GRICE E PEANO: L'IMPLICATURA CONVERSAZIONALE -- FILOSOFIA ITALIANA -- LUIGI SPERANZA

 

Grice e Peano: l’implicatura conversazionale – filosofia italiana – Luigi Speranza (Spinetta di Cuneo). Filosofo italiano. Grice: “As I reduce “the” to “every,” I am of course following Peano, who predates Russell!” -- important Italian philosopher. Linceo. P.’s postulates, also called P, axioms, a list of assumptions from which the integers can be defined from some initial integer, equality, and successorship, and usually seen as defining progressions. The P. postulates for arithmetic are produced by P. He takes the set N of integers with a first term 1 and an equality relation between them, and assumed these nine axioms: 1 belongs to N; N has more than one member; equality is reflexive, symmetric, and associative, and closed over N; the successor of any integer in N also belongs to N, and is unique; and a principle of mathematical induction applying across the members of N, in that if 1 belongs to some subset M of N and so does the successor of any of its members, then in fact M % N. In some ways P.’s formulation was not clear. He had no explicit rules of inference, nor any guarantee of the legitimacy of inductive definitions which Dedekind established shortly before him. Further, the four properties attached to equality were seen to belong to the underlying “logic” rather than to arithmetic itself; they are now detached. It was realized by P. himself that the postulates specified progressions rather than integers e.g., 1, ½, ¼, 1 /8,..., would satisfy them, with suitable interpretations of the properties. But his work was significant in the axiomatization of arithmetic; still deeper foundations would lead with Russell and others to a major role for general set theory in the foundations of mathematics. In addition, with Veblen, Skolem, and others, this insight led in the early twentieth century to “non-standard” models of the postulates being developed in set theory and mathematical analysis; one could go beyond the ‘...’ in the sequence above and admit “further” objects, to produce valuable alternative models of the postulates. These procedures were of great significance also to model theory, in highlighting the property of the non-categoricity of an axiom system. A notable case was the “non-standard analysis” of Robinson, where infinitesimals were defined as arithmetical inverses of transfinite numbers without incurring the usual perils of rigor associated with them. Fu l'ideatore del latino sine flexione, una lingua ausiliaria internazionale derivata dalla semplificazione del latino classico. Nacque in una modesta fattoria chiamata "Tetto Galant" presso la frazione di Spinetta di Cuneo. Fu il secondogenito di Bartolomeo P. e Rosa Cavallo; sette anni prima era nato il fratello maggiore Michele e successivamente nacquero Francesco, Bartolomeo e la sorella Rosa. Dopo un inizio estremamente difficile (doveva ogni mattina fare svariati chilometri prima di raggiungere la scuola), la famiglia si trasferì a Cuneo. Il fratello della madre, Giuseppe Michele Cavallo, accortosi delle sue notevoli capacità intellettive, lo invitò a raggiungerlo a Torino, dove continuò i suoi studi presso il Liceo classico Cavour. Assistente di Angelo Genocchi all'Torino, divenne professore di calcolo infinitesimale presso lo stesso ateneo a partire dal 1890.  Vittima della sua stessa eccentricità, che lo portava ad insegnare logica in un corso di calcolo infinitesimale, fu più volte allontanato dall'insegnamento a dispetto della sua fama internazionale, perché "più di una volta, perduto dietro ai suoi calcoli, [..] dimenticò di presentarsi alle sessioni di esame".  Ricordi del grande matematico (e non solo della vita familiare) sono raccontati con grazia e ammirazione nel romanzo biografico Una giovinezza inventata della pronipote Lalla Romano, scrittrice e poetessa. Aderì alla massoneria, iniziato nella loggia Alighieri di Torino guidata dal socialista  Lerda.  Morì nella sua casa di campagna a Cavoretto, presso Torino, per un attacco di cuore che lo colse nella notte.  Il matematico piemontese fu capostipite di una scuola di matematici italiani, tra i quali possiamo annoverare Vailati, Castellano, Burali-Forti, Padoa, Vacca, Pieri e Boggio. Peano precisa la definizione del limite superiore e fornì il primo esempio di una curva che riempie una superficie -- la cosiddetta "curva di Peano", uno dei primi esempi di frattale -- mettendo così in evidenza come la definizione di curva allora vigente non fosse conforme a quanto intuitivamente si intende per curva.  Da questo lavoro partì la revisione del concetto di curva, che fu ridefinito da Jordan (curva secondo Jordan).  Fu anche uno dei padri del calcolo vettoriale insieme a Levi-Civita. Dimostra importanti proprietà delle equazioni differenziali ordinarie e idea un metodo di integrazione per successive approssimazioni.  Sviluppa il Formulario mathematico, scritto dapprima in francese e nelle ultime versioni in interlingua, come chiama il suo latino sine flexione, contenente oltre 4000 tra teoremi e formule, per la maggior parte dimostrate. Da un eccezionale contributo alla logica delle classi, elaborando un simbolismo di grande chiarezza e semplicità. Da una definizione assiomatica dei numeri naturali, i famosi assiomi di P. che vennero poi ripresi da Russell e Whitehead nei loro Principia Mathematica per sviluppare la teoria dei tipi.  I contributi di Peano sulla logica furono osservati con molta attenzione da Russell, mentre i contributi di aritmetica e di teoria dei numeri furono osservati con molta attenzione da Vailati, il quale sintetizzava in Italia il passaggio tra l'esame delle questioni fondamentali e l'applicazione di metodiche di analisi del linguaggio scientifico, tipica degli studi logici e matematici, e anche specifica gli interessi di storia della scienza, allargando la prospettiva anche agli studi sociali. Per questo P. ha dei contatti molto stretti con il mondo degli studiosi di logica e di filosofia del linguaggio nonché gli studiosi di scienze sociali empiriche (Cfr. Rinzivillo, P., Vailati. Contributi invisibili in Rinzivillo, Una Epistemologia senza storia” (Roma Nuova Cultura). Ha ampi riconoscimenti negli ambienti filosofici più aperti alle esigenze e alle implicazioni critiche della nuova logica formale. E affascinato dall'ideale leibniziano della lingua universale e sviluppa il "latino sine flexione", lingua con la quale cercò di tenere i suoi interventi ai congressi internazionali di Londra e Toronto. Tale lingua e concepita per semplificazione della grammatica ed eliminazione delle forme irregolari, applicandola a un numero di vocaboli "minimo comune denominatore" tra quelli principalmente di origine latina rimasti in uso nell’italiano. Uno dei grandi meriti della sua opera sta nella ricerca della chiarezza e della semplicità. Contributo fondamentale che gli si riconosce è la definizione di notazioni matematiche entrate nell'uso corrente, come, per esempio, il simbolo di appartenenza (“x ∈ A”) e il quantificatore esistenziale "∃".  Tutta l'opera di P. verte sulla ricerca della semplificazione, dello sviluppo di una notazione sintetica, base del progetto del Formulario, fino alla definizione del latino sine flexione. La ricerca del rigore e della semplicità lo portano P. ad acquistare una macchina per la stampa, allo scopo di comporre e verificare di persona i tipi per la “Rivista di Matematica” da lui diretta e per le altre pubblicazioni. Raccolge una serie di note per le tipografie relative alla stampa di testi di matematica, uno per tutti il suo consiglio di stampare le formule su righe isolate, cosa che ora viene data per scontata, ma che non lo era ai suoi tempi. Cavaliere dell'Ordine della Corona d'Italia Ufficiale della Corona Commendatore della corona L'asteroide  P. è stato battezzato così in suo onore.  Il dipartimento di Matematica di Torino è a lui dedicato. Molti licei in Italia portano il suo nome, come ad esempio a Roma, Cuneo, Tortona, Monterotondo, Cinisello Balsamo o Marsico Nuovo, così come la scuola di Tetto Canale, vicina alla sua città natale. Saggi: “Aritmetica”; “Algebra” (Torino, Paravia,); “Forma matematica” (Torino, Bocca); “Calcolo differenziale”; “Calcolo integrale” (Torino: Bocca); “Analisi infinitesimale” (Candeletti); “Calcolo infinitesimale e geometria” (Torino: Bocca), “Logica della geometria” (Torino: Bocca)”; “Principio dell’arimmetica” (Torino, Paravia); “Giochi di aritmetica e problemi interessanti” (Paravia, Torino). Provai una grande ammirazione per lui quando lo incontrai per la prima volta al Congresso di Filosofia, che e dominato dall'esattezza della sua mente. Russell. Amico, Storie della scuola italiana. Dalle origini (Zanichelli, Bologna); Celebrazione, Luciano e Roero Torino); “Storia di un matematico” (Boringhieri).  L. Romano, “Una giovinezza inventata” (Torino, Einaudi); Racconta episodi del rapporto con il prozio Giuseppe.  Assiomi di P., Glottoteta, Lingua artificiale, Matematica, Latino sine flexion, Cassina Calcolatori ternari M. Gramegna  Treccani Dizionario biografico degli italiani, Istituto dell'Enciclopedia Italiana. E P. stregò Russell. The third kind of term, things, are only the entities indicated by proper names, but they have no additional relation with other terms. This leads Russell to consider the sole denoting concept which presupposes uniqueness -- "the.” Russell admits the great importance of this term, recognizes the merit of P.'s notation, and attributes to P. the capacity to make possible genuine mathematical definitions defining terms which are not concepts, the meaning of a word with its indication-reference and the meaning of a denoting concept with its denotation. P. does something more than provide the standard notation. The pre-eminence of a description over other forms of denotation is definitive. The notation for a description is inspired in the Peanesque symbolism (i.e. "laeb"). Membership to a class is replaced by a propositional function (i.e. (l£)(<I>X)). A propositional function is explained as a certain denoting function of <l>x, which, if <1>£ is true for one and only one value ofx, denotes that value, but in any other case denotes (P).p. Perhaps most interestingly for us is the insistence on the indefinability of "the" – P.'s inverted iota is already used -- together with the notion of denotation. The article, as published, adds the expression of the main definition in terms of propositional functions together with the previous manuscript definition in P.'s terms of existence and uniqueness, albeit if not in symbolic form. The two essential definitions are Principia, * 14.01.02: . \jI(IX)(epX) • =.(3b) : epx •=~ .x=b : \jib E ! ( 7 X ) ( e p X ) • = . ( 3 b ) : 4 > x . =• . x = b. This expresses the conditions of existence and uniqueness essentially with P.’s resources, i.e., in terms of quantification and identity, although adding propositional functions. P. has different vresources to eliminate the definite article – his inverted iota -- from a proposition. P. actually recommends this line in cases where the required conditions of existence and uniqueness are doubtful, precisely through a sort of definition in use. The descriptor is by no means indefinable in his system.  Russell:  "I read Schrader on Relations and found his methods hopeless, but P. gave just what I wanted (Letter to Jourdain, in Grattan-Guinness). If, as Russell maintains in Principia, following P., that a definition is to be always nominal, the definienda is only an abbreviation. Russell formulates his principle to preserve the admissible part of Bradley’s analysis -- (his methodological and analytical resourses -- and almost the entire Moore, in so far as they were compatible with the requirements of Peano's logic. Some of the mostti mportant ideas and symbolic devices that made Russell's theory of descriptions possible are already present in essays Peano that Russell knows well. We may proceed by a detailed comparison between the relevant parts of Russells theory -- including manuscripts now published-and some of Frege and , . . ht as well as a discussion of numerous possible obJectlons that P.’s mSig s, . . fl db could be posed to the main claim. Even if Russell was not actually influenced by those insights, the parallelisma are close enough to be worth analyzig, especially in the case of P., whose writings are not very well known.  (r) can be clearly found in Frege and Peano, that (2) was almost admitted by Frege and was admitted explicitly-including the symbolic expression by P.. THE SYMBOLIC ELIMINATION OF "THE" IN P.. The source in P. of the symbols relevant to Russell's theory of descriptions have been noted and sometimes explained (see, .for instance, 1988a and 199Ia, Chap. 3). I will confine myself to recalling that they were the letter iota (i) for the unit class, and the same letter inverted (1), or denied ("fa), for the only member of thiS class, i. e., the definite article of ordinary language. P.'s ideas evolved in three stages towards greater precision in the treatment of a description. This last P. starts from the definition in terms of the unit class. He then adds a series of possible definitions (the ones allowing an alternative logic al order), one of which offers this equivalence. P. introduces his fundamental d~fimt~on ~f the u:l1t class as the class such that all of its members are identical. In P.’s symbols, tx =ye (y =x). Likewise, P. defines indirectly the.unique member of such a class: x = "fa • = • a = tx. However, concerning the definability of the definite article, P. adds the crucial idea that any proposition containing “the” can be reduced to. the for,? ta eb, and thiS, again, to the inclusion of the referr~d .um~ class in the oth~r class (a ~ b), which already supposes the elimination of the symbol t: Thus, P. says, we can avoid an identity whose first member contams thiS symbol. Here we find the assertion that the only individual belonging to a unit II As an anonymous referee pointed out to me, one ~aj~rdifferenc~between P. and Russell's treatment of classes in the context of descnption theolJ' is that, while, for Peano, a description combines a class abstract with the inverse of the umt class operator, for Russell the free use of class abstracts is not available due to the discovery of paradoxes. P. does not explicitly state that the mentioned expression would be meaningless, but rather "nous ne donnons pas de signification a ce symbole si la classe a est nulle, ou si elle contient plusieurs individus.” But this is equivalent in practice, given that if we do not meet the two mentioned conditions, the symbol cannot be used at all. There are, however, other ways of eliminating the same symbols according to P.. One which is very similar and depends on the same hypothesis: laE b. = : a = tx. :Jx • Xc b(ibid).   class (a) such that it belongs to another class (b) is equal to the existence of exactly one element such that this element is a member of that class (b). In other words: "the only member of a belongs to b" is to be the same as "there is at least one x such that (i) the unit class a is equal to the class constituted by x, and (ii) x belongs to b" (or "the class of x such that a is the class constituted by x, and that x belongs to b, is not an empty class"). This seems to be equivalent to Russell's definition. P., of course, speaks in terms of classes instead of a propositional function, i. e., in terms of the property or the predicate, which define a class – P. often read the membership symbol as "is" -- which expresses the same idea in a way where any reference to the letter iota has been eliminated. We can read now "the only member of a belongs to b" as the same as "there is at least one x such that (i) the unit class a is equal to all the y such that y =x, and (ii) x belongs to b" (or "the class of x such that they constitute the class ofy, and that they constitute the class a, and that in addition they belong to the class b, is not an empty class"). Thus, the full elimination underlies the definition, although P., in lacking a specifically explicit philosophical goals, shows no interest in making this point. Peano is totally aware of the importance of this device as a way to reduce the definite article to more primitive logical concepts, i.e. to eliminate it, as a result of which the symbol would cease to be primitive. That is why P. adds that the above definitions "expriment la P[proposition] 1a Eb sous une autre forme, OU ne figure plus Ie signe 1; puisque toute P contenant le signe 1a est reductible a la forme 1aEb,OU bestuneCIs, on pourra eliminer Ie signe 1 dans toute P.” Therefore, the general belief according to which the symbol "1" was necessarily primitive and indefinable for P. is wrong. By pointing out that in the "hypothesis" preceding the quoted definition it is clearly stated that the class "a" is defined as the unit class in terms of the existence and identity of all of their members (i.e. uniqueness): Before making more explicit the parallelism with Russell's theory, let us consider possible objections against this rather strong claim. All of these objections are either misconceptions or simply have no force with regard to P.’s main claim. This is why"a"is equal to the expression ''tx'' (in the second member). The objection could still be maintained by insisting that since"a" can be read as "the unit class", P. did not really achieve the elimination of the idea he was trying to define and eliminate, as it is shown through the occurrence of these words in some of the readings proposed above. However, as I will explain below, the hypothesis preceding the definition only states the meaning of the symbols which are used in the second member. Thus, "a" is stated as "an existing unit class", which has to be (1) It is true that the symbol "1" has disappeared, but in the definiens we still can see the symbol of the unit class, which would refer somehow to the idea that is symbolized by ''tx'', so the descriptor has not been really eliminated. The answer is very simple: for P. there were at least two forms ofdefining this symbol with no need for using the letter iota (in any of its forms). However, the actual substitution would lead us to rather complicated expressions,14 and given P.'s usual way of working (which can be First, by directly replacing tx by its value: y 3(y = x), as defined above. Making the replacement explicit, we have: 14 Starting from this idea, we can interpret the definition as stating that "la Eb" is only an abbreviation for the definiens and dispensing with the conditions stating exist- ence and uniqueness in the hypothesis, which have been incorporated to their new place. Thus, the new hypothesis would contain only the statement of"a" and"b" as being classes, and the final entire definition could be something like the following: la Eb • =:3x 3{a =y 3(y =x) • X Eb}, a, bECls.::J :. ME b. =:3XE([{3aE[w, zEa. ::Jw•z' w= z]} ={ye(y= x)}] •XEb), a E Cis. 3a: x, yEa. ~x.y.X = y: bE CIs •~ : ... (Ibid.) understood in this way: " 'a' stands for a non-empty class su~h that all of its members are identical." Therefore, we can replace "a", wherever it occurs, by its meaning, given that this interpretation works as only a purely nominal definition, i.e. a convenient abbreviation.  characterized as the constant search for shorter and more convenient formulas), it is quite understandable that he preferred to avoid it. In fact, the operation is by no means necessary, for the symbolic expression above was already enough to obtain the full elimination of the descriptor. We must not forget that the important thing is not the intu- itive and superficial similarity between the symbols "la" and ''tx'', caused simply by the appearance of the letter iota in both cases, or the intuitive meaning of the words "the unit class", but the conditions under which these expressions have been introduced in the system, which were completely clear and explicit in the first definition.IS "k e K" as "k is a class"; see also the hypothesis from above for another example). But this by no means involves confusion with i~clusion,as. it is shown by the fact that P. soon added four defimte properties precisely distinguishing both notions, which made it po~siblefor.hi~~.~ for Russell himself, to preserve the useful and convenient readmg is (2) The supposed elimination is a failure, for (i) it depends upon Peano's confusion of class membership and class inclusion, so that (ii) a singleton class (la) and its sole member (lX) are not clearly distinct notions; it follows that (iii) "a" is both a class and, according to the interpretation of the definition, an individual (iv), as is shown by joining the hypothesis preceding the definition and the definition itself This multiple objection is very interesting because it can be taken as proceed- ing from the received view on P., according to which his logic not only falls s~ort ofstrict logical standards, but also contains some import- ant confuSions here and there. However, the four points can easily be s~own t? be mistaken. (Incidentally, I think this could have been recog- mzed With pleasure by Russell himself, who always thought of P. and his school as being strangely free oflogical confusions and mistakes.) . Fir~t, it ~n hard~y be said that P. confused membership and mcluslOn, given that it was he himselfwho introduced the distinction through his symbol "e" (previously to, and therefore independently of, Frege). If the objection means (which is rather unlikely) that P. would admit the symbol for membership as taking place between two classes, it is true that this was the case when he used it to indicate the meaning of some symbols, but only through the reading "is" (e.g. full clarity that"1" (T) makes sense only before individuals, and ''t'' before classes, no matter which particular symbols we use for these notions. Thus, ''ta'', like "tx", both have to. be read as "the class consti- tuted by ...", and" la" as "the only member of a". Therefore, although P., to my knowledge, never used "lX" (probably because he always which could be read as " 'a and b being classes, "the only member of a belongs to b" is to be the same as "there is at least one x such that (i) 'there is at least one a such that for eve~,': and z belonging to a,.w = z' is equal to 't~ey such that y =. x' , and (ii) x belongs to b ,where both the letter Iota and the words the unit class" have disappeared from the definiens. aeCis.3a:x,yea.-::Jx,y. x=y:beCIs•~:. . l a e b . = : 3 x 3(a = t x . x e b), 15 There is a well-known similar example in the apparent vicious circle of Frege's famous definition ofnumber. the reply to objection (1). There are other, minor objections as well. Second, "la" does notstand for the singleton class. P. stated with thought in terms of classes), had he done so its meaning, of course, would have been exactly the same as "la", with no confusion at all. Third, "a" stands for a class because it is so stated in the hypothesis, although it can represent an individual when preceded by the descriptor, and together with it, i.e. when both constitute a new symb.ol as a w.hol~. Here P.'s habit could perhaps be better understood by mterpretmg it in terms of propositional functions, and then by seeing" la" as being somewhat similar to <!>x, no matter what reasons ofconvenience led him to prefer symbols generally used for classes ("a" instead of"x"). There is little doubt that this makes a difference with Russell. It could even be said that while, for Peano, the inverted iota is the symbol for an operator on classes, which leads us to a new term when it flanks a term, for Russell it was only a part of an "incomplete symbol". I am not sure about P.'s answer to this, but at any rate for him the descriptor could be eliminated only in conjunction with the rest of the full express- ion "la e b", so that the most relevant point of similarity again can be found in P.. Last, there is no problem when we join the original hypothesis and the definition: as I have pointed out in the interpretation contained in the last part of (3) If, as it seems, "a" is affected by the quantifier in the hypothesis, then it is a variable which occurs both free and bound in the formula (if it is a constant, no quantifier is needed). I am not sure about the possible reply by P. himself Perhaps he did not always distinguish with present standards o f clarity between the several senses o f "existence" (or related differences) involved in his various uses of quantifiers,r6 but in principle there is no p'roblem when a variable appears both bound and free in the same expression, although in different occurrences. At any rate, I cannot see how this could affect my main claim; the important thing here is to recognize the fundamental similarities between the elim- ination of the descriptor in P. and Russell. However, in the several readings I proposed I hope to have clarified a little the role of ".3" in P. . (5) P. could hardly have thought that he was capable of eliminat- ing the descriptor, for he continued to use the symbol and his whole system depended on it as a primitive idea.IS The only additional reply is that only reasons ofconvenience can explain the retaining ofa symbol in a system in cases where the symbol can be defined, i.e. eliminated. (After all, Russell- himself continued to use the descriptor after its elimination by means of his theory of descriptions.) But, as we have seen, there is no doubt P. thought that the descriptor could easily be eliminated from propositions. (4) Russell rejected definitions under hypothesis, therefore he would have rejected the Peanian definition of the descriptor. Of course, we must admit that Russell (like Frege) rejected this kind ofdefinition, but this took place especially in the context of the unrestricted variable of Principia.I ? Besides, he himself used this kind of definition for a long period once he mastered P.'s system. It was because he interpreted these definitions as P. did, i.e. merely as -a device for fixing the meaning of the letters used in the relevant symbolic expressions. Thus, when for instance one reads, after whatever symbolic definition, things like" 'x' being ..." or" 'y' being ...", this would really be a definition under hypothesis, but, of course, only because the meaning of the sym- bols used always has to be determined somehow. Anyway, there is no point in continuing the discussion ofthis objection, given that it is hard- ly relevant to my main claim. Even if P.'s original elimination of the descriptor does not work because of its taking place in the framework of a merely conditional definition, the force of his original insight could well have influenced Russell; at any rate, it is worth knowing in itself (6) The reduction mentioned, even if it really took place, was by no means followed by the philosophical framework which made Russell's theory of descriptions one of the most important logical successes of the century. Thus, P. did not realize the importance of the elimination. This last point can hardly be denied, but P.'s goals were very different from Russell's, so I think that to point out a "lack" like this makeslittle sense from a historical point ofview. 16 I would like to recall here that it was P. himselfwho discovered the distinction between bound and free variables (which he respectively called "apparent" and "real"), and probably-and independently of Frege-also the existential and universal quantification (see my I988a and I99Ia for a detailed account of both achievements). Quine wrote that "1" was a primitive and indefin- able idea in P. However, now that we have exchanged several letters concerning an earlier version ofthis article, I must say he has changed his mind. His letter to me ofII October 1990 contains the following passage: "I am happy to get straight on P. on descriptions. I checked your reference and I fully agree. P. deserves all the credit for it that has been heaped on Russell (except perhaps for Russell's elaboration of the philosophical lesson of contextual definition)". As for the sense in which the philosophical consequences of the elimination of the descriptor were not very important for P., I have faced the problem in my reply to an objection. And also in previous stages, through the (finally unsuccessful) attempt at a substitutional theory based upon propositions, with no classes and no propositional functions. . For according to him the descriptor cannot be defined in isolation, but only in the context of the class (a) from which it is the only member (la), and also in the context of the clas~ from which that class is a member, at least to the extent that the class a is included in the class b, although this supposes no confusion between membership and inclusion; see the second point of my reply to objection (2) above. I think this is just the right interpretation ofthe whole expression"1a Eb". In any case, I cannot help being convinced that none of these objec- tions seems to have any force against my main claim: that the elimin- ation of the descriptor was present in P. with essentially the same symbolic resources as in Russell. This is equivalent to the first two claims at the beginning of this paper: P. clearly stated the conditions of existence and uniqueness as providing the true significance of the descriptor; and (2) he had enough symbolic techniques for dispensing with it, including those required for constructinga definition in use. We have a few relevant passages, but the clearest one occurs. There we can read that" Ta" is meaningless if the conditions of existence and uniqueness are not ful- filled. Thus, even the third claim was made by P.. Perhaps under certain different interpretations of P.'s devices it could be shown that his elimination of the descriptor was not exactly equivalent (in the tech- nical sense) to Russell's. Yet even if so, I think that from the historical viewpoint, which means to do justice both to P. and Russell, it is important to know that P. had these resources at his disposal,' and that they may have influenced Russell. However, we can see the heritage from P. in a clearer way if we compare the definition with the version for classes in the same letter: . The parallelism is therefore complete, but before finishing this paper I want to insist on my main claims by resorting now to one of Russell's manuscripts, "On Fundamentals. First, we find there a definition stated in terms similar to P.'s, and with almost exactly the same symbolic resources: Finally, I am not accusing Russell of plagiarism. I only affirm that some ofthe ideas and devices which are important for the eliminative definition of the descriptor were already present in Frege and P., including the conceptual and symbolic resources, and that these works are ones that Russell had studied in detail before his own theory was formulated. Second, the later improvement of this definition is precisely in the sense of making clearer that, although the method of the propositional function was preferable to the one of class membership, the symbolic expression of the conditions of existence and uniqueness is preserved. Even the idea -- also coming from P. -- according to which we cannot define the expression “la" alone, but always in the context of a class (which in Russell became the form of a propositional function), appears here. Benacerraf, and Putnam, Philosophy of Mathematics  (Cambridge). The first appearance of Russell's definition, under the form which was adopted as final, took place, not in "On Denoting", but in a letter to Jourdain: According to that, all other influences must be regarded as secondary. Concerning Meinong's influence, for Russell the principle of subsistence disappears as a consequence of the eliminative construction of the definite article, which was a result of the new semantic monism. Russell's later attitude to Meinong as a "main enemy" was only a comfortable recourse (v. however, Griffin). As for Bacher, Russell himself admitted some influence from his nominalism.  In fact, Bacher describes mathematical objects as "mere symbols" and he advises Russell to follow this line of work in a letter (only two months before Russell's key idea): "the 'class as one' is merely a symbol or name which we choose at pleasure" (quoted by Lackey [Russell). Finally, for MacColl it is necessary to mention his essay where he spoke of "symbolic universes", which include things like round squares, and also spoke of "symbolic existence". Russell pub- lished his essay as a direct response to this author, and there we can see some conclusions from the unpublished manuscripts, although still by solving peculiar cases in a Fregean context. I agree with Grattan-Guinness that MacColl was an important part of the context of Russell's ideas on denoting (personal communication), but I have no room here to devote to the matter. There is, however, a previous occurrence of this definition in the,manuscript "On 'JI(lX)(<I>x)•=•(:3b):<j>x.=x.X =b:'JIb. (Grattan-Guinness  Substitution"  written with only slight symbolic differences. I am indebted to Landini for the historical point. 'JI(t'u)•=:(:3b):xEU.=x.X =b:'JIb. Peano, G., as. Opere Scelte, ed. U. Cassina, Roma: Cremonese, Studii di logica matematica". Repr. Logique mathematique". Repr. Analisi della teoria dei vettori". Repr. Formules de logique mathematique". Giuseppe Peano. Peano. Keywords: implicatura, l’operatore iota. Refs.: Luigi Speranza, “Peano e Grice sull’articolo definito,” -- Luigi Speranza, “Peano e Grice sull’operatore ‘iota’, Deutero-Esperanto, l’errore di Quine, il carattere non primitive dell’operatore iota. --  H. P. Grice, “Definite descriptions in Peano and in the vernacular,” Luigi Speranza, "Grice e Peano: semantica filosofica," per il Club Anglo-Italiano, The Swimming-Pool Library, Villa Grice, Liguria, Italia.