a ) = g ( a ) true . Not only Roman letters have implicit typing assumptions, and,indeed, here we have another example with the Gothic lettersthat are used to mark the variables bound by a quantifier. This1convention is patently clear in
Grundgesetze I §
20 where thetheorem given as ∀ ( f )( ∀ ( a ) f ( a )) Ą f ( x )actually stands for the fully explicit theorem schema, which, forconvenience, I will henceforth write without Gothic letters, x : ι ⊢ ∀ ( f : ι → ι )( ∀ ( a : ι ) f ( a )) Ą f ( x ) true because x : ι and f ( x ) : ι can only mean that f : ι → ι . Now,if we apply the same strategy to Basic Law V as well, we areable to bring it to its definite form f, g : ι → ι ⊢ ` ǫf ( ǫ ) = ` αg ( α ) ↔ ∀ ( x : ι ) f ( x ) = g ( x ) true . On close inspection, one can see that there is one fundamentalassumption concerning the nature of value-ranges that is notstated in this formulation, namely, the condition (1) that value-ranges be objects. As it stands, there is nothing inherentlycontradictory about Basic Law V per se and this is reflectedby the fact that this law has been rediscovered in type theoryunder the name of function extensionality (see Section 4.1).
On an equal footing with Frege’s classification between firstand higher level is his distinction between unary and binaryfunctions in the ideography. Clearly, unary functions can beonly possibly annotated as f : ι → ι , but how should we writethe type of a binary function? Binary functions act on a pairof arguments but may also be partially saturated with a singleterm, resulting in a unary function:So far only functions with a single argument havebeen talked about; but we can easily pass on tofunctions with two arguments. These stand in needof double completion insofar as a function with oneargument is obtained after their completion by one2 argument has been effected. Only after yet anothercompletion do we arrive at an object, and this objectis then called the value of the function for the twoarguments. ( Grundgesetze I § g actually has the type g : ι → ( ι → ι )However, in general, a binary function may be partiallysaturated in two different ways, and, to make the argumentorder explicit, Frege always writes a binary function term g as g ( ξ, ζ ) instead, meaning that, given a term a : ι , he canindicate the two resulting unary functions as g ( a, ζ ) and g ( ξ, a ).In the lambda calculus, the use of explicit argument-marks areunnecessary because function abstraction leaves no room forambiguity in the determination of the arguments of a function,and in this way the unary functions above may be represented as λx.g ( a )( x ) and λx.g ( x )( a ), respectively. Simons (2019) explainsin detail that Frege’s ideography had a very similar feature forhandling application in curried functions with the use of thedouble value-ranges introduced in Grundgesetze I §
36. Since,for instance, g ( a, ζ ) is a unary function, we can form its value-range ` ǫg ( a, ǫ ) : ι . Now, by removing a : ι from this term, a termformation method explained in §
30, we can form a new unaryfunction ` ǫg ( ζ, ǫ ), whose value-range is ` α ` ǫg ( α, ǫ ) : ι , the doublevalue-range of the binary function g ( ξ, ζ ).In view of the fact that value-ranges are objects, it is notpossible to apply them to objects. Instead, Frege has a special3purpose application function, introduced in §
34, which is not abuilt-in primitive operation on a par with the ordinary functionapplication, which from f : σ → τ and a : σ results in f ( a ) : τ ,but, instead, is a definable binary function of the ideography,derived using the definite description function of §
11. I shallwrite this application function as an open term x, y : ι ⊢ x ∩ y : ι in order to avoid the implicit convention of adopting lowercaseGreek letters for argument-places. Now, x ∩ y refers to f ( x ) if y is a value-range ` ǫf ( ǫ ) and to the false otherwise. In fact, wehave an explicit equality that holds for value-ranges f : ι → ι, x : ι ⊢ x ∩ ` ǫf ( ǫ ) = f ( x ) true (2)and, given that a double value-range is just a value-range witha doubly-iterated function abstraction, for any binary function g : ι → ( ι → ι ), and terms a, b : ι , the following equality holds b ∩ a ∩ ` α ` ǫg ( α, ǫ ) = b ∩ ` ǫg ( a, ǫ )= g ( a, b ) . This eliminates the ambiguity in the order of application ofa binary function, allowing Frege to explicitly differentiatebetween the terms ` α ` ǫg ( α, ǫ ) and ` α ` ǫg ( ǫ, α ), which generally referto distinct objects when g is not commutative, since b ∩ a ∩ ` α ` ǫg ( α, ǫ ) = g ( a, b ) but b ∩ a ∩ ` α ` ǫg ( ǫ, α ) = g ( b, a ) . There has been one instance, pointed out by Simons (2019),where Frege articulates a notion of simultaneous application ofdouble value-ranges with ordered pairs. The idea is put forwardlater in § x ; y via application iteration as ` ǫ ( x ∩ y ∩ ǫ ) so that Actually, when y is not a value-range, x ∩ y refers to the value-rangeof a function whose value for every argument is the false, which, accordingto the stipulations of §
10 is the false itself (see Section 4.2). α ` ǫg ( α, ǫ ) ∩ a ; b = ` α ` ǫg ( α, ǫ ) ∩ ` ǫ ( a ∩ b ∩ ǫ )= b ∩ a ∩ ` α ` ǫg ( α, ǫ )= b ∩ ` ǫg ( a, ǫ )= g ( a, b ) . In type theories with a product type σ × τ , whose terms areordered pairs ( a, b ) : σ × τ for a : σ and b : τ , currying iscommonly expressed as a logical equivalence between the types σ × τ → υ and σ → ( τ → υ ), which states the existence of onefunction that transforms a binary function into its curried formand one function that takes a curried function and transformsit back into its binary form. Frege’s conception of curryingcould not possibly be better stated with his identification ofsimultaneous and iterated applications, except that since x ; y is a value-range, ordered pairs are not terms of a product type.They are objects like any others, and since a function f : ι → ι has to be defined for all objects (see Section 4.1), any such f can take an ordered pair as argument and still be unary. Recall that the real purpose of the restrictions of a type systemis to ensure that operations are applied only to argumentsof the intended domain, making sure that well-typed termsare always well-behaved in a certain sense. Given that everywell-formed term is well-typed and vice-versa, for Frege, goodbehavior means that a term definable in the ideography isreferential, and, as mentioned in the previous section, Fregegoes to great lengths in
Grundgesetze I §§ ǫf ( ǫ ) ∩ ` ǫf ( ǫ ) whichis coreferential with f (` ǫf ( ǫ )) for any f : ι → ι . Now, ifwe instantiate the application theorem (2) with the function ¬ x ∩ x and its corresponding value-range, we immediately arrive5at a contradiction, a closed term of the form ¬ a = a thatsimultaneously refers to both the true and false.It is therefore fair to assume that Frege’s theory of typesturned out to be inconsistent due to his flawed conception ofvalue-ranges as objects (1), which undermines all his efforts toseparate objects from functions. If the most prudent way out ofthe contradiction is to view value-ranges not as objects but asfunctions, the only problem is that value-ranges are themselvesdetermined by functions for Frege, so such a stipulation wouldmake the distinction between functions and value-ranges loseits purpose: f : ι → ι ⊢ ` ǫf ( ǫ ) : ι → ι It was Church (1940) who first realized with his lambda calculushow to adequately dissolve the dichotomy between functionsand value-ranges and capture Frege’s intuition that we formvalue-ranges by abstraction on unsaturated expressions. Theidea simply involves a functional abstraction on open terms,which are then regarded as closed function terms via theintroduction of an abstract binding operation that ranges overall their free occurrences of variables. That is, instead ofabstracting functions to form saturated objects, we abstractunsaturated objects to form functions. If we were to expressthis view in a Fregean style, it would be as x : ι ⊢ f ( x ) : ι ⊢ ` xf ( x ) : ι → ι . (3)Instead of Frege’s smooth-breathing, however, which isrecognized by Church (1942) himself as one of the precursorsof his lambda-notation, I shall stick to Church’s λx.f ( x ).Moreover, as the explicit use of hypothetical judgments alreadydetermines what variables an open term may depend on, (3)has a certain redundancy which may be completely eliminatedby rephrasing it as x : ι ⊢ f : ι ⊢ λx.f : ι → ι . (4)6 As a result of the identification of value-ranges withfunctions in the above sense, the ideography no longer wouldneed two distinct forms of function application, the primitive f ( x ) for functions and the derived x ∩ f for value-ranges.Instead, we need only one notion of application app ( f, x ), aunifying operation that comes with the same typing structureas f ( x ) in being restricted to a function and an object f : ι → ι, x : ι ⊢ app ( f, x ) : ι (5)and that inherits, at the same time, the computation rule of a ∩ ` ǫf ( ǫ ) which in §
34 is stipulated to denote the same as f ( a ),as stated in (2). In the lambda calculus, this computation isthe so-called β -reduction rule, which roughly states that theapplication of a lambda term to a closed term results in aclosed term where all occurrences of the abstracted variableare replaced with the term applied x : ι ⊢ f : ι ⊢ a : ι ⊢ app ( λx.f, a ) = f [ a/x ] : ι (6)Notice that this computation rule expresses the same idea asFrege’s stipulation of §
34, apart from the fact that it deals withfunction rather than object terms. Even another main rule ofChurch’s lambda calculus, α -conversion, the stipulation thattwo lambda terms that use different variable names are still thesame, was already envisioned by Frege (1891), who famouslydeclares that we can write a function like ‘ x − x ’ as ‘ y − y ’ without altering its sense. For the sake of brevity, I shallsimply write f ( x ) for app ( f, x ).Except by their restriction to the domain of the type ofindividuals ι , the rules described here are all present in thesimple type theory of Church (1940), an extended version ofthe lambda calculus with a type system composed of a type ofindividuals, functions, and truth values. It is remarkable thatthe two turning points that determine the success of Church’s I have stressed this point in Bentzen (forthcoming). For a more focusedinvestigation of Frege’s theory of sense and reference in the setting of typetheory see Martin-L¨of (2001) and Bentzen (2020b). α - and β -rules, I would like to consideranother important rule not found in Church (1940), but studiedextensively in Curry and Feys (1958). Considering that Frege’svalue-range terms are formed by abstraction on function ratherthan open terms, one may argue that the direct representationof the value-range ` ǫf ( ǫ ) should be the lambda-term λx.f ( x ),where f is a function. However, if x does not occur in f , thedistinction between λx.f ( x ) and f turns out to be unnecessaryextensionally speaking, since both functions will have the samevalue for the same arguments. This intuition is captured by therule known as η -reduction f : ι → ι ⊢ λx.f ( x ) = f : ι → ι (7)which was not originally included in the untyped lambdacalculus for the semantic reason that while the left side of theequality is a function, the right side may not be in Church’sintended interpretation (Curry and Feys, 1958, p.92). Clearly, η -reduction cannot have a representation in the ideography,because if value-ranges are objects, so it would not make senseto ask whether they could be functions.Since for Frege value-range terms always refer to objectswhile for Church lambda terms should be interpreted asfunctions, and, moreover, that for Frege function applicationmust result in an object while for Church functions are allowedto be values, it can be argued, following Potts (1979), that theirapproaches to function abstraction are fundamentally different.I am inclined to disagree with Potts on this point, for I do notsee the two reasons given above as compelling motivations fordistinguishing value-range and lambda terms. I shall address8the second reason first. As I have indicated in Section 3.2,Frege’s binary functions are essentially of second level. Fregeeven considers third-level functions in the ideography and, ascorrectly pointed out by Quine (1955), the only reason whyFrege does not adopt a hierarchy of higher functions is becausehe sees no need for it: his conception of value-ranges as objectsallows him to always reduce higher-level functions to objects.Finally, Klement (2003) calls attention to the fact that Fregehas once entertained the idea of having a function abstractiondevice for function terms in a letter to Russell of 13 November1904, where Frege employs a rough-breathing notation ´ ǫ ( ǫ = 1)for the function term that he would otherwise write as ξ = 1:But this notation would lead to the same difficultiesas my value-range notation and in addition to a newone. For a range of values is supposedly an objectand its name a proper name; but ‘´ ǫ ( ǫ = 1)’ wouldsupposedly be a function name which would requirecompletion by a sign following it. ‘´ ǫ ( ǫ = 1)1’would have the same meaning as ‘1 = 1’, andaccordingly, ‘´ ǫ ( ǫ = 1) Ą ’ would have to have thesame meaning as ‘ Ą = 1’, which, however, wouldbe meaningless. ‘´ ǫ ( ǫ = 1)’ would be defined only inconnection with an argument sign following it, andit would nevertheless be used without one; it wouldbe defined as a function sign and used as a propername, which will not do. (Frege, 1980, p.161–162).Although Frege clearly anticipates the developments of lambdacalculus in this passage, he quickly abandons the proposalof indicating function terms by abstraction because it wouldbe incompatible with his resolve that the nature of functionconsists in its unsaturatedness. For him, it is possible to use´ ǫ ( ǫ = 1) in isolation as an object term because it has nooccurrences of argument-places while ξ = 1 is a proper functionterm because the expression itself requires completion.Put differently, in the quotation above Frege suggests thatfunction terms must be unsaturated because they are formed byincomplete expressions and I believe that this leaves no grounds9for doubt that he completely confuses function terms f : ι → ι with open terms x : ι ⊢ f ( x ) : ι . If we think of Frege’s functionterms as open terms, whose argument-places are specified astyping assumptions, then the terms that should be assignedto the function type are his value-range terms, when properlyreinterpreted as lambda terms in the sense described earlier.It would seem to me that this confusion is one of the factorsthat motivates Frege to make the bold claim that value-rangesare objects (but see the Section 4.2). On the other hand,I should mention that this reading of Frege’s function termsas open terms is not fully consistent with all the aspects ofthe ideography I have discussed so far. In particular, Frege’saccount of functions of second and third level is not amenable tothis interpretation, since open terms, which are represented ashypothetical judgments, cannot be part of other hypotheticaljudgments in any way. In the final analysis, it seems thatFrege vacillates between the treatment of function terms as openterms and closed terms of a function type.
Finally, before I conclude, I would like to discuss what I seeas the main reason why Frege could not endorse the view thatvalue-ranges are functions, an observation that goes beyond hisconviction that value-range terms are saturated. Of relevance tothis is Frege’s conception of identity as a first-level relation, hislack of a direct identity criterion for functions, and convictionthat no identity statements can ever be made about functions.
Frege is known for conceiving identity as an all-inclusive relationin the domain of all objects, or, more precisely, for holdingthat an identity statement can be formed for any two objectsterms in the ideography. This happens to be a generalization But note that in (4) I rendered function abstraction, a second-levelfunction, as an inference rule that takes an open term to a closed term.
Grundgesetze II §
65, Frege is clear that this principle ofcomplete determination is expressed by the requirement thatevery function term must have a reference. It goes withoutsaying that this applies to the identity function as well x, y : ι ⊢ x = y : ι Frege’s criterion of referentiality of
Grundgesetze I §
29 statesthat to determine the reference of a function term x : ι ⊢ f : ι is to determine the reference of f ( a ) : ι for a closed a : ι . More precisely, to determine the reference of this function term,which is in fact a binary relation, it suffices to determine thetruth value of a = b for any two closed terms a, b : ι .That identity is restricted to the domain of objects is veryclearly expressed in Frege’s writings. Bearing in mind thatone should never be allowed to speak of two functions as beingthe same according to Frege, when a mathematician expressesthe view that two functions are identical he or she is, strictlyspeaking, incurring in a type mismatch error. Actually, whathe or she should have in mind is the idea of two functionsbeing coextensional, as noted in Ruffino (2003), which we mayarticulate as the fact that f and g always have the same valuefor the same arguments, that is, in the form of the first-levelrelation of pointwise function equality. Still, since the twohalves of Basic Law V are taken to express the same sense, butin a different way (Frege, 1891, p.27), and the value-range of afunction is an object, its graph, we can express coextensionalitymore directly in first-order terms via function abstraction, asan identity statement between the value-ranges of f and g .As a matter of fact, even when Frege seems to be explicitlyspeaking of an identity criterion for functions he recognizes itas a relation of second level that must be distinguished from The same strategy appears in Martin-L¨of’s (1982) meaning explana-tions of the hypothetical judgments of his type theory, but it is unclearwhether it was inspired by Frege’s approach. Either way, this puts moreweight to my allegation that Frege mistreats function terms as open terms. x = 1 has thesame (truth-) value as the function ( x + 1) =2( x + 1) i.e. every object falling under the concept less by 1 than a number whose square is equal to itsdouble falls under the concept square root of 1 andconversely. If we expressed this thought in the waythat we gave above, we should have α = 1 α ≍ ( α + 1) = 2( α + 1)What we would have here is that second levelrelation which corresponds to, but should notbe confused with, equality (complete coincidence)between objects. If we write it ∀ ( a )( a = 1) =( a + 1) = 2( a + 1), we have expressed whatis essentially the same thought, construed as anequation between values of functions that holdsgenerally. (Frege, 1979, p.121) Put another way, this means that Frege’s second-level relationterm for function correspondence f ( α ) α ≍ g ( α ) expresses, viaBasic Law V, the same sense as the first-level identity statementbetween value-range terms ` ǫf ( ǫ ) = ` αg ( α ). Here we see that theconception of value-ranges as objects is used in a crucial wayas a technical device to escape the restriction that one is notallowed to speak of identical functions.Notice that if we were to follow the developments of theprevious section of using lambda terms for value-range termsthen we would have to make some adjustments to Basic Law Vaccordingly, as we would have an illegitimate identity statement It is believed that Frege’s explanation of this new notation was givenin the lost first part of the manuscript (see Frege (1979, p.121, fn.1)). Itook the liberty to modernize Frege’s quantifier notation. f, g : ι → ι ⊢ λx.f ( x ) = λy.g ( y ) ↔ ∀ ( x : ι ) f ( x ) = g ( x ) true or, equivalently, in the presence of η -reduction, as f, g : ι → ι ⊢ f = g ↔ ∀ ( x : ι ) f ( x ) = g ( x ) true . The latter principle is known as function extensionality independent type theory, and, as it can be seen, it is nothingmore than a paraphrase of Basic Law V. Dependent typetheory is flavor of type theory developed by Martin-L¨of (1975)that extends simple type theory with the introduction ofdependent types, a concept that allows for an elegant treatmentof quantifiers and identity, and universe types, types whoseterms are themselves types. Identity is conceived as a sortalrelation that is limited to two terms of a same type. Comparingincomparables is not allowed, for it does not make sense to askwhether two terms of a different type are equal. Actually, just asenvisaged by Frege, identity of objects and identity of functionsare regarded as different relations. The difference is that independent type theory no identity relation is taken to be morefundamental than the other. But it must be emphasized that in the ideography thereis no sortal identity and a first-level relation of samenesscannot be confused with a second-level one. Although theabove formulation of function extensionality is meaninglessfor Frege, we can certainly avoid the so-to-speak fallaciousidentity statement f = g on its left-hand side if we state Surprisingly, many forms of dependent type theory are unable toprove function extensionality, even though the principle is validated byits intended semantics, the meaning explanations (Bentzen, forthcoming).Homotopy type theory (UFP, 2013) has been gaining acceptance as afoundational language for mathematics strong enough for proving notonly function extensionality but also that isomorphic objects are equal.However, as the theory has to abandon the meaning explanations as itsinformal interpretation, its philosophical coherence is open to question(Ladyman and Presnell, 2016; Bentzen, 2020a). f ( α ) α ≍ g ( α ) instead, which, is not only co-referential but hasthe same sense as the pointwise function equality statement ∀ ( x : ι ) f ( x ) = g ( x ). Since there is no violation of the principlethat identity is a first-order relation, it is fair to regard thisprinciple as Frege’s conception of function extensionality. For Frege, numbers are extensions of concepts, a supposedlylogical conception of class defined as value-ranges of predicates.Indeed, recall that Frege’s whole program presupposes thederivation of the concept of number from purely logical means.This definition is first envisaged in §
68 of
Grundlagen , Frege’sphilosophical masterpiece, but only after his proposal of twowell-known tentative definitions. His first definitional attemptis not seriously considered, and appears to serve only tomotivate his claim that numbers are self-subsistent objects.In contrast, there is no denying that Frege does seem tostruggle to establish the legitimacy of his second attempt, acontextual definition via the so-called Hume’s Principle, thatsays that the number that falls under the predicate f is equalto the number that falls under the predicate g iff f and g arein one-to-one correspondence, a relation of second level that issketched informally in §§ Grundlagen . We thus have xf ( x ) = xg ( x ) ↔ f ≈ g. (HP)Frege eventually rejects this tentative definition because it doesnot rule out the possibility of Julius Caesar being a number, andI have argued elsewhere that this strange objection is raised forthe reason that it cannot ensure our epistemic grip on numbersas logical objects (Bentzen, 2019). Surely, it can hardly beargued that Hume’s Principle is a candidate logical law and, ina letter to Russell of 28 July 1902, Frege expressly states thathe sees the notion of value-range as the only possible foundationfor our apprehension of logical objects:I myself was long reluctant to recognize value-ranges4 and hence classes; but I saw no other possibility ofplacing arithmetic on a logical foundation. But thequestion is, how do we apprehend logical objects?And I have found no other answer to it than this, weapprehend them as extensions of concepts, or moregenerally, as value-ranges of functions. (Frege, 1980,pp. 140–141)Frege’s transsortal identification of truth values with value-ranges proposed in Grundgesetze I §
10, one of the mostextensively studied sections of the book, makes it clear thatin his view value-ranges are the fundamental logical objectsthat populate the universe of arithmetic. To resolve areferential indeterminacy affecting the notion of value-range,Frege stipulates that the true is equal to the value-range of afunction that always has the true as value for every argumentand a similar specification is given for the false.This is a curious section that has caused considerableconfusion among Frege scholars because the Julius Caesarobjection is generally regarded as the semantic problem thatHume’s Principle does not determine the truth value of mixed-identity statements of the form xf ( x ) = a . This commoninterpretation of the Julius Caesar objection, however, isincapable of explaining why, in §
10, when a similar problemof indeterminacy is encountered, now with respect to BasicLaw V and the reference of mixed-identity statements of theform ` ǫf ( ǫ ) = a , which means that their truth value is yetto be decided, Frege feels entitled to restrict his solution tothe domain of logical objects with his stipulation that sometruth values are value-ranges, completely ignoring the questionof whether a value-range could be identical to an urelement.This dilemma can be resolved by noting that, since theJulius Caesar objection in Grundlagen just calls into questionwhether Hume’s Principle succeeds in establishing beyond alldoubt that we apprehend numbers are logical objects, oncevalue-ranges are already accepted as logical objects, there isno need to worry about urelements anymore. All that we needto do is to determine the reference of the function ` ǫf ( ǫ ) = x ,and, according to §
29, to do so is to determine the truth value of5the closed sentence ` ǫf ( ǫ ) = a for every closed term a : ι . Seeingthat, prior to the transsortal identification of §
10, every closedobject term in the ideography is supposed to refer to either avalue-range or truth value, and Basic Law V already takes careof value-range terms, it is enough to decide whether ` ǫf ( ǫ ) = a is true or false for a closed sentence a : ι .The position that value-ranges are functions may be ableto prevent the occurrence of paradox threats in the ideographyand, considering that it is founded on the assumption that weperform function abstraction on open terms, it may even bein line with Frege’s tendency to consider his function terms asopen terms, as I have already mentioned. Still, this positionwould be of no use to Frege. More than anything else, Fregefelt he had to commit himself to the existence of value-ranges inorder to define numbers logically and make his logicist programplausible, but to conform to his thesis that numbers are objects,value-ranges have to be objects as well. Frege came remarkablyclose to the formulation of simple type theory as we know it, butin the end he failed to express his theory of types consistentlydue to his logicist ambitions. References
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