The Logical Research by Giuseppe Peano through the ... · Metalogicon (2006) XIX, 2 75 The Logical Research by Giuseppe Peano through the Analysis of his Works (1888-1894) Giuseppe

Metalogicon (2006) XIX, 2

75

The Logical Research by Giuseppe Peano through the Analysis of his Works

(1888-1894)

Giuseppe Sicuranza

1.1.1 The first tribute by Giuseppe Peano to logic is contained in his work Calcolo geometrico secondo l’Ausdehnungslehere di H. Grassmann, preceduto dalle operazioni della logica deduttiva, published in Turin in 1888.

It deals with a 20-page introduction made which in the author’s mind wants to represent a sort of Summa of the logical research till his time : «La logica deduttiva, la quale fa parte delle scienze matematiche, non ha finora molto progredito, benchè sia stata oggetto di studii di Leibnitz, Hamilton, Cayley, Boole, H. e R. Grassmann, Schroeder, ecc. Le poche questioni trattate in questa introduzione costituiscono già un insieme organico che può servire in molte ricerche».1 This note shows clearly the idea developed by Peano about logic: it is a branch of mathematics. Since his first work about this subject, Peano takes on a line that will characterize all his research: he will continue the research since Leibniz, whose aim had been to modify the traditional approach to logic, by making it closer to mathematics or better using the mathematical instrument as a foundation, changing logic in a “calculus”, with the rules of a general mathematical notion of calculus, adopting as a model the Boolean algebra.2

1 G. Peano, Calcolo geometrico secondo l’Ausdehnungslehere di H. Grassmann, preceduto dalle operazioni della logica deduttiva, Fratelli Bocca, Torino 1888, p. VII. 2 He mentions, in fact, in a note to p. X of the preface to the above-mentioned work, the authors and the writings he refers to, that is Boole, Schroeder, Peirce, Jevons, and others who have, as a matter of fact, studied the way of rendering, through the application of a binary algebra model, the ideas of the logic. See G.


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The idea is to use a particular algebraic structure nowadays called ring, applying it to logic. The idea of the ring, actually, was developing really in the XIX century in the field of mathematics as a generic notion of calculus, with the intention of eradicating algebra and mathematics from those material conditions, or better geometrical so distinctive till that time. Without getting into too many details, the definition of ring is the following: a ring is a set characterized by two operations and by their properties, identified among its several elements: the “nature” simply operational of the ring allows to work it independently from all the elements grouped within the reference set, reaching such a flexible structure that it can be applied to every set, even to the set of logical “objects”, that are the concepts of the logic. Peano begins his work following this assumption. The calculus in the mentioned work goes on as follows: the starting point is a system of any element to be identified in classes whose symbols are Latin capital letters A, B, etc.), an assumption with no definition. The division of this set in classes is done by putting into the system the equivalence relation “=”: in the modern mathematical theory it is used together with the order relation, to structure a set, by entering a classification, therefore a division.3 The equivalence relation is particular, because characterized by properties like the reflexive (x = x), symmetrical (x = y . y = x), transitive (x = y . y = z . = . x =z). The sign “=” can be understood through its properties and this is the starting idea by Peano: in the second edition of his Formulario, as it will be noticed afterwards and in other works

Peano, op. cit. p. X. 3 This type of formulation is successive to the time in which Peano lived: it is used simply in order to render the idea of the importance of the said relation.


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published during the first years of 20th century he will develop a generalized definition of the discussed concept.4 The concept of class, system, set, may be classified only through circular definitions, even if, he himself in the following speech will arrive to the definition of class as “created” by the symbol “:” (likely), as a dominion where certain conditional propositions are satisfied. As concerns the calculus, he introduces the operations among classes and their properties in the reference set: they represent the logical product and the logical sum “

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∪”, besides the symbols of negation “–“, of the empty set “ ” and universe “ ” lastly, the signs “<” and “>” major and minor, that is of content and containing.5 It is interesting to note how the symbols adopted by Peano do not belong to the traditional logic algebra. He will write as follows to explain them: “I considered useful to replace the signs

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∩ ,

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∪ , –A, , , to the signs of logic , + ,

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A1 , 0, 1, used by Schröder, with the aim of avoiding a possible confusion between the logic signs and the mathematical ones”,6 in this way he justified his choice which needed to be in accordance with Peano’s view about logic. The two main operations define distinctively: 1. “A B C,… la massima classe contenuta nelle classi A, B, C… ossia la classe formata da tutti gli enti che sono ad un tempo A e B e C”,7 the logical product corresponds to the connector et of the classical logic; 2. “A B C,… la minima classe che contiene le classi A, B, C… 4 See G. Peano, Formulaire de mathématique, Carré & Naud, Paris 1901, vol. 1, p. 35. 5 Also, in this phase, not influential: “introdussi i segni di logica < e >, i quali, benché non necessarii, sono assai utili”, G. Peano, Calcolo geometrico secondo l’Ausdehnungslehere di H. Grassmann, preceduto dalle operazioni della logica deduttiva, Torino, Fratelli Bocca 1888, abstract with the title Operazioni della logica deduttiva, published in G. Peano, Opere scelte, Ed. Cremonese 1958, vol. II, pp. 3-19. See p. 18. 6 G. Peano, Operazioni, cit., p. 18. 7 G. Peano, Operazioni, cit., p. 4.


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ossia la classe formata dagli enti che sono o A o B o C”,8 the logical sum corresponds, therefore, to the connective vel of the classical logic: this operation however can be interpreted in an exclusive way like for example in Boole, i.e. as the connective aut, but the disciples of the British thinker showed that the vel use turned out more productive in the logic application of this operation and Peano follows this interpretation. Once the operations are introduced and also all the other symbols, the author lists all the properties belonging to the operations: 1) AB = BA9 1’) A

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∪ B = B

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∪A 2) A(BC) = ABC 2’) A

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∪ (B

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∪C) = A

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∪ B

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∪C 3) AA = A 3’) A

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∪ A = A 4) A(B

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∪C) = AB

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∪AC 4’) A

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∪ BC = (A

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∪ B)(A

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∪C) 5) A = A 5’) A

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∪ = A 6) A = 6’) A

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∪ = 7) -(-A) = A

8) -(AB) = (-A)

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∪ (-B) 8’) -(A

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∪B) = (-A)

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∩ (-B) 9) A

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∩-A = 9’) A

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∪-A = 10) - = 10’) - = 10

The formulas just listed represent the commutative (1 e 1’), associative (2 e 2’), distributive (4 e 4’) properties of both operations: the logical sum and the logical product. The formulas 3 and 3’ are the so-called (even if the author doe not call them so openly), laws of simplification, that Boole considered the real innovation in this kind of approach to logic: as a matter of fact, they give the system the possibility of being

8 Ibid. 9 AB is another way for symbolize the logical product between A and B. 10 G. Peano, Operazioni, cit., p. 5.


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associated to only two values (in Boole’s case 0 and 1), that represent true and false. The other formulas embody the double negation laws (7), De Morgan’s (8 e 8’), of the module existence for the two operations (5 and 5’) and the interchangeability of the signs and . Behind this presentation, although not clearly expressed, there is the law of duality, which grants to get other laws concerning one operation (1’, 2’, 3’, etc.) starting from the others (1, 2, 3, etc.). To these identities we have to add the following: 11) A

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∪ AB = A 11’) A (A

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∪ B) = A Laws of absorption due to Stanley Jevons.11 To complete the description we have to remind also these formulas: 12) AB < A 12’) A

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∪ B > A 13) <A12 13’) >A 13 Through these operations and their properties it is possible to structure a range (set of reference) ring,14 in such a way to justify the laws of syllogism or better, of its generalization or improvement obtained thanks to the calculus derived from these techniques. Before dealing with the propositional calculus assumption, as it is presented in the work, which is the real novelty in Peano’s 11 The Peanian system is not just the Boolean one: he, as already noticed, acknowledges all the adjustments which followed the work of the British logician; its main text of reference is, for its admission, in fact, “Der operationskreis der Logikkalkuls” by Schroeder. 12 Remarkable, formulation of the most important theorem of the Pseudoscotus is one. 13 G. Peano, Operazioni, cit., p. 5-7. 14 Once again, it is to be said that also the ring idea is employed just as an analogy, since it is never used by Peano; the ring itself, moreover, is not structured just in this way.


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thought, it is convenient to remember how he accepts (but only in this work) also the Boolean theories about the formulas development. He introduces the concept of function as an operation on a class with a separated form: and so, the function f(X), can be written in separated form:

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f (X) = PX∪QX ,

where P and Q are independent classes, and, through it we can indicate the different meanings of every expression (development). This theory as said before won’t be taken again by Peano, nevertheless it is one of the foundations in the Boolean logical method. The propositional calculus, the other current of the logic tradition, is here dealt with as a derivation calculus of classes: this is the great novelty above mentioned, introduced by Peano.15 Actually, after discussing the difference among categorical propositions that are true or false and conditional too, containing one or more variables on which their value of truth depends, Peano presents the symbol, “:” ( *cui), that “creates” the class of solutions in a well definite conditional proposition : if x, y, … are variables of the proposition � “colla scrittura (x, y, …) : � intenderemo la classe formata da tutti gli enti (x, y, …) per cui è vera la �…, la classe (x, y, …): � è una classe ben determinata ed una proposizione che affermi qualche proprietà di questa classe è categorica”.16 Peano’s words are satisfactory to expose the idea of the symbol meaning “:”. The innovation given by the sign just shown is of great value as through it we can quantify the operations of the propositional calculus with no use of other symbols. It is enough to observe that, if � and � are conditions (conditional propositions): � < � e � > � means that the class made with �

15 He takes, in fact, explicitly the paternity of it. See G. Peano, Operazioni, cit. p.18. 16 G. Peano, Operazioni, cit. p. 9.


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solutions is in the class of � solutions, that is � has as a consequence �; � = � expresses the equality of classes solutions � and �, rather of the propositions � and �; �

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∩ � is the condition obtained when we suppose as true both propositions (intersection between the two corresponding classes); �

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∪ � means that it is supposed as true one of the two conditions (union);–� is � negation; is an absurd condition; is an identical condition (tautology). As concerns the propositions conceived in this way, the same properties of the operations among classes are considered as valid: in so doing both calculi are closely related, reaching the unification between the two classical logical traditions never so evident till now. There are anyway other laws as necessary as them: [1] (A = B) = (B = A) [2] (A < B) = (

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AB = ) [3] (A > B) = (B < A) [4] (A = B) = (A > B)

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∩ (A < B); expressing the way of changing propositions and exchanging between them the signs =, <, >. [5] (A = B) < (AC = BC) [6] (A = B) < (A

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∪ C = B

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∪ C) [7] (A = B) < (-A = -B), from where we get: [5’] (A = B)

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∩ (A’ = B’) < (AA’ = BB’) [6’] (A = B)

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∩ (A’ = B’) < (A

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∪ A’ = B

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∪ B’), and [5’’] (A < B) < (AC < BC) [6’’] (A < B) < (A C < B C) [5’’’] (A < B)

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∩ (A’ < B’) < (AA’ < BB’) [6’’’] (A < B)

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∩ (A’ < B’) < (A

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∪ A’ < B

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∪ B’);


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which are different ways to indicate the possibility of adding, multiplying the members of a logic equation17 to any proposition or multiplying by the addition of a member to the other one, with no variations in the meaning of the expression . To complete the description we need to add: [7’] (A = B) = (-A = -B) [8] (A < B)

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(A > B ) ; which inform about the feasibility of the equation transformation through the exchange occurring between the signs = <, >, and the sign -; lastly the notable:

[9] (A

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∪ B = ) = (A = )

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∩ (B = ).18 All these identities show how different formulas of the same proposition can be obtained: that implies a greater adaptation to real situations by the calculus, where it is likely that a well definite proposition is expressed in different ways among them though preserving their meaning. Within a paragraph aside also syllogisms have been dealt with, till then the most relevant section of the classical logical theory (it is enough to think about the Kantian “prejudice”). Peano himself will consider the syllogism as an essential method of reasoning, including it in the logical theory of modern mathematics. The assumption in fact begins with the presentation of four fundamental propositions:

[I]

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AB = Every A is B [II] AB = No A is B [III] - (AB = ) Some A is B [IV] - ( =

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AB ) Some A is not B;19 17 That is combinations of the introduced logical signs, that constitute “the applicative” part of the calculation, on the model of the algebraic equations. 18 G. Peano, Operazioni, cit. pp. 11-12. 19 Op. cit,. p. 14.


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that is the universal propositions (I e II), particular (III e IV), affirmative (I e III) and negative (II e IV) of the traditions. Equally the propositions I and IV, II and III are said contradictory, while the I and II, III and IV are contrary. The novelty which is shared by the whole logic developed in the XX century, consists in the approach to the deductive part of the syllogism, how the mentioned propositions are combined with the aim of creating the modes and figures, the varied forms of the syllogism. The classical tradition implied a set of rules for the deduction applied in a way that was “external” to the initial propositions: in the discussed theory, instead, they result to be a consequence of the calculus property, and in this way, the syllogism becomes a particular case. The rule of inversion belonging to the propositions universal negative and particular affirmative, becomes, for example, a consequence of the commutative property typical of the logical product.20 The choice of setting it as an algebraic logical calculus, allows also to get results in contrast with the tradition of the syllogistics: the application, for example, of the property expressed by [9], of the propositional calculus, shows in some cases a possible coexistence between two contrary propositions (when one of the two reference classes is null).21 As we have mentioned, the modes and the figures of the classical syllogism have been created by simply working on the basic propositions, through the operations of the calculus and their properties, thus obtaining in this way all cases quoted by the tradition but also correcting mistakes, or emphasising some combinations.22 In closing his first work on logic, Peano illustrates some rules, referring to the logical equations, which recall the

20 Op. cit. loc. cit. 21 Op. cit. p.14-15. 22 Op. cit. p. 16.


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mathematical model of an equation solution, allowing to get into the system possible unknown classes (the equalization of the equation towards , the elimination of an unknown).23 This process which characterizes the Boolean approach, has not to be misunderstood or confused with potential rules of deduction, the intention is not, therefore, to deduce from a system of elements a possible consequence, but to show the several opportunities where a certain proposition can be expressed in such a way as to allow the choice of the one true. The remaining written text about the argument (the section devoted to the logic is only the introduction), concerns the presentation of the geometrical calculus , as H. Grassmann had planned it. Logic does not represent in Peano organization, as already said, an instrument to make stricter and less ambiguous the possible propositions of mathematics : this idea will be constant in all the work by Peano about logic. 1. 1. 2. In his works written in 1889, where the question of logic is discussed again, it is once more confined to the introductory section, though without making the important tribute by Peano about the question less noteable: these works are Arithmetices principia, nova methodo exposita24 and I Principii di geometria logicamente esposti. The care in the choice of symbols or in the presentation of the theory shows out in these works as in the next ones the importance that logic had in Peano’s thought; so that, despite the “instrumental” approach of the research or thanks to it, he has obtained appreciable results in this field. With reference to the mentioned works, and particularly to Arithmetices principia, nova methodo exposita (in I Principii di geometria logicamente esposti the logical system used by Peano is

23 Op. cit. p. 16-18. 24 Written in Latin: it is a sign of the importance attributed from the Author to this work.


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the same of the previous work), about the analysis we can notice interesting changes: after the Praefatio, in which the author shows his aim of using the logical device to present the principles of arithmetic,25 appears the Signorum Tabula, the index of the used symbols. They are not properly the same contained in the 1888 writing: P propositio K classis

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∩ et

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∪ vel - non

absurdum aut nihil

deducitur aut continetur = est equalis � est [ ] inversionis signum

qui vel [�].26 The sign of the empty set or of the absurd condition has changed in “ “, (instead of ), and the symbol of deduction that in the previous work had the form “<”, becomes “ ”:27 these seem little important changes, but on the contrary they have a precise motivation. The signs used, in fact, he specifies: “Aut ad logicam pertinent, aut proprie ad arithmeticam”;28 the aim purely functional of logical needs a clear distinction between the logical signs and the mathematical ones.

25 The main fault of the previous discussions, in fact, he says is the lack of rigour: “Hic difficultas maxime ex sermonis ambiguitate oritur”; G. Peano, Arithmetices principia, nova methodo exposita, Fratelli Bocca, Torino 1889, repr. in G. Peano, Opere Scelte, cit., vol. 2, pp. 20-55. See p. 21. 26 G. Peano, Arithmetices principia, cit. p. 23. 27 The deduction symbol had been introduced in logic by Peirce. 28 G. Peano, Arithmetices principia, cit. p. 21.


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In The geometrical calculus according to Ausdehnungslehere di H. Grassmann, preceded by operations of deductive logic, Peano had meant to change some symbols to avoid a probable confusion between logic and mathematics: more precisely in this work he will be able to get his purpose. From the presentation of the symbols already used we move to the real calculus with a great innovation to be developed in the following works so much to become a true axiom:29 the use of brackets. The brackets will be very interesting as they will be used to separate formulas in such a way to avoid every possible ambiguity: this also explains the use of dots, more effective than the ordinary signs of bracket and more suitable for the aim. As concerns the calculation, instead, it is based on the same system contained in one of his 1888 writings and is about the operations and their properties: what changes is the order of presentation and it is not by chance. If in the previous work devoted to logic the calculus of classes is presented for first, on the contrary in this case it is the propositional calculus to have the priority: Peano won’t ever reveal the prevailing one between the two, since he considered them complementary, although one of his pupils, Beppo Levi, will show that there must exist a priority criterion and it is given to propositional logic. In this case the author wants to deal with arithmetic logically and as it is made mainly of propositions, it is urgent to develop the propositional calculus. That is why he uses so frequently the symbol “ ”: the arithmetical propositions are of the kind “if… then…”, and the symbol becomes fundamental to express them. We need to specify that it has not properly the meaning of implication (if… then…): the symbol therefore is presented by Peano as deduction with a metatheorical meaning. It deals with an interpretation of implication, shown by the propositions presented, 29 The use of the axiomatization in the enunciation of the calculation is not, however, still present here.


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afterwards, as “ laws” of calculus: the subdivision between theory and metatheory is not contained in the work by the Italian mathematician. 30 About the symbol “ ” we have to notice how he introduces the convention of placing the propositional variables (the conditions) as subscript of the same symbol, to replace and mean, the universal quantifier: “a x,y,... b significat, quaecumque sunt x, y…, a propositione a deducitur b”,31 a rule used also for the symbol“=”. Now Peano illustrates the Logicae propositiones, with number 43, which represent the real “laws” of calculus : they are , in fact, a set of tautologies discovered till his period, that Peano thinks to adopt as deduction rules. A proof according Peano view is a transformation of known propositions from propositions called primitive (a distinction revealed in Formole): “to demonstrate a proposition means getting it through a suitable combination of admitted past propositions”.32 E.g., given the primitive propositions:

4. ab ba 5. abc acb 6. a b . ac bc 7. a . a b : . b

by substitution in P4. and by P6, P7, he deduces 9. abc bac.33

30 See as an example, the observations in purpose of D. Palladino, in AA. VV. Logica matematica e logica filosofica, ed. Editrice La Scuola, Milano, 1990, p. 88. 31 G. Peano, Arithmetices principia, cit., p.25. 32 G. Peano, Formole di logica matematica, in “Rivista di matematica” (1891), 1, pp. 24-31, repr. in G. Peano, Opere Scelte, cit., vol. 2, pp. 101-113. See pp. 103-104. 33 G. Peano, Formole, cit. p. 105.


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The explanation of rules for deduction make him closer to Frege and to the particular form that logic will have after him, but the shift to this kind of process, does not seem completed: the tautologies are not able to explain logically logically the transformation occurring from a proposition in another one as there is no formal process to connect the resultant propositions. For this reason it is appropriate to provide some examples of Logicae propositiones, also to show the graphical variations in the symbols, mentioned above and more:

2. a b . b c : : a c: 12. ab = ba 30. a (b

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∪ c) = ab

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∪ ac 40. a.34 They are some propositions proposed by Peano which are enough to indicate the news appeared: among them we can observe the properties of operations, the syllogism of Barbara the theorem by Pseudoscotus, all included in a different context but revealing the move towards the ideas of the next logic. The properties, the great tautologies will become without any distinction “rules of derivation” for the propositional calculus. Without dealing with the mentioned argument, we’ll talk now about the further step done by the Italian mathematic related to the link between the propositional calculus and the calculus of classes : the presentation of the sign “�” that means “belonging to” is x�a where x “belongs”, it is an element belonging to class a. The importance held by this sign is remarkable: through it we can modify a well defined class into a proposition, connecting closer the classes calculus and the propositional one. It is clear how this sign represents the reversal of the sign “:”, described in 1888 writing that now becomes “[�]” (o “ “). These two calculus become even the two faces of a medal, two ways of approaching the same question.

34 See G. Peano, Arithmetices principia, cit., pp. 25-26.


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The calculus of classes actually is not shown out in this writing: the formulas related to the sign “�” and of the reversal symbol “[ ]”. The propositions which rule the use of these signs are about 20 and together the others contained in Logicae propositiones allow the translation of the propositional calculus in classes’, giving the opportunity of using both with no distinction.35 In more details they inform us of how “�” (and “[ ]”) behaves towards the logical operations before presented: 47. a, b � K .

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∴ x � . ab : =x : x � a . x � b . 50. a, b � K.

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∴ a b: = : x � a. x . x � b . 52 . a � b . . b � K . 57. a � P . : [x �] a . � K . 58. a � K .

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∴ [x �] . x � a : = a . 62. [x �] (�

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∪ �) = [x �] �

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∪ [x �] � .36 These few examples are enough to emphasize the role of signs’ “translators”: particularly the 57 and the 58 that make easy to move from a proposition to a class and vice versa just using two symbols keeping, the operations, the same meaning (as shown by other examples).37 The use of reversal symbol is obviously not limited in the application to �: it is possible by it to consider the reversal of every operative symbol. It has to be noticed the last marked proposition: it presents another important use of the sign [], that is to work as a substitute of variable: “Sit � formula indeterminatum continens x.. Tunc scriptura x’ [x] �, quae legitur x’ loco x in � substituto,

35 Actually the translation is not completely faithful: in at least one case it is not possible. The problem, but, simply is overlooked by the Author, demonstrating, once again, how he is far away from the philosophical problems connected to the logical theory. 36 G. Peano, Arithmetices principia, cit. pp. 27-29. 37 Ibid..


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formulam indicat quae obtinetur si in �, loco x, x’ legimus. Deducitur x [x] � = �”.38 The concept of substitution of variables reveals all its importance in logic as it permits to reduce the proofs, by using known theorems, with no necessity of continuous explanations. Before discussing about the ending part of this work, devoted to the theory of functions, (very important in the Peanian view), it is worthy to notice another thing linked to the use of sign “�”: in fact it could be misunderstood with “ “, but not properly. Peano is aware of that: he will emphasize it in this work and so as in the following works. The theory of functions is described in the last paragraph of the mentioned work and all along the text it will have a greater importance with reference to the arithmetical foundations. The Author writes about a function, which is “Sit � signum, sive signorum aggregatus, ita ut si x est ens classis s, scriptura �x novum indicet ens; supponimus quoque aequalitatem inter entia �x definitam; et si x et y sunt entia classis s, et est x = y, supponimus deduci posse �x = �y. Tunc signum � dicitur esse functionis praesignum in classi s, et scribemus � � F’s. Verum si, cum sit x quodlibet ens classis s, scriptura x� novum indicet ens, et, ex x = y deducitur x� = y�, tunc dicimus � esse functionis postsignum in classi s, et scribemus � � s’F”.39 This long quotation can explain very well the idea of a function: it has the role of relating two units, of any class, x and �x (o x�), in the case that the two units are resulting equal to each other x and y, the units to which they correspond (�x and �y o x� and y�) must result the same. A function represents a correlation between two units belonging to the same class, in such a way to keep it even taking any unit of it : to give an example the function of successor a+, (introduced in the second part of the work that will be presented in the following), means that anyway we take a, a+ it will be always 38 Op. cit. p. 29. 39 Op. cit. p. 30.


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the immediate following unit to a, and in case a is equal to b, b+ it will be equal to a+, in this way we avoid repetitions. This theory is really important for the justification, through the symbol of the function +, the range of natural numbers, it is possible to concatenate all the elements of a class by obtaining in this way a continuous range of new elements and denying the possibility of replications. After introducing the symbols of functions the speech goes on with the emphasis on their properties, that is the product of two functions, the iteration of a function, the reversal (through the sign “[ ]”) and the combination with the logical operations and, lastly, a quotation about the functions so- called similar (that is �x = �y . . x = y). E.g. see: s � K . � � F‘s : : [�] y . = . [x�] (�x = y);40 s � K. a, b � K. a s. b s. � � F‘ s : . �(ab) (�a) (�b);41 s � K . � � F‘ s . � � F‘ �s . x � s : . ��x = �(�x).42 Peano will deal with this argument in other works since it will hold a relevant position in the following development of the mathematical principles. At the end of the work there is the theory of definitions, considered fundamental in his study and this will be taken again to be completed in some booklets. The theory of definitions will be developed more and more and it is necessary to refer to the first edition of the Formulario. In short, a definition is an expression of the kind x = a, where x represents the sign to be defined and a a group of known signs, to become a useful abbreviation in the following applications. 40 Op. cit. loc. cit. p. 30. 41 Op. cit. p. 32. 42 Ibid.


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The necessary condition to get a correct and even definition is the homogeneous aspect of the two members of equality as they have to contain the same variables. He says that we often need to consider a hypothesis, under which we create the equality and the definition in this case becomes:

h . . x = a.43 The hypothesis however is a restriction of the defined field and does not influence the meaning of the function itself. There is another kind of definition, which assumed a great value in mathematics: the definitions for abstraction, where the definition does not belong to the sign, rather a function on it or better, the equality between functions of two objects,

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hu,v . : �u = �v . = .

€

pu,v . 44 A definition by abstraction, therefore, he says in case of a defined hypothesis h, the equality �u = �v , that is a relation p, between two objects which satisfy the properties of reflexivity, symmetry, transitivity, of the relation in equivalence. The author will assign great importance to the theme of definitions to settle the foundations of a theory and will include them in the primitive ideas of a set, a wide range of booklets, where the argument will be analyzed with attention: what has been said, can give an idea of definition according to Peano. The Italian mathematician is in search of a theory of definitions purely nominal, disputing both the classical theory, the Aristotelian, and the new theory by Poincaré: the intention is to

43 G. Peano, Notations de logique mathématique, Introduction au Formulaire de mathématiques, Ch. Guadagnini, Torino, 1894, repr. In G. Peano, Opere Scelte, cit. vol .2 pp. 123-176. See p. 167. 44 G. Peano, Notations, cit. p. 168.


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consider a definition as an abbreviation, in order to avoid to create circular definitions with no real advantage.45 1. 1. 3. In the same year was published the work I principii di geometria logicamente esposti (1889) whose introduction is again devoted to logic, but it does not offer new elements in comparison to the approach in Arithmetices principia, nova methodo exposita: it is anyway important how within this work it is mentioned for the first time the possibility of interdefining the logical signs, making some of them superfluous. He says: “The signs and

€

∪ are useful but not necessary, as we can place

a b writing

a-b = , and in the place of

a

€

∪ b writing

–((-a)

€

∩ (-b))”.46 This observation47 will allow a deeper analysis of the logical theory, aiming to the reduction of the symbols, making the logical device more effective than ever.

45 See G. Peano, Le definizioni in matematica, in Opere Scelte, cit. vol. 2 p. 433 ssg. , and G. Peano, Super theorema de Cantor-Bernstein et additione,in Opere Scelte, cit. vol. 1 p. 337 foll. 46 G. Peano, I principii di geometria logicamente esposti, Fratelli Bocca, Torino 1889, repr. in G. Peano, Opere Scelte, cit., vol. 2, pp. 56-91, See p. 60. 47 To tell the truth, Peano had already noticed the possibility to interdefining the logical operations, but here he does it in a more explicit way, so to hope for the possibility of a axiomatic study of logic.


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This does not mean that the direction taken goes towards the elimination of some signs but the idea is now to adopt the mathematical method presented in these works too, dividing the propositions in primitive ideas (axioms) and derivatives (theorems),48 with the aim of a reduction of axioms and the consequent simplification of the calculus. In Principii di geometria he presents three important rules to be used in proofs though they appear too much tied with the Boolean method, which on the contrary Peano will include in other works as more general rules, or he will follow another approach of the same calculus. It deals with rules adopted to eliminate the unknowns as follows: if there are propositions containing an unknown x, three cases are possible or better they can assume three forms, that is a . x � h : . x � k;

a . x � h . x � k . x � l : = ; a . x � h . x � k : . b.

From them we can eliminate the x, then by observing how it may occur a transformation in: a . . h k ;

a . h

€

∩ k

€

∩ l = ; a . h

€

∩ k - = : . b.49 1. 1. 4. As regards the reduction of the logical symbols to be used, we can notice how this was his intention when in

48 “… sarebbe uno studio interessante, e che finora manca, il distinguere le (proposizioni) fondamentali, che si debbono ammettere senz’altro, dalle rimanenti, contenute nelle fondamentali. Questa ricerca porterebbe ad uno studio, sulla Logica, analogo a quello qui fatto per la Geometria, e nel precedente opuscolo per l’Aritmetica”, G. Peano, I principii di geometria, cit. p. 81. 49 For these and previous formulas see op. cit. pp. 85-87. In all these formulas, a not containing proposition represents one x, e h, k, l classes.


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1891, he published the Principii di logica matematica. However before this work, it is necessary to mention a booklet, published in the previous year about mathematics, Démonstration de l’intégrabilité des équations différentielles ordinaries (1890). This work (mainly with a mathematical content) represents a primary source for logic as within it we find not only the known symbols but also the new sign “�”. This kind of symbol indicates a class made by one member: thus �a is a class containing only the element a. All of this allows to solve a relevant question: the possibility to build classes of classes, in the view of basic classes as individuals. For example: x � a

€

∪ b : = : x � a .

€

∪ . x � b 50 x � �a

€

∪ x � �b : = : x � �a .

€

∪ . x � �b : = : x = a .

€

∪ . x = b 51 mean: “a

€

∪b la classe dont les individus sont les individus de a ou b ; �a

€

∪ �b désigne la classe don’t les individus sont a et b ; �a

€

∪ �b désigne la couple des droite a et b, a

€

∪ b désigne l’ensemble des points qui se trouvent sur l’une ou sur l’autre des droites a et b ; les individus de a

€

∪ b sont des points; ceux de �a

€

∪�b sont des droites”.52 Through this symbol in the mentioned case it is possible to create the class �a, which represents a straight line, that is a class of points: in so doing there is a noteworthy simplification and above all we can operate without misunderstandings on classes of classes. All along this argument we find also a reflection about the syllogism of Barbara kind:

50 G. Peano, Notations de logique, cit. p. 140. 51 G. Peano, Démonstration de l’intégrabilité des equations différentielles ordinaires, in “Mathematische Annalen”, Bd. XXXVII,(1890), pp. 182-228, repr. in G. Peano, Opere Scelte, vol. 1, Cremonese, Roma, 1957, pp. 119-170. See p. 131. 52 Op. cit. loc. cit..


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a b . b c : . a c , a � b . b c: . a � c, 53 are the two ways, both correct of writing the syllogism Barbara. What may seem something irrelevant, is instead a very important discovery for two main reasons: a) for the first time it has been noticed how the debated syllogism is actually the sum of two, emphasizing again the way in which the theory of classes reproduces or better contains the classical syllogistic; b) all this makes clear the difference between“ ” and “�“ the argument mentioned before. In conclusion, it is important to highlight a further graphical change of two symbols: the sign of function becomes “/”54 and the sign of inversion “ ” (placed on the related symbols). Coming back to Principii di logica matematica, it has to be said that it represents a change in the Peanian prospect: in this work dated 1891, he outlines a systematic theory of the logical operations, applying the method used in the mathematical proofs, even if the calculus obtained was the same contained in the preceding works. As said before the method is just drafted: within this brief paper he illustrates the logical symbols and their properties, analysing them in a separate way. In the previous works the presentation was almost the same for all the signs, they were shown all together with their properties, which in a way were common. Here the prospect is another one: both the signs of deduction and conjunction and the symbols of membership, negation, 53 Ibid. 54 Actually / is only used here in Sul concetto di numero, but it witnesses the continuous adjustments by Peano in his works, showing the enormous importance that he gave to logic, even though in a purely instrumental perspective: the relevant theory, however is different, but we will deal with it later on. See Sul concetto di numero, in “Rivista di matematica” (1891), 1, pp. 87-102, 256-267, repr. in G. Peano, Opere scelte, Cremonese, Roma 1959, vol. 3, pp. 80-109.


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disjunction and absurd are presented one differently from one another and analysed with all their properties.55 Every symbol is expressed in all its possibilities and modes of application, commented step by step through a common language, creating also a sort of specific vocabulary. It is worthy to notice, en passant, the frequent quotation by the author of Leibniz: he considered himself one of his followers and actually in all his works he referred to the German thinker many times; to tell the truth he saw himself as the one who had completed Leibniz work: “Risulta così che la questione proposta da Leibnitz è completamente, se non ancora perfettamente, risolta”.56 This represents a very important work for the future plan of Peano’s ideas about logic till now developed, in a prospect of a complete system. 1.1.5. In the same year Peano published Formole di logica matematica,57 the real first attempt to give a system to logic following the method adopted for the mathematical foundations. The calculus is based on four primitive ideas corresponding to the signs ,

€

∩ , , –, and on twelve primitive propositions:58 1) a a 2) a aa 3) ab a 4) ab ba 55 That does not mean that the dimension of the interdependence of the symbols gets lost, but that, for the first time, they are analyzed as they were different. 56 G. Peano, Principii di logica matematica, in “Rivista di logica matematica” (1891) 1, pp. 1-10, repr. in G. Peano, Opere Scelte, cit. vol. 2 pp. 92-101, See p. 100. 57 Both the works, like almost all, are published on the “Rivista di matematica”, the periodical he founded in 1891: the papers we are discussing appear right in the first issue. 58 The primitive propositions were initially twelve: in the additions published in appendix to this work one of them is demonstrated.


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5) abc acb 6) a b . . ac bc 7) a . a b : . b 8) a b . b c : .a c 9) b . a ab 10) a b . . -b -a 11) -(-a) = a 12) a -a = .59 Moving from these symbols and these propositions Peano thinks that it is possible to differentiate all the logical propositions he needs. However a little reflection seems necessary: supposing that the distinction between primitive and derivative propositions is arbitrary, because we could assume as primitive any set of propositions and afterwards differentiate the others, (the choice follows “economical” principles), there is still Peano’s theory of demonstrations to be presented. “Dimostrare una proposizione significa ottenerla combinando convenientemente le proposizioni già ammesse”:60 the criterion is very simple. We take the primitive propositions or the ones already demonstrated and we combine them with each other by adding or taking away, to obtain new propositions. The only allowed “external” rule is the replacement of the variables or of the propositional letters, as said previously: the only difference is that the symbol is the brackets “( )” and no more the reversal sign “[ ]”. It was observed how the proof set lacked a formal process with rules of derivation: the Peanian method could, therefore, appear confused since the combination among propositions are in a sense arbitrary.

59 G. Peano, Formole di logica matematica, cit. pp. 105 ssg. 60 Op. cit. p. 103.


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In Peano’s view it could not be different: he wants to create a synthesis of the logic in his time, a universal device to be consulted, without caring about the means to explain the system. The criterion followed is simple: the propositions are all interconnected, so the only aim to achieve is to find the smallest set of propositions to make easier the use of logic. Once we have ended this topic whose content will be developed later, we can complete our discussion about the exposition of the logical theory, its development with the works written between 1888 and 1894 (the year when Peano will begin to publish his ambitious Formulario), with a note to the review produced in 1891 and published together with the works I have just analysed directed to Vorlesungen ueber die Algebra der Logik by Schroeder. 1. 1. 6. This brief writing is a presentation of the work produced by Schroeder but contains some new reflections about Peano’s view about logic. First of all, it is again underlined again the lack of interest by the author towards the philosophical questions emerging within the theory, which has always been evident in all the assumptions by Peano, following the instrumental plan of the logic. He was however very attentive to the logical theory getting important results: in this work Peano praises the Schroederian presentation but in a critical way; in spite of these criticisms, he does not move much from him studying particularly a formula that generalizes the distributive property of a product in comparison with the logical sum. Actually, Peano shows how it is not suitable for all the possible interpretations of the used signs and it has to be admitted as primitive: whereas the author in his work Formole di logica matematica conceives it as something deriving from other propositions since he had not considered as primitive the sign . He is not able to give a real solution to the problem as we refer to the previous observations about the proofs. With this written work a cycle of Peano research ends: in 1891 issue of “Rivista” he informs or better plans in his article Sul concetto di numero the work that will make him very busy till


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1908, the Formulario: “Sarebbe pure cosa utilissima il raccogliere tutte le proposizioni note, che si riferiscono a certi punti della matematica, e pubblicare queste raccolte. Limitandoci a quelle dell’aritmetica, non credo si possa trovare difficoltà ad esprimerle in simboli logici; ed allora esse, oltre all’acquistare nella precisione, acquistano pure in concisione”.61 The Formulario should be a collection of all the known mathematical formulas, the peak of the whole research made by the Italian mathematician: the logic is the device to write and order them and for this reason the logic takes on a relevant place.

61 G. Peano, Sul concetto di numero, cit. p. 109.

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