frascolla - tractatus arithmetic

PASQUALE FRASCOLLA

THE TRACTATUS SYSTEM OF ARITHMETIC

ABSTRACT. The philosophy of arithmetic of Wittgensteins Tractatus is outlined and thecentral role played in it by the general notion of operation is pointed out. Following which,the language, the axioms and the rules of a formal theory of operations, extracted fromthe Tractatus, are presented and a theorem of interpretability of the equational fragment ofPeanos Arithmetic into such a formal theory is proven.

1. INTRODUCTION

In my (1994) I put forward a conjectural overall reconstruction of thepropositions of the Tractatus Logico-Philosophicus devoted to mathemat-ics. Its main conclusions can be summarized as follows: (i) Wittgensteinsfamous claim A number is the exponent of an operation (TLP 6.021) isto be interpreted as stating the thesis that each numeral represents a certainformal property of linguistic expressions belonging to any series which isgenerated by the successive application of an operation (any series whosefirst term is an expression not generated by means of the operation andevery other term of which is obtained from the immediately preceding termin the series by one application of the same operation): i.e. the number oftimes the operation is applied to generate an expression of such a kind.Wittgenstein viewed his treatment of the subject as an alternative to thelogicist explication of the primitive arithmetical notions of zero, successorand natural number in terms of the notion of class and his conceptioncan be matched to the basic insight on which the representation of naturalnumbers in Churchs -calculus is founded (apart from the obvious differ-ences); (ii) there is enough textual evidence in the Tractatus supporting theexegetical hypothesis that Wittgenstein though implicitly endorsed thefollowing tenet: every arithmetical identity can be mapped onto a corre-sponding equation of the general theory of operations, in such a way thatthe former is a theorem of arithmetic if, and only if, the latter is a theoremof the operation theory.

Here I am going to present the theory of operations which can be extract-ed from the Tractatus and to give a proof of the statement regarding theexistence of such a mapping, with reference to the equational fragment of

Synthese 112: 353378, 1997.c

1997 Kluwer Academic Publishers. Printed in the Netherlands.

354 PASQUALE FRASCOLLA

Peanos Arithmetic (Sections 3 and 4). Some peculiar features of Wittgen-steins translation of numerical arithmetic into the theory of operations arestressed in Section 5. Before the accomplishment of the main task of thepaper, the general philosophical setting of Wittgensteins interpretation ofarithmetic is outlined, with a view to the justification, supplied by the Trac-tatus, for the choice of the axioms and rules of the formal theory which Ipropose as a realization, as faithful as possible, of his intentions (Section2).

2. THE PHILOSOPHICAL BACKGROUND

Wittgenstein, whose early reflections on the foundations of logic and thenature of meaning were prompted as he himself says in the Preface ofthe Tractatus by Freges great works and by Russells writings, couldnot escape the problem of the status of mathematics and its relationshipwith logic. It is well known that in the FregeRussell logicist programmeof reduction of arithmetic (and, via the arithmetization of analysis, of allmathematics) to logic, logical calculus was understood in a broad sense, i.e.as including the theory of classes.1 For this very reason, the two blocks ofpropositions of the Tractatus devoted to mathematics (propositions 6.026.031 and propositions 6.26.241) may be, at first sight, rather surprising:on the one hand, Wittgenstein states that mathematics is a logical method(TLP 6.2 and, with a slight stylistic variation, TLP 6.234); on the other, inthe concluding proposition of the first group, one finds the drastic assertion:The theory of classes is completely superfluous in mathematics (TLP6.031). When Wittgenstein numbers mathematics among the methods oflogic, he seems to be saying something which is close to the core of thelogicist position; at the same time, proposition 6.031 proves that the senseof his matching mathematics with logic substantially differs from thatwhich is entailed by the FregeRussell thesis that mathematics is logic.

In the second part of proposition 6.031, Wittgenstein overtly states thereason for his rejection of the theory of classes. I will dwell on this themelater. For now it is expedient to expound along general lines the view whichthe Austrian philosopher sets against the FregeRussell approach to therelation between mathematics and logic. To put it in a nutshell, the notionof class is replaced, in its basic role, by the notion of logical operation; as aconsequence of this theoretical move, the primitive arithmetical conceptsof zero, successor and natural number are traced back to certain formalproperties of language. Thus the label no-classes logicism turns out totally with the Tractatus view of arithmetic.

THE TRACTATUS SYSTEM OF ARITHMETIC 355

Before undertaking the examination of the textual evidence supportingthe claim that it is the general notion of operation which plays the roleof basic notion in the philosophy of arithmetic of the Tractatus, a crucialpoint concerning the right interpretation of the word operation should bestressed. In the context under scrutiny, one may easily misunderstand itsmeaning and fall into the trap of considering natural numbers as essentiallyconnected to the notion of operation, inasmuch as they could be charac-terized as what can be obtained from 0 by any finite number of iterationsof the operation of adding 1 (+1). The risk of falling into this trap is reallygreat, when reading the Tractatus, owing to proposition 6.03 (The generalform of an integer is [0, , +1]), which seems to suggest the above char-acterization, framed in the usual Wittgensteinian style, i.e. as supplyingthe general term (the variable) for the series of natural numbers. But, if thisinterpretation is adopted, then it actually becomes impossible to accountfor some of the most telling features of the Tractarian treatment of arith-metic: first of all, for the inductive definition given by Wittgenstein in 6.02and for the related claim that a number is the exponent of an operation(TLP 6.021); second, for several propositions belonging to the group 6.26.241 (e.g. for proposition 6.241, where Wittgenstein, instead of definingthe function of product of two numbers by the usual recursive procedure,introduces complex arithmetical terms of the form as exponentsof the variable ). Plainly, it is reasonable to undertake the analysis ofproposition 6.03 only after having explained propositions 6.02 and 6.021which (with some other propositions) precede it in the sequence. In fact, itis in 6.02 that the meaning of the symbols 0 and + 1, which occurr inthe variable [0, , + 1], is fixed; following this path, one is eventuallyled to the general notion of logical operation and to its properties.

Proposition 6.02, which opens Wittgensteins treatment of arithmetic,begins as follows: And this is how we arrive at numbers. The italicsused for the pronoun this (in the German text, for the adverb so)witnesses Wittgensteins intention to point out the relationship betweenwhat he is going to say and what precedes it. In the immediately precedingproposition 6.01, the Greek capital letter occurs for the first time inthe Tractatus; it is used soon after in 6.02, where Wittgenstein introducesstandard numerals (terms of the form 0 + 1 + 1 + + 1) as its exponents.The exegetical conjecture that he employs as a variable representingthe formal concept of operation as a means to refer to an operation ingeneral is strongly corroborated by the pertaining textual evidence.2

Let us recall the fundamental features of the Tractarian notion of oper-ation: (i) an operation is a uniform procedure by the application of whichan expression (in particular, a proposition) can be generated from giv-


en expressions (again, propositions) whenever there is a formal relationbetween the former (the result of the operation) and the latter (its base).For instance, if the proposition A is formally related to the proposition Bin such a way that A is true ifB is false, and is false if B is true, thenA canbe represented as the result of the application of an operation (negation)to the base B, namely as B. Besides negation, instances of operationsexplicitly mentioned by Wittgenstein are: logical sum, logical product and,of course, joint negation, the generalized version of Sheffer stroke-functionused by the Austrian philosopher to generate all the truth-functions of theelementary propositions (TLP 5.25.23, 5.2341, 5.5); (ii) no object corre-sponds to an operation sign as its semantic value: therefore, the occurrenceof an operation sign in a proposition does not contribute to the determi-nation of the sense of the proposition in the same way as the occurrenceof names does (TLP 5.25); (iii) an operation can always be applied tothe result of its own application, i.e. the result of any application of theoperation is a suitable base for a new application of the operation (TLP5.2551). Wittgenstein calls successive application of an operation anyfinite sequence of applications of the operation, starting from an expres-sion which has not been generated by means of the operation (an initialexpression or initial symbol) and consisting in iterated applications of theoperation to the result of its own application (TLP 5.2521).

The clarification of both the role of the letter as an operation variableand the meaning that the word operation has in the Tractatus enablesus to answer the question of the true import of the inductive definitionof the expressions of the form 0+1+1++1x, laid down by Wittgensteinin 6.02. It has to be construed as a definition by induction on the lengthof a standard numeral (the number of occurrences of the symbol +1)and it is given the task of putting the abstract notion of application of anoperation at the bottom of the construction of arithmetic.3 By virtue ofthe inductive definition, an expression of the form 0+1+1++1x, (withn 0 occurrences of +1) has the same meaning as the string : : :

x (with the same number n of occurrences of ). Since the lattershows, at the utmost level of generality, the form of an expression yieldedby the n-th successive application of an operation (whatever this may be),one can safely say that a standard numeral is attached as an exponent to thevariable in order to represent the formal property which is common toall the expressions having that form: the number of times the operation isapplied to generate any one of them.

The shift from the notion of number as the cardinal number of a class(of the extension of a concept) to the notion of number as the numberof applications of a symbolic procedure reveals the distance of Wittgen-


steins logicism from that of Frege and Russell. According to the Austrianphilosopher, both the notion of zero and successor can be reduced to theabstract notion of application of an operation, at least in the sense that themeaning of each numeral 0 + 1 + 1 + + 1 is shown by the definiens ofthe corresponding expression in which it occurs as exponent of the opera-tion variable . This reductionistic claim is expressed by Wittgensteinsfamous statement that a number is the exponent of an operation (TLP6.021) and justifies, in my opinion, the matching of his view to the insighton which the representation of natural numbers in Churchs -calculus isfounded.

We will return to the relation between Wittgensteins system of arith-metic and the representation of natural numbers within -calculus in Sec-tion 5 below. As for the third primitive arithmetical notion of Peanosaxiomatic system (that of natural number), Wittgenstein decidedly rejectsthe FregeRussell theory of the so-called proper ancestral relation of a (giv-en) relation. In his opinion, the theory misunderstands the formal nature ofthe concept of arbitrary term of the formal series of propositions aRb,(9x): aRx . xRb, (9x, y): aRx . xRy . yRb and so on, whose logicalsum is asserted when the statement that b is a successor of a (with respectto the relation R) is made. For this very reason, it falls into a vicious circle(TLP 4.1273).4

Once that the FregeRussell definition of the predicate natural numberhas been discarded, the treatment which the predicate undergoes in the parsconstruens of Wittgensteins theory can be dealt with. First, it expressesa formal concept: every instance of the schematic sentence n falls underthe concept of natural number is nothing but the result of an ill conceivedattempt to say something that belongs to the semantic domain and that canonly be shown by the use of the symbol which replaces n in the schema.As in all other cases of formal concepts, the relation between the conceptof number and the objects falling under it is radically different from theapparently analogous relation between any material concept (eigentlichBegriff) and the objects which fall under it: the concept of number issimply what is common to all numbers, the general form of a number(TLP 6.022). By this, Wittgenstein means that the concept of a naturalnumber is the formal characteristic which is common to all numerals instandard notation, i.e. the possibility of obtaining any one of them byadding a finite number of times the symbol +1, starting from 0. Sincethis totality is an endless series, a perspicuous variable for the formalconcept of natural number ought to exhibit the first term of the seriesand the uniform procedure to obtain any term, other than the first, from itsimmediate predecessor: the variable [0, , +1] fulfils the requirements.


However, one further comment is needed for a proper understanding of sucha variable: as, in the variable [p, , N()], that represents the generalform of a proposition, the meaning of the symbol p, which denotes the setof elementary propositions, and that of the symbol N, which denotes theoperation of joint negation of a set of propositions, are taken for granted, sois the meaning of the symbols 0 and +1 in the variable [0, , + 1]:and it is the inductive definition in 6.02 which establishes this meaning,along the lines suggested above.

The true import of Wittgensteins conception of arithmetic is far fromexhausted by the reduction of the meaning of each numeral 0 + 1 + 1 + + 1 (with n 0 occurrences of +1) to the formal property shownby the corresponding string of the operation theory language

x (with the same number of occurrences of ). A further develop-ment is suggested by the definition of the arithmetical function of productof two numbers and by the proof of the equation (22)x = 4x,which translates the arithmetical identity 2 2 = 4 into the operationallanguage (TLP 6.241). Though meagre, the textual evidence supplied byproposition 6.241, together with the results of the interpretation achievedso far, gives reasonable support to my attempt of systematic reconstructionof the Austrian philosophers sketchy outline.5 Here I limit myself to recallthat the decisive step in this direction is constituted by the interpretation ofthe expression ()x, which occurs in Wittgensteins aforementionedproof. Whereas the expression x shows the form of the result of theapplication of an operation to the result of its own application to an initialsymbol, the expression ()x shows the form of the result of the appli-cation of the composition of an operation with itself (the second iterationof an operation) to an initial symbol. Once the notion of composition oftwo operations (and the related notation) is available, the expressions ofboth the forms (rs)x and (r+s)x can be defined within the theory ofoperations and the translation of the equational fragment of arithmetic intothat theory becomes feasible.

Proposition 6.241 apart, the second block of propositions of the Trac-tatus devoted to mathematics (TLP 6.26.241) is mainly concerned withphilosophical issues. According to Wittgenstein, propositions of mathe-matics are to be identified with equations (expressions of the form tx =

r

x, where t and r stand for arbitrary arithmetical terms). An imme-diate consequence of such an identification is that they are confined amongthe pseudo-propositions (Scheinsatze), which do not express a thought(TLP 6.26.21). In order to clarify the reasons for Wittgensteins somewhatdisconcerting claim, a close comparison between equations correspondingto true arithmetical identities and tautological formulae of logic may be


useful. As known, he maintains that neither tautologies are able to expressa thought, since they are true in all possible configurations of the world.Nonetheless, the difference of status between correct equations of opera-tion theory and tautologies of logic is, in a crucial respect, more importantthan their similarity. In general, a Scheinsatz is, in the terminology of theTractatus, a linguistic construct yielded by the attempt to say somethingthat can only be shown by language (its formal properties).6 On the con-trary, tautologies though devoid of sense are not pseudo-propositions:they are sinnlos, they lack sense, but are not unsinnig (non-sensical) as,according to Wittgenstein, equations are.

The difference can be easily illustrated by comparing the logical formulap q :: q p with the equation (22)x = 4x. Thetautologousness of p q :: q p shows that any two complexpropositions which are constructed from the same pair of propositionsand whose forms are exhibited, respectively, by p q and q p are tautologically equivalent (hence, according to Wittgensteins purelyextensional criterion of propositional synonymy, that they have the samesense). Needless to say, the symbol , occurring in the logical formulap q :: q p, is the sentential connective called biconditionaland does not mean the metalogical relation of tautological equivalence.Now, even the correctness of the equation (22)x = 4x shows theidentity of meaning of any two expressions generated by one and thesame operation, starting with the same initial symbol whose forms areexhibited, respectively, by ()()x and x. But theidentity sign = which occurs in the equation different from the sententialconnective plays the forbidden role of asserting the existence ofthis semantic relation: as in all similar cases, the attempt to say that ametalogical relation obtains, yields a pseudo-proposition.

According to the Tractatus theory of sense, equations, like tautologies,do not express thoughts: nonetheless, whereas the latter are limiting casesof genuine propositions, the former are, so to speak, vanishing entitiesbecause, if language were regimented in accordance with the strict rulesof logical syntax, they would disappear, as would all the other pseudo-propositions. What Wittgenstein says in the propositions following 6.21brings about a further weakening of the precarious status of equations.He speaks of the identity of meaning (Identitat der Bedeutung) and of theequivalence in meaning (Bedeutungsgleichheit) of the expressions occur-ring on the two sides of the sign = in a correct equation and this inevitablyleads him to a comparison with Freges solution to the so-called paradoxof the informative value of identities (as far as mathematics is concerned).Propositions 6.231 and 6.232 (first part) run as follows: It is a property of


affirmation that it can be construed (da man sie : : : auffassen kann) as dou-ble negation. It is a property of 1 + 1 + 1 + 1 that it can be construed as(1 + 1) + (1 + 1). Frege says that the two expressions have the samemeaning (Bedeutung) but different senses (Sinn). Wittgenstein, on theother hand, rejects the view that there exists an ideal object one and thesame, namely the number 4 which is identified in two different waysby the arithmetical terms in question. An expression where an arithmeti-cal term occurs as exponent of the variable is short for a symbolicconstruct which directly shows the arithmetical structure of an operationalmodel of sign construction, i.e. the arithmetical structure of a form. Anequation should be considered as an attempt to assert the mutual reducibil-ity of two such forms: since a form is not an object denoted or describedby tx but is that which is shown by the corresponding expressionwithout arithmetical exponents (the expression into which tx can betransformed by means of the relevant definitions), stating the identity ofmeaning, for instance, of x and (()())x simplyamounts to asserting that the latter is a different grouping of the occurring in the former.

The recognition of the formal possibility of the process which trans-forms one configuration of a certain string of into another (formal,since calculation is not an experiment, TLP 6.2331) is set by Wittgen-stein against Freges vindication of the informative value of identities interms of recognition of the sameness of the objects denoted by two expres-sions with different senses. This quasi-formalistic tenet is formulated byWittgenstein in the second part of 6.232: But the essential point about anequation is that it is not necessary in order to show that the two expressionsconnected by the sign of equality have the same meaning, since this canbe seen from the two expressions themselves.

To be quite honest about it, the problem of the very existence of equa-tions is far from being solved by the above interpretation. Indeed, if whatreally matters is the ineffable content of an equation the mutual reducibil-ity of two forms and if this can be directly seen from the two symbol-ic constructs which exhibit the forms, then the questions: why does theequality sign have to be introduced in the notation? why do mathemati-cal pseudo-propositions (equations) have to be formulated? still demand,with even greater force, a plausible answer. In order to find it, one hasto take into account the distinction between what is shown by languageand what, in any given empirical circumstance, users of language are ableto see; or, in equivalent terms, between a god who has a complete visionof the formal domain and an individual who is not endowed with logicalomniscience. Whereas there are no formal properties and relations (shown


by language) which God does not see, their immediate visibility is, forus, only an ideal. God would have no reason to formulate those pseudo-propositions which equations are, and even less, to use them in that stepby step procedure of substitution of expressions with the same meaningwhich, in Wittgensteins view, constitutes calculation. It is only the empir-ical limitations of our skill in grasping the relations between forms thatmakes it indispensable to resort to equations and to arithmetical calculus.By extending to arithmetic what proposition 6.1262 says of the applica-tion of logical techniques, one can conclude that calculation is merely amechanical expedient to facilitate the recognition of the correctness ofequations in complicated cases. The existence of arithmetic is exclusivelydue to the contingent limitations in our capability of direct recognitionof formal properties and relations. However, the effective decidability ofnumerical identities which, not by chance, are the only mathematicalpropositions dealt with by Wittgenstein in the Tractatus ensures thatthe difference between Gods knowledge and ours is merely empirical,extensional: in every specific case, the gap between God and us can bebridged, in principle, by the application of calculating techniques.

We have reached a position which enables us to clarify the relationbetween the Tractatus philosophy of arithmetic and Freges and Russellslogicism and to justify the proposal of the label no-classes logicism asa characterization that tallies with Wittgensteins view. Mathematics iscounted among the methods of logic since it deals with formal propertiesand relations: as seen, equations mark the recognition of the identity ofmeaning of expressions generated by processes (of growing complexity)of iteration and composition of logical operations.7 The idea that forms oflinguistic expressions have an arithmetical structure is the cornerstone ofthe Tractatus conception of arithmetic and determines the kind of connec-tion existing between mathematics and the world. What holds true for logicholds true for mathematics as well: first, the possibility of ascertaining thecorrectness of an equation by calculation, i.e. by means of a procedureof symbolic manipulation which does not go outside language, provesthat such correctness has nothing to do with the actual configuration of theworld (TLP 6.2321); second, a correct equation, being concerned with theidentity of meaning of linguistic expressions, grounded on their abstractoperational structure, shows a formal feature of the world (i.e. a featurewhich it shares with every possible world), shows the logic of the world(TLP 6.22). Just as with logic, mathematics comes into contact with the(one and only) world in virtue of its peculiar relationship with the formsof picturing-facts language.


The theme of the relation between mathematics and the world is alsoinvolved in Wittgensteins rejection of the FregeRussell foundation ofmathematics on the theory of classes. He explains the reason for his rejec-tion by invoking the fact that the generality required in mathematics isnot accidental generality (TLP 6.031). Plainly, Wittgenstein has in mindthe type-theoretical version of the theory of classes: this is charged withthe accusation that its critical axioms (Axiom of Reducibility, Multiplica-tive Axiom, Axiom of Infinity), despite their complete generality, are nottautological: even if they are true, they are merely accidentally true truebecause of the fortuitous circumstance that a certain possible configura-tion of the world happens to be its actual configuration (TLP 6.1232). Thetranslation of arithmetic into the theory of operations should afford theformal purity of arithmetic, its essential generality, which, according toWittgenstein, the type-theoretical translation fails to preserve.

In conclusion, I cannot pass in silence over what soon appeared asa very serious weakness of the Tractarian approach to mathematics. Inthe frame of his general theory of operations, Wittgenstein covers onlya small portion of arithmetic: any reference to the entire remaining partof mathematics, even to the part of number theory beyond its equationalfragment, is eschewed. In his Introduction to the Tractatus, Russell notesthat Wittgensteins theory is only capable of dealing with finite numbersand that it stands in need of greater technical developments in order thattransfinite numbers can be dealt with.8 Ramsey, reflecting upon this quiteunsatisfactory situation, remarks that Wittgensteins view, as it stands, isobviously a ridiculously narrow view of mathematics.9 One can conjecturethat Wittgensteins somewhat astonishing laconism on the subject provesthat he took for granted the possibility of extending his interpretation toall of mathematics (no further philosophical achievement would have beenneeded, in his opinion, for the accomplishment of this task). In a differentdirection, one can wonder if his rejection of the theory of classes witnessesthe existence of a revisionary attitude to mathematical practice: but only acritical examination of Wittgensteins overall reflection on mathematics from the Tractatus up to his latest writings might give a clue to answerthe question.

3. WITTGENSTEINS THEORY OF OPERATIONS (W)

The language, the axioms and the rules of the formal theory of operationsthat I have extracted from the Tractatus are presented in this section.It is a logic free, or purely equational, theory: logical connectives andquantifiers are not included in its language and, accordingly, no logical


axioms and rules belong to its deductive machinery. This perfectly squareswith Wittgensteins claim that equations are pseudo-propositions and henceare not suitable bases for the application of logical operators.

3.1. The Language of the Theory W (LW)(I) Primitive symbols of L

W

.

I;; ; ; x;=; (; ); S; 0;+;:(II) Definition of the category of the arithmetical terms of L

W

.

(IIa) Definition of the category of the numerals of LW

.

(i) 0 is a numeral of LW

;(ii) If U is a numeral of L

W

, then SU is a numeral of LW

.

We use n (in boldface) as an abbreviation for SS : : : S0, with noccurrences of S.

(IIb) (i) Every numeral of LW

is an arithmetical term of LW

;(ii) If r and s are arithmetical terms of L

W

, then (r + s) and (r s)are arithmetical terms of L

W

.

(III) Definition of the category of the operational terms of LW

.

(i) I and are operational terms of LW

;(ii) If t is an arithmetical term of L

W

and O is an operational term ofLW

, then tO is an operational term of LW

.

(iii) If O1 and O2 are operational terms of LW

, then (O1O2) is anoperational term of L

W

.

(IV) Definition of the category of the operational expressions of LW

.

(i) and x are operational expressions of LW

;(ii) If O is an operational term of L

W

and t is an arithmetical term ofLW

, then Otx is an operational expression of LW

;(iii) If E is an operational expression of L

W

and O is an operationalterm of L

W

, then OE is an operational expression of LW

.

(V) Definition of the equations of LW

.

(i) If O1 and O2 are operational terms of LW

, then O1 = O2 is anequation of L

W

;(ii) If E1 and E2 are operational expressions of L

W

, then E1 = E2 isan equation of L

W

.

3.2. Axioms and Rules of the Theory WAXIOMS


(I) The Axiom for II =

(II) The Axiom-Schema for For every pair of operational terms O1, O2, the equation (O1O2) =O1O2 is an axiom of W.

(III) Definitions of the operational terms t for every arithmetical term tof L

W

.

(1a) 0 = I .(1b) For every n 0, sn = (n).(2a) Let t be an arithmetical term (r + s); then, (r+s) = (rs).(2b) Let t be an arithmetical term (r s); then, (rs) = sr.

(IV) Definitions of the operational expressionstx for every arithmeticalterm t of L

W

(1a) 0x = x.(1b) For every n 0, snx = nx.(2a) Let t be an arithmetical term (r + s); then (r+s)x = (rs)x.(2b) Let t be an arithmetical term (r s); then (rs)x = rsx.

DERIVATION RULES

For every pair of operational terms O1, O2, the equation O1= O2 can be derived from the equation O1 = O2.

(R1)

For every operational term O and every pair of operationalexpressions E1, E2, the equation OE1 = OE2 can be derivedfrom the equation E1 = E2.

(R2)

By subst/O [E] the result of the uniform substitution of with the operational term O in the operational expression E isto be meant; then: for every operational term O and every pairof operational expressions E1, E2, the equation subst /O[E1]= subst /O[E2] can be derived from the equation E1 = E2.

(R3)

By subst /E[E] the result of the substitution of withthe operational expression E in the operational expression Eis to be meant; then: for every triple of operational expressionsE, E1, E2 such that is the end symbol of both E1, E2, theequation subst /E[E1] = subst /E[E2] can be derived from theequation E1 = E2.

(R4)

Reflexivity of =.(R5)


Symmetry of =.(R6)

Transitivity of =.(R7)

3.3. Some CommentsAs for the language of the theory W, the most important changes withrespect to the Tractatus are: (i) the introduction of the operational constantI, which denotes the identical operation (Axiom I); (ii) the introductionof operational terms in which an arithmetical term occurs as an exponenton the left of the operational variable (Definition III). As clearlyshown by Definition III, the meaning of such operational terms is fixed byresorting to the notion of composition of two operations: Axiom-Schema IIstates the usual fundamental property of the composition of two operations.The notation (O1O2) for the composition of the operations O1 and O2is borrowed from proposition 6.241 of the Tractatus. Even the variable, which occurs in Axiom II, is borrowed from the Tractatus. It can besubstituted by any operational expression, without the restrictions that thereplacement of the variable x would undergo (as explained in Section 2,the latter shows the form of an initial expression of any series generatedby the successive application of a logical operation).

4. THE INTERPRETATION THEOREM

Let PE be the equational fragment of Peanos Arithmetic. The set ofthe (closed) arithmetical terms of the language of PE coincides with theset of the arithmetical terms of L

W

; I am going to prove the followingmetatheorem: let t and r be any two such arithmetical terms; then:

`

W

t

x =

r

x if and only if `PE

t = r:

The proof of the existence of this theoretical mapping should providea reasonable justification for Wittgensteins unproven assumption (in theTractatus) that the equational fragment of arithmetic can be translated intothe general theory of logical operations.

Along general lines, the proof of the theorem goes as follows: first, thefollowing result is proven: for every arithmetical term t of L

W

(and thenfor every closed arithmetical term of L

PE

), there exists one and only onenumeral n such that `

W

t

x=n

x and `PE

t = n. Once this pointis reached, two further results are needed in order that the main theoremcan be easily proven: (i) for every arithmetical term t of L

W

, `W

t

x =


t

x; (ii) for a certain function D(E) (degree of an operational expressionE), suitably defined in metalanguage, if `

W

E1 = E2, then D(E1) = D(E2)(the identity sign = is used in metalanguage too).

LEMMA 1. For every numeral n, `W

n

= n

.

Proof: By induction on the number of occurrences of S in n:(i) Base: n is 0. We have:(1) 0 = I A III 1a(2) 0 = I R1, (1)(3) I = AI(4) 0 = R7, (2), (3)(5) 0 = R4, (4)(6) 0 = R2, (4)(7) 0 = 0 R7, (5), (6)(ii) Inductive step. By hypothesis,`

W

n

= n

for an arbitraryn. Then:

(1) n = n R2, ind. hyp.(2) (n) = sn AIII1b, R1, R7(3) (n) = sn R4, (2)(4) (n) = n AII(5) (n) = n R4, (4)(6) sn = n R7, (3), (5)(7) (n) = sn R2, (2)(8) (n) = n R2, (4)(9) n = sn R7, (7), (8)(10) sn = sn R7, (6), (1)(11) sn = sn R7, (10), (11)

THEOREM 1. For every pair of numerals m and n, there exists one andonly one numeral k such that:

`

W

(m+n)

=k

and`PE

m+ n = k:

Proof:(1) Existence: by induction on the number of occurrences of S in n.(i) Base. n is 0. We have(1) 0 = I A III 1a(2) 0 = I R1, (1)(3) I = AI(4) 0 = R7, (2)(3)(5) m0=m R2, (4).


(6) (m+0) = (m0) AIII 2a, R1(7) (m0) =m0 AII(8) (m+0) =m R7, (5), (7)Since`

PE

m + 0 = m, m is the numeral whose existence should be proven.(ii) Inductive step. By hypothesis, there exists a numeral k such that

`

W

(m+n)

=k

and `PE

m + n = k, for an arbitrary n.Then:

(1) sn = (n) AIII 1b, R1(2) (n) = n AII(3) sn = n R7, (1), (2)(4) msn =mn R2, (3)(5) m =m Lemma 1, R6(6) mn = mn R4, (5)(7) (m+n) = k R2, ind. hyp.(8) (m+n) =mn AIII, R1, AII(9) (m+n) = mn R2, (8)(10) mn = k R7, (7), (9)(11) (k) = k AII(12) sk = (k) AIII 1b, R1(13) msn = mn R7, (4), (6)(14) msn = k R7, (13), (10)(15) k = sk R7, (11), (12)(16) msn = sk R7, (14), (15)(17) (m+sn) = (msn) AIII 2a, R1(18) (msn) = msn AII(19) (m+sn) = sk R7, (16),(17),(18)By inductive hypothesis, k is a numeral; then Sk is a numeral as well.Furthermore, by inductive hypothesis, `

PE

m + n = k; then `PE

m + Sn= S(m + n) = Sk and thus Sk is the numeral whose existence should beproven.

(2) Uniqueness:

Suppose that there exist two distinct numerals k and j such that, for a givenpair of numerals m and n:

`

W

(m+n)

=k

; `PE

m+ n = k

and

`

W

(m+n)

=j

; `PE

m+ n = j:


Then `PE

j = k; this holds only if j = k and this entails that j and k areone and the same numeral, which is contrary to hypothesis.

THEOREM 2. For every pair of numerals m and n there exists one andonly one numeral k such that:

`

W

(mn)

=k

and `PE

m n = k:

Proof:(1) Existence: by induction on the number of occurrences of S in n.(i) Base. n is 0. We have:(1) 0 = I AIII 1a(2) 0 = I R1, (1)(3) I = AI(4) 0 = R7, (2) (3)(5) 0m = R3, (4)(6) 0m = 0 R7, (4), (5)(7) (m0) = 0m AIII 2b, R1(8) (m0) = 0 R7, (6), (7)Since `

PE

m 0 = 0, 0 is the numeral whose existence should be proven.(ii) Inductive step. By hypothesis, there exists a numeral k such that `

W

(mn)

=k

and `PE

m n = k, for an arbitrary n. Then:(1) sn = (n) AIII1b, R1(2) snm = (mnm) R3, (1)(3) (mnm) =mnm AII(4) (mn) = nm AIII 2b, R1(5) m(mn) = mnm R2, (4)(6) m(mn) = (mnm) R7, (3), (5)(7) snm =m(mn) R7, (2), (6)(8) m(mn) =mk ind. hyp., R2(9) snm =mk R7, (7), (8)(10) (m+k) = (mk) AIII2a, R1(11) (mk) = mk AII(12) (m+k) = snm R7, (9), (10), (11)By theorem 1, there exists one and only one numeral j such that:`

W

(m+k)

=j

and `PE

m + k = j. Then:(13) snm = j R7, (12), Th 1(14) (msn) = snm AIII2a, R1(15) (msn) = j R7, (13), (14)


From `PE

m n = k and `PE

m + k = j it follows that `PE

m Sn =m + (m n) = m + k = j. Then j is the numeral whose existence shouldbe proven.

(2) Uniqueness:

As in proof of Theorem 1.

LEMMA 2. For any arithmetical term t, if `W

t

=m

, then`

W

n

t

=n

m

for every numeral n.Proof: Let t be an arbitrary arithmetical term; we proceed by induction

on the number of occurrences of S in n.

(i) Base. n is 0. We have:(1) 0 = I AIII1a(2) 0 = I R1, (1)(3) I = AI(4) 0 = R7, (2) (3)(5) 0t = R3, (4)(6) 0m = R3, (4)(7) 0t = 0m R7, (5), (6)

(ii) Inductive step. By hypothesis, `W

n

t

=n

m

for an arbitraryn. Then:(1) snt = (tnt) AIII1b, R1, R3(2) (tnt) = tnt AII(3) tnt = tnm Ind. Hyp, R2(4) tnm =mnm Lemma Hyp., R4(5) tnt = mnm R7, (3), (4)(6) mnm = (mnm) AII, R6(7) (mnm) = snm AIII1b, R1, R3, R6(8) tnt = snm R7, (5), (6), (7)(9) snt = snm R7, (1), (2), (8)

THEOREM 3. For every arithmetical term t there exists one and only onenumeral k such that: `

W

t

=k

and `PE

t = k.Proof:

(1) Existence. By induction on the logical complexity of the term t.(i) Base. t is a numeral n. Then the numeral looked for is n iself (by R5).


(ii) Inductive step.(ii1) t is (r + s).By inductive hypothesis: (a) `

W

r

=m

and `PE

r = m for onegiven numeral m; (b) `

W

s

=n

and `PE

s = n for one givennumeral n. Then we have:(1) (r+s) = (rs) AIII2a, R1(2) (rs) = rs AII(3) rs = rn Ind. Hyp. (b), R2(4) rn =mn Ind. Hyp. (a), R4(5) (m+n) = (mn) AIII2a, R1(6) (mn) =mn AII(7) (r+s) = mn R7, (1), (2), (3), (4)(8) (r+s) = (m+n) R7, (5), (6), (7)By Theorem 1, there exists a numeral k such that `

W

(m+n)

=k

and `PE

m + n = k; then `W

(r+s)

=k

and from the inductivehypothesis it follows that `

PE

r + s = k.

(ii2) t is (r s).

The inductive hypothesis is the same as in (ii1). Then:(1) (rs) = sr AIII2b, R1(2) sr = nr Ind. Hyp. (b), R3(3) nr = nm Ind. Hyp. (a), Lem 2(4) nm = (mn) AII2b, R1, R6(5) (rs) = (mn) R7, (1), (2), (3), (4)By Theorem 2, there exists a numeral k such that `

W

(mn)

=k

and `PE

m n = k; then: `W

(rs)

=k

and from the inductivehypothesis it follows that `

PE

r s = k.

(2) Uniqueness.

As in the proof of Theorem 1.

THEOREM 4. For every arithmetical term t, `W

t

x =t

x.

Proof: By induction on the logical complexity of the term t. Now wedefine (in metalanguage) the function degree of an operational term O,D(O), by induction on the complexity of an operational term O:(a) O is I: then D(O) = D(I) = 0;

O is : then D(O) = D() = 1;


(b) O is tO1

for some arithemetical term t and some operational term O1

;then:

D(O) = D(tO1

) = D(O1

) t;(c) O is (O

1

O2

) for two operational terms O1

, O2

; then:D(O) = D((O

1

O2

)) = D(O1

) + D(O2

).Notice that, whereas t (in bold) is a syntactic variable, t is a numerical

variable belonging to metalanguage. The correspondence between the val-ues of the two variables is the natural one (for instance, if the term SS0is the value of t, the number 2 is the value of t).

The function D being on hand, we can define the function degree ofan operational expression E, D(E), by induction on the complexity of anoperational expression E:(a) E is x; then D(E) = D(x) = 0;

E is ; then D(E) = D() = 0;(b) E is Otx for some arithmetical term t and some operational term O;

then:D(E) = D(Otx) = D(O) t;

(c) E is OE1

for some operational term O and some operational expres-sion E1; then:

D(E) = D(OE1

) = D(O) + D(E1).Some lemmas are needed to prove that the two operational expressions

occurring on the two sides of = in any provable equation (in the theoryW) have the same degree. Only the proof of the first one of them will begiven for illustrative purposes.

LEMMA 3. For every pair of operational terms O and O1

:

D(subst /O1

[O]) = D(O1

) D(O).

Proof: By induction on the complexity of the operational term O:(i) O is I; then:

D(subst /O1

[O]) = D(subst /O1

[I]) = D(I) = 0= D(O

1

) 0 = D(O1

) D(I).(ii) O is ; then:

D(subst /O1

[O]) = D(subst /O1

[]) = D(O1

)= D(O

1

) 1 = D(O1

) D().


(iii) O is tO2

for some arithmetical term t and some operational term O2

;then:

D(subst /O1

[O]) = D(subst /O1

[tO2

]) =D(tsubst /O

1

[O2

])) = D(subst /O1

[O2

]) t;by inductive hypothesis: D(subst /O

1

[O2

]) = D(O1

) D(O2

);hence:

D(subst /O1

[O2

]) t = (D(O1

) D(O2

)) t = D(O1

) (D(O

2

) t) = D(O1

) D(tO2

) = D(O1

) D(O).(iv) O is (O

2

O3

) for two operational terms O2

, O3

; then:D(subst /O

1

[O]) = D(subst /O1

[(O2

O3

)]) =D((subst/O

1

[O2

]subst/O1

[O3

])) = D(subst/O1

[O2

])+ D(subst /O

1

[O3

]);by inductive hypothesis:

D(subst /O1

[O2

]) = D(O1

) D(O2

) andD(subst /O

1

[O3

]) = D(O1

) D(O3

);hence:

D(subst /O1

[O2

]) + D(subst /O1

[O3

]) = (D(O1

) D(O

2

)) + (D(O1

) D(O3

)) = D(O1

) (D(O2

) +D(O

3

))) = D(O1

) D((O2

O3

)) = D(O1

) D(O) LEMMA 4. For every operational term O and every operational expres-sion E,

D(subst /O[E]) = D(O) D(E).

Proof. By induction on the complexity of the operational expression E.

LEMMA 5. For every pair of expressions E and E1, if is the end symbolin E,

D(subst /E1

[E]) = D(E1

) + D(E).

Proof. By induction on the complexity of the operational expression E.

LEMMA 6. For every pair of operational terms O1

, O2

, if `W

O1

= O2

,

then D(O1

) = D(O2

).Proof. By induction on the length of the proof of O

1

= O2

.

THEOREM 5. For every pair of operational expressions E1, E2, if `W

E1

= E2

, then D(E1) = D(E2).Proof. By induction on the length of the proof of E

1

= E2

.


(i) E1

= E2

is Axiom (I), i.e. is the equation I = . Then:

D(I) = D(I) + D() = 0 + 0 = 0 = D().

(ii) E1

= E2

is an instance of the Axiom-Schema (II), i.e. it is an equation(O1O2) = O1O2, for two operational terms O1, O2. Then:

D((O1O2)) = D((O1O2)) + D() = D((O1O2)) + 0 =D((O1O2)) = D(O1) + D(O2); but D(O1O2) = D(O1) +D(O2) = D(O1) + (D(O2) + D()) = D(O1) + (D(O2) +0) = D(O1) + D(O2).

(iii) E1

= E2

is an instance of Definition (IV).

E1

= E2

is 0x = x.(a)

Then: D(0x) = D() 0 = 0 = D(x).

E1

= E2

is snx = nx for some n 0.(b)

Then:

D(snx) = D() (n + 1) = 1 (n + 1) = n + 1;

but:

D(nx) = D() + D(nx) = 1 + D(nx) = 1 + (D() n) = 1 + n.

E1

= E2

is (r+s)x = (rs)x for two arithmetical terms tand s.

(c)

Then:

D((r+s)x) = D() (r + s) = 1 (r + s) = r + s;but:

D((rs)x) = D((rs)) + D(x) = D((rs)) + 0 =D((rs)) = D(r) + D(s) = (D() r) + (D() s)= r + s.

E1

= E2

is (rs)x = rsx for two arithmetical terms t ands.

(d)


Then:

D((rs)x) = D() (r s) = 1 (r s) = r s;

but:

D(rsx) = D(r) s = (D() r) s = (1 r) s = r s.

(iv) E1

= E2

is derived from O1

= O2

by R1.Then E

1

is the expression O1

and E2

is the expression O2

. ByLemma 6 it follows that

D(O1

) = D(O2

);

hence

D(E1

) = D(O1

) = D(O1

) + D() = D(O2

) + D() = D(O2

)= D(E

2

).

(v) E1

= E2

is derived from E3

= E4

by R2.Then E

1

is the expression OE3

and E2

is the expression OE4

; byinductive hypothesis we have:

D(E3

) = D(E4

);

then:

D(E1

) = D(OE3

) = D(O) + D(E3

) = D(O) + D(E4

) = D(E2

).

(vi) E1

= E2

is derived from E3

= E4

by R3.Then E

1

is the expression subst /O[E3

] and E2

is the expressionsubst /O[E

4

]; then we have:

D(E1

) = D(subst /O[E3

]) = D(O) D(E3

) [Lemma 4]= D(O) D(E

4

) [inductive hypothesis]= D(subst /O[E

4

]) [Lemma 4]= D(E

2

).

(vii) E1

= E2

is derived from E3

= E4

by R4.


Then E1

is the expression subst /E[E3

] and E2

is the expression subst/E[E

4

]; hence we have:D(E

1

) = D(subst /E[E3

]) = D(E) + D(E3

) [Lemma 5]= D(E) + D(E

4

)[inductive hypothesis]= D(subst /E[E

4

]) [Lemma 5]= D(E

2

).

(viii) E1

= E2

is introduced by R5.Then E

1

is the same expression as E2

and thus D(E1

) = D(E2

).

(ix) E1

= E2

is derived from E2

= E1

by R6.By inductive hypothesis D(E

2

) = D(E1

) and thus the result follows.

(x) E1

= E2

is derived from E1

= E3

and E3

= E2

by R7.By inductive hypothesis we have: D(E

1

) = D(E3

) and D(E3

) = D(E2

);hence the result follows.

Notice that from Theorem 5 the consistency of the Theory W follows(as usual, W is said to be consistent if, and only if, not every equation isderivable from its axioms and rules): for instance, since D(x) = 2 andD(x) = 3, the equation x = x is not derivable in W.

THEOREM 6. (Interpretation Theorem) For every pair of arithmeticalterms t and r,

`

W

t

x =

r

x if and only if `PE

t = r.

Proof.(1))Let us suppose that `

W

t

x =

r

x for any two arithmetical termst and r. By Theorem 4 we have that `

W

t

x =

t

x and `W

r

x =

r

x; hence: `W

t

x =r

x. By Theorem 5 it follows that

D(tx) = D(rx) [*].By Theorem 3 there exists one numeral n such that `

W

t

=n

and`

PE

t = n, and one numeral m such that `W

r

=m

and `PE

r =m. Then by R4:

`

W

t

x =n

x and `W

r

x =m

x.

By Theorem 5 and the definition of the function D: D(tx) = D(nx)= n and D(rx) = D(mx) = m; from [*] it follows n = m. Hence`

PE

n = m and from this and `PE

t = n, `PE

r = m we get: `PE

t = r.


(2)(

Let us suppose that `PE

t = r. By Theorem 3 there exists one numeral nsuch that `

W

t

=n

and `PE

t = n, and one numeral m such that`

W

r

=m

and `PE

r = m. Hence `PE

n = m and then n = m.This means that n and m are one and the same numeral and thus, by (R5),we have: `

W

n

=m

. From this by (R6) and (R7) we get:`

W

t

=r

.

By (R4) it follows: `W

t

x =r

x and then `W

t

x =

r

x (byTheorem 4).

5. CONCLUDING REMARKS

As pointed out at the end of Section 3 above, there is no textual evidence,in the Tractatus, which can be appealed to to justify the introduction ofthe identical operation sign I (in the definiens of the definition of theoperational term 0). On the contrary, proposition 5.23 of the Tractatusseems to affirm that an operation always turns a linguistic expression intoa different one. However, there is no reasonable alternative to my choice,if the attempt to systematically reconstruct Wittgensteins theory is made.

A related remark concerns the form of the Interpretation Theorem: it isdetermined by the intention of supplying a theory which is as far as possiblefaithful to Wittgensteins original positions, first of all to his idea that it isDefinition IV (1a) which is given the task of fixing the meaning of 0. Ifthis claim were abandoned and the identical operation I were allowed toplay that role, then a slightly simpler theory than W could be constructed,since only Definitions III would be needed to prove the metatheorem:`

W

t

=r

if and only if `PE

t = r, which would replace the aboveformulation of the Interpretation Theorem.

Within my tentative reconstruction, it is only the introduction of theoperational expressions t and the corresponding derivation rule (R4),which provides the theory W with enough deductive strength (as alreadystressed, the variable x could not be substituted freely because it is intend-ed to show the form of a linguistic expression which is an initial expressionof a series generated by successive application of a logical operation, i.e. anexpression which is not yielded by applying the operation). This explainsthe reason why the proof of the Interpretation Theorem goes through theproofs of theorems such as Theorem 1, 2 and 3, which concern operationalexpressions where an arithmetical term occurs as exponent on the left ofthe variable , and the variable occurs as end symbol.


These considerations can be readily generalized. As one can see fromDefinitions III and IV, numerals are introduced in the theory W with adouble role: as exponents on the left of in expressions of the form n,they represent the property constituted by the number of times an operationis composed with itself; as exponents on the right of in expressionsof the form nx, they represent the property constituted by the numberof times an operation is applied in a procedure of successive application,starting from a base which is not generated by applying the operation. Thedouble role of numerals corresponds to the double role played by the singleinverted comma , which is used both as sign for the composition oftwo operations in operational terms of the form (O

1

O2

) and as signfor the application of an operation to a given base.

As explained in Section 2, it is the occurrence of numerals in opera-tional expressions of the form nx which, in the Tractatus, representsnatural numbers. Nonetheless, the occurrence of numerals as exponentson the left of and the related use of the sign in contexts such as(O

1

O2

) appears to be an indispensable condition for the realization ofWittgensteins project to elaborate by resorting to the abstract notion ofoperation an alternative to the logicist reduction of natural numbers to thenotion of class. This consideration again suggests that simplification of thetheory which would consist in introducing numerals only via DefinitionsIII, i.e. by exclusive reference to the notion ofn-th iteration of an operation(which, as known, is the path followed in -calculus).10

NOTES

1 The fact that, in Principia Mathematica, class expressions are introduced by contextualdefinitions and are analysed away in favour of propositional functions does not modifythe situation in essence: indeed, as we shall see, Wittgenstein equally rejects the ideaof a foundation of mathematics on the notion of propositional function (in his originalterminology of eigentlich Begriff, as opposed to formal concept).2 See Anscombe (1959), p. 124 and Frascolla (1994), Chapter 1, pp. 14.3 The relevant notes added by Wittgenstein in the margins of Ramseys copy of the Tractatusin 1923 (The beginning of logic presupposes calculation and so number. Number is thefundamental idea of calculus and must be introduced as such. The fundamental idea ofmath[ematics] is the idea of calculus represented here by the idea of operation (see Levy(1967), Wittgensteins italics) easily square with the above interpretation on condition thatthe notion of calculus be construed in the way Wittgenstein himself takes care to explain(as equivalent to the general notion of operation).4 For a comparison of Wittgensteins accusation and the analogous objection raised, froma constructivistic point of view, against impredicative definitions, see Frascolla (1994),Chapter 1, pp. 3437.5 See Frascolla (1994), Chapter 1, pp. 1320.6 On the central theme of the ineffability of the formal domain and, in general, of semantics,


in the Tractatus, see Hintikka, M. B. Hintikka, J. (1986), Chapter 1.7 What Wittgenstein says in TLP 6.211 about the role of mathematics in life (by meansof its theorems, meaningful propositions can be inferred from meaningful propositions) iscoherent with the view I have attributed to him: if two expressionsA and B have the formshown, respectively, by tx and rx, and if the equation tx = rx is correct,then A and B can be substituted for each other salva veritate in any given meaningfulproposition. Unfortunately, there is no further textual evidence which could be referred tofor a more detailed explanation of Wittgensteins view.8 See Russell (1922), p. XX.9 Ramsey (1931), p. 17.10 Comments of an anonymous referee of Synthese on the draft of the paper led to substantialimprovements.

REFERENCES

Anscombe, G. E. M.: 1959, An Introduction to Wittgensteins Tractatus, Hutchinson Uni-versity Library, London.

Frascolla, P.: 1994, Wittgensteins Philosophy of Mathematics, Routledge, LondonNewYork.

Hintikka, M. B. and Hintikka, J.: 1986, Investigating Wittgenstein, Blackwell, OxfordNewYork.

Lewy, C.: 1967, A Note on the Text of the Tractatus, Mind XXVI, 41623.Ramsey, F. P.: 1931, The Foundations of Mathematics and Other Logical Essays, Routledge

& Kegan Paul, LondonRussell, B.: 1922, Introduction, in L. Wittgenstein (1922).Wittgenstein, L.: 1922, Tractatus Logico-Philosophicus, Kegan Paul-Trench-Trubner, Lon-

don (quotations are from the Routledge & Kegan Paul edition, London 1969, translationfrom the German text by D. F. Pears and B. F. McGuiness).

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