The Logic in the Different Editions of Giuseppe Peano’s ... · Giuseppe Peano’s Formulario...

Metalogicon (2007) XX, 1

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The Logic in the Different Editions of Giuseppe Peano’s Formulario Mathematico (1894-1908) and in its Works of Integration

Giuseppe Sicuranza 1. 1. 1. The Formulario mathematico represents the most important work by Giuseppe Peano that will take him a long period, about 20 years, publishing five editions of it. The aim is to collect inside it all the formulas regarding the different branches of mathematics: however, what makes really great this work, is the idea of writing all the propositions expressed through a symbolic form. Actually every edition of the Formulario, contains an introduction devoted to logic, a useful device in Peano’s view, which he had “created” or better, improved in the previous years. So is it in the first edition of the mentioned writing, published in different periods between 1893 and 1895: in 1894 he produced Notations de logique mathématique, that will be included as an introduction to the first book Formulaire de mathématiques, first edition in French of this work. This paper can be considered as the greatest critical illustration of the logical symbols used till that time by Peano, and what is more it shows a lot of innovations with regard to the past theory. It begins, not by chance, with the word Leibnitz: the German thinker, is the continuous point of reference for Peano since he had been the first to believe in the opportunity of making logic a precise instrument to be used in all the possible applications. We have also to mention some references to 19th logical algebraists, that somehow influence Peano theory. Analysing the work, we see that it opens up with a presentation of class calculus: this choice has been made for its utility as there is no difference between the class calculus and the propositions’.


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As usual the classes are not defined, but the discussion about them is characterized by lots of examples, drawn out from mathematics, whose most recurrent symbol is given by K. Afterwards he introduces the logical operations between classes and properties: even in this case the symbols are the previous ones, as the properties about their application are already explained in his first work in 1888, together with the corresponding use of brackets. The connection with the calculus of propositions is ensured by symbols (they are not different from the previous ones) of belongings (ε) and inversion (–). It is worthy noticing how through the illustration of ε properties, Peano highlights the difference between it and : for the first time he refers to the distinction, so peculiar of medieval logic, between sensus compositus (ε) and sensus divisus ( ). At the same way, but only once, Peano will describe a table of truth1 for the deduction: in Peano reasoning, the logic has no need to prove (we will deal with this topic later on). The propositional calculus has neither variations in symbols nor in its rules: the only innovation lies in the concept of adoption and in its expansion, due to Peirce. The author explains it as follows: “Affirmer que de a on déduit que de b on déduit c, c’est comme affirmer que de ab on déduit c”, and through symbols:

a . b c : = . ab c.2 Its meaning is clear, like its importance for the demonstrations: in fact thanks to it we can modify a proposition by deleting or adding one of its deductions.

1 However, the concept of truth table will be introduced only by Wittgenstein in the 30’s. In that period, however, it was already present, implicitly, in the search of Boole and Frege. 2 G. Peano, Notations de logique mathématique, Introduction au Formulaire de mathématique, Turin, Guadagnini 1894, repr. in G. Peano, Opere Scelte, Ed. Cremonese, Roma 1958, vol .2, p. 136.


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Afterwards the reasoning moves to the use of variables where Peano works out a distinction between real and apparent variables; in so doing he gives the question a deeper meaning. According to him there are cases in which the letter, or the letter variables are independent and can be replaced with any else: the Peanian example, drawn out from the analysis, reports in writing (fx)x = a the x is of little importance to find out the value of a function, and it can be replaced by y, z,… .3 The use of setting variables as inclination of the sign deduction, conceived to replace the symbol of universal addition, concerns, therefore, only the apparent variables, by completely justifying the solution adopted. This solution respects a fundamental rule: the abolition of variables. This principle, as mentioned in the previous works, important for the simplification of logical equations, is now presented in a more detailed way: its demonstration in the example by Peano, is rather long, but shows us the idea of the relevance to be given.4 From this part on which represents an illustration of different concepts already explained in other works, even though with a wide range of examples, the Author moves to the theory of functions which is completely new. Its symbols are new: and . The concept of function is also set in a different way:5 in Arithmetices principia, nova methodo exposita, a work written about this topic long time before, it is conceived as a relation between elements belonging to the same class, while in this work he points out belongings between elements related to two different classes. Anyway, on “Rivista di matematica” in 1891, he wrote the article Sul concetto di numero, a writing which seems to anticipate the Formulario, by taking again the interpretation of Démonstration de l’intégrabilité des equations différentielles ordinaires, he used the sign“\”, to indicate the functions, with the meaning of belongings 3 See G. Peano, op. cit., p.137. 4 See G. Peano, op. cit., p.143. 5 The idea of function, however, is not definable, since it is simple and primitive. See. G. Peano, Sul concetto di numero, in “Rivista di mtematica”, vol. I, (1991), pp. 87-102, 256-267, repr. in Opere Scelte, cit. vol. 3, p. 81.


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between two elements of two classes : a\b is “sign placed after a produces b”,6 that is to say

a, b � K . � � a\b . = : x � a . x . x� � b.

This could be in reference with the new idea of number comes out within the mentioned writing where the number7 becomes a function that joins classes, as specified: “il segno num è un segno di operazione che fa corrispondere ad ogni classe o un N, o 0, o l’∞”,8 for example num a = p or a �

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num p (“the class a contains p individual”).9 The question about the number will be treated later, with this anticipation to justify a change in perspective (in fact the illustration of the theory begins from the concept of num), and looking back to Notations, it is to be said the way of writing the theory of functions revealing a lot of new elements, first of all the one explained before which is described in deeper. It deals with a detailed survey about the sign‘, (meaning of), never explained before : it indicates a function and allows not to misunderstand a class with an operation or, better, their names. The reciprocal confusion is probable: in the case of log, for example, it is necessary to distinguish between log x, that represents a class of logarithms, and log ’x the operation to the solution of logarithm in comparison to a number x that means two different concepts, absolutely unchangeable, although similar in graphics. This specification is certainly important for the idea of a precise order, the symbols of parts in a theory, to which the Author will give more attention in the further development of his reasoning. The symbol is important also for another reason: it is used to indicate, together with the signs “

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∩”, “

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∪”, “la plus petite classe 6 Ibid. 7 That does not mean, however, that this is a definition of number: Actually, Peano will point out many times that it is not possible to give a rigorous definition, at least beginning from the logical signs used, anyway he will discuss later on about such a topic. 8 G. Peano, op. cit. p. 102. 9 Op. cit. loc. cit.


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contenant toutes les classes du système u” and “la plus grande classe contenue dans toutes les classes du système u”,10 that is the logical sum (

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∪’ u) and the logical product(

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∩’u) of classes belonging to a system of classes u (subclasses).11 With the sign ‘, however, that symbolises generic operations and a function, Peano introduces the symbols “ ” and “ “, that express the two ways of writing a function, before and after the corresponding sign c : b a, therefore, it is a function, written before the corresponding sign, that changes an a in a b, b a is ,on the contrary, a function whose symbol is written after the related sign. The distinction is contained in all his works devoted to this theory: it is helpful to present it because in mathematics it is well-known how rich is the use of these two writings is (for example in log case, which is function presign or of !, function postsign. In the following presentation about the relative properties of the functions, anyway, the Author will always adopt the sign “ “: 1. a, b � K . h � b a. x � a . . hx � b ;

2. a, b � K . h � b a. x, y � a . x = y . . hx = hy ;

3. a, b, c � K . h � b a . b c . . h � c a;

4. a, b, c � K . h � b a . c a . . h � b c.12 These formulas express widely the idea of a function, as conceived by Peano: first of all the correspondence, a word used as a synonym of function. The 1, in fact, says that if h is a function connecting two elements, whereas one of them belongs to a class a, its corresponding is part of a class b: so the connection is between two elements belonging to two different classes. The 2, instead,

10 G. Peano, Notations de logique mathématique, cit. p. 147. 11 In effects it is the same Author to say that this is its only true use, always being able to create signs of operation in substitution of it in several cases: it was, however, necessary to speak of it for the above reasons. 12 G. Peano, op. cit. p. 150-151.


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represents a generic part of a speech contained in Arithmetices principia concerning the actual theory: while in the previous work it was this property to convey a sign of function by making uniform its use, here it deals with a way of substituting similar objects, specifying the meaning of operations signs. That is, two elements of a class are equal, so as their corresponding that allows a switch without any change in the result of the “correspondence”. The remaining ones can represent applications of the syllogism, making possible a passage from an operation to another one contained in it . This speech concerns general functions: but there are some categories of functions which have peculiar properties. The functions called similar (univocal, reciprocal), symbolized through sim, for example : these correspondences have the characteristic in which every element of the departure class a, is in relation with only one element of destination class b, and viceversa.13 The similar functions can be also of other kinds (Sim), if an element in a corresponds to a b, to the elements different from the one in class a correspond different elements in class b: “Donc h � (b a)sim signifie que chaque b est le correspondant d’un et un seul a; h � (b a)Sim signifie qu’il n’y a pas de b correspondant à deux valeur différentes de a; c’est-a-dire, chaque b est le correspondant d’un seul, ou d’aucun a... Il est chair que, si h � (b a)sim, elle est aussi (b a)Sim ; mais non réciproquement”.14 Besides these functions, we find others as increasing, decreasing, continuous, peculiar cases, but the Author gives only mentions of it. The similar functions, anyhow, will be treated again in the following works, getting particular importance in the definition of the equality among numbers. 13 A function that goes from a to b does not have necessarily only a corresponding: it is possible, indeed, that an element is in correspondence with an entire class, for example. 14 G. Peano, op. cit. p. 154.


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After these particular case, the Author illustrates the theory about the inversion of functions: the concept of inversion is really significant because in this context it is described in general terms, not only in reference to the particular application of the � sign. If a, b � K, h � b a, and x � a, then y = hx. In fact:

1. x =

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h y . = . y = hx, in case h � sim, and

2. x �

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h y . = . y = hx In a more general case in which to x of a correspond two or more elements of b. As said before, it is about cases of inversion applied to every function, even if its use is limited to this application, the sign of membership.15 We have not to neglect an important observation concerning the concept of inversion: through it, we can justify the removing of variables, of course the appearing ones, for the first time studied in deep.

1. a, b, u � K . u a. h� b a . hu =

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yε (x � u . y = hx . - =x );

2. a, b, u � K. u a . h�b a.

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∴y � hu . = : x � u . y = hx . - =x ;

3. hu v. = : x � u x . hx � v ; 4. (hu)

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∩ v = . = : x � u . hx � v . =x ; 5. (hu)

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∩ v - = . = : x � u . hx � v . - =x .16

15 Without, by this, denying the importance of the general case: it, indeed, allows to better specify the particular use of it. 16 G. Peano, op. cit. p. 158. In the last three propositions u and v represent classes and h A function.


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This shows how it is possible the remove of apparent variables: through the symbol of inversion, we “translate” a proposition in another one, for example the 1, in which the variable use of x is of little importance or better, it becomes “apparent” since its value is not such to get a solution of the proposition. This procedure is different from the delete principle before presented, about variables: in that case, in fact, it implied to delete properly an unknown (variable), making the calculation easier, here instead, the problem is to realize when a variable is independent within the calculation, and can be “replaced” with any other or, simply, deleted without intruding the working out of a formula. After this annotation, the Author introduces a further complement to logic, completely new : the theory of the connections, although briefly described, as he won’t pay the attention that it will have later with the work by Russell. In this writing a connection is considered on the same level of a function: “Soit x � y une relation entre x et y; etant donné y, il résulte déterminée une classe de x qui satisfont à la relation proposée. Cette classe est une fonction de y”,17 i.e. x � y = x � �y. A connection � 18 can be broken out into two signs � and �, it is only a way to reduce the composition of the mentioned symbols. It is really interesting the explanation related to the sign = which reports a connection (equivalence), however considered as the composition of signs � and �.19 The symbol �, function, associated with x represents the class made only of a single entity, in this case x, or the class of entities are the same as x (x itself, therefore). The factorization allows a fundamental observation: in logic, following a tradition, it isn’t possible to talk about single entities, 20

17 G. Peano, op. cit. p. 160. 18 All that is worth, obviously, also for the negation –� of the same relation. 19 The said decomposition, actually, had already been used previously, but here it is justified since it is a particular case of the same theory.


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unless as elements of a proposition like x � a, through it we can compare two single entities. The single entities anyhow, need to be connected, above all in the prospect of a presentation related to the theory of numbers that remains the main aim of the Peanian work. To say, for example, x = 0 is possible on condition that the symbol is doubled = in ��: x � 0, indeed, could have no meaning, as 0 represents the class null, such as conceiving x a class, being an element included in N series of the numbers.21 This “expedient” aims to get the possibility of comparing elements of a class, or, generally, to admit a connection (of order for example) between the same ones, with no need to consider a part a theory of connections, assuming the usefulness for the classes, sometimes considered as single entities. As an integration of the theory, Peano introduces other three symbols: |,

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↑,

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↓ , which respectively indicate, inversed, all, some.. The symbol | has not to be misunderstood with : the idea of reversal expressed by the two, is not the same, since y � x . = . x = �y, and y � � | x . = . x � � y do not represent the same proposition. In one case the formula expresses the reversal of a function, in the other one of a connection that, though, closely related, are not the same concept: a connection between x and y, creates a link or, more precisely an element x belongs to a class of corresponding �y.22 About the other two symbols, they are aimed to limit the value of a connection towards all or some of the elements classes concerned with the same connection: 20 Aristotle already excluded that it could be dealt with individuals in the presentation of the syllogism: that is why, in fact, one always speaks of classes (concepts). 21 It is obvious that here we speak about the characterization of the series as it had been introduced by Peano, and of which we will discuss later on: the idea of number as class will always be refused or not much considered by the Italian mathematician. 22 It is however always possible to connect the two meanings of the reversal by observing that the inverse one of a function is equal to (��) |.


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1. x � � v . = : y � v . y . x � y ;

2. x �

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↓ v . = : y � v . x � y . - =y ; 3. u ↑ � y . = : x � u . x . x � y ; 4. u

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↓ � y . = : x � u . x � y . - =x . The meaning of these formulas is : x can link with every element of the class v, x is linked with some elements of class v, and so on. Their use is of intensifiers23 both existence and universal: a

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↓ � b signifies ab - = , a ↑ � b signifies a b). They are, for example, other ways of writing basic elements of a syllogism (particular and universal affirmative).24

1. 1. 2. After this introduction about logic and its applications, the first edition of Formulario continues with the chapter concerning Logique mathématique: within this writing there are all the formulas that set up the rules of calculation, that is the propositions whose combination allows proofs.25 The calculation develops from the primitive concepts of deduction “ ”, conjunction “

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∩” and negation “-” (besides the inevitable symbol �, and the connected idea of class), with the conventions of using Latin small letters (a, b, c...) to indicate propositions or

23 The quantifiers, in effects, never had up to now not been used by Peano, but the convention to affix as a pedix to the symbols of deduction and equality to simulate the universal quantification, as pointed out previously. 24 G- Peano, op. cit. p. 163. 25 G. Peano, Logique mathématique, in G. Peano, Formulaire de Mathématique, I, Turin, Bocca Frères 1895, pp. 1-7; 127-129, repr. in Opere Scelte, cit. vol. 2, pp. 177-188. See p. 185.


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whatever classes, on the use of points (parenthesis) and the symbol

expressing the substitution of variables

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a', b'a, b

.

The theory is the same in Formole dated 1891 with the exception of a little difference. The first is the symbol of identity (=), considered determined according to the original concepts:

a = b . = . a b . b a. This proposition represents the definition of the sign “=” by means of “ ” and “.”. Note that such a formula is the definition of “=”, but we need to point out a difference: the symbol here appears two times, but only in the first occurrence it indicates the equality among classes or propositions.26 At the same way there is a definition of the sign “ ”, by means of the proposition

a-b = ,27 reducing 28 the original symbols into three (in comparison with the above four Formole). The original propositions of the real calculus are twelve, introduced in Peano’s writing dated 1891, which remain unchanged. The exposition consists of five paragraphs, the first given up to the original symbols “ ” and “

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∩” , the second to “

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∪” and “-”, the

26 We will discuss about this in detail below. 27 The concerned proposition existed already in the Formole, but here the definition of the absurdity symbol is considered, even if in a non completely explicit way. 28 The “economical” criterion of the Peano’s formulation also emerges here clearly: in the review to the work of Frege Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, of 1895, he puts in evidence just the merit of its particular point of view (apart from, obviously, to reassert the greater simplicity of the symbols that he used). See G. Peano, Recensione, in “Rivista di matematica” vol. V (1895), pp. 122-128, repr. Opere Scelte, cit. vol. 2, p. 189-195.


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third 29 to the introduction of the sign “ ” and the fourth to classes: the last one follows the same contained in Formole, except some slight changes in their disposition. The fifth paragraph, instead, is completely new: it presents the propositions regulating the calculus of functions (multiplication, power, sum, etc.).30

1. 1. 3. As it was said before, the idea of reducing the necessary ideas to express all the logical propositions is a continuous thought in Peano: in 1897 he will publish the paper Studii di logica matematica, whose aim has already been exposed.31 The original ideas has changed them into six: the letters a, b,...x,y,z, signifying any objects (classes, propositions, elements), the points or parenthesis, the idea of class (K), the concept of belongings from a single entity to a class (x � a), the convention adopted for the symbol of the deduction (apparent variables), the idea of a logic conjunction (p

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∩ q or pq) and of an ordered couple (x ; y), here is exemplified for the first time.32 Starting from these ideas, therefore, we can draw all the others with suitable definitions, which take more and more the characteristic of brief ideas someway expressible.

29 In this paragraph it is also made reference to the symbol °, indicating the exclusive sum: also accepting it, already in the previous works, among the logical signs, Peano will never make use of it, and is therefore possible to omit it from the theory: it, however, means a ° b = a-b

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∪ b-a. 30 The symbol used for them, however, is still \: this booklet, in fact, even if the continuation of the Notations, had been already published, a year before, on the “Rivista di Matematica”. 31 “Così, con definizioni opportune, si può ridurre le idee di logica ad un numero sempre più piccolo di idee fondamentali, o idee primitive... In questa Nota tratto della riduzione delle idee di logica al minimo numero”, G. Peano, Studii di logica matematica, in “Atti della Reale Accademia di Torino”, vol. XXXII (!896-97), pp. 563-583, repr. in Opere Scelte, vol. 2 cit., pp. 201-217. See p. 204. 32 See G. Peano, op. cit. pp. 204-205. It is important to observe that these ideas in issue are extremely simple, representing, in effects, a level of “still greater primitivity” than the common language.


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So we have the double meaning of the symbol “ ” (among propositions and classes), the symbol of inversion “ ”, equality “=”,33 the sign “

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∪” (logic sum), the negation “~”, the absurd “ “, or“ = ”, the sign “� “ (class of a single element) and its inversed “ ”, the sign “

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∃”(exists), new, and the correspondence “ ”. The method adopted and the observations related to it make this written text particularly important, maybe the most explicit of the Peanian reflection. It seems useful to follow some reasoning, beginning from the sign “ ” : it is taken as original as it deals with propositions, but defined in the calculus of classes as index “ x “. The aim is obvious : “il segno fra classi si legge è contenuto, mentre fra proposizioni si legge si deduce. Il fatto che esso si può leggere in più modi non prova che esso abbia più significati, ma solo che il linguaggio ordinario ha più termini per rappresentare la stessa idea”.34 This observation brings a very important consequence: the two calculations closely connected become one single entity. The same reasoning concerns the other operators used in the calculation , both primitive and derivative.35 The sign “=” in fact, can be defined in several ways, it depends on its use,36 anyway Peano sets up a general law

x = y . = : a � K . x � a . a . y � a.

33 For the first time here is explicitly discussed the distinction previously mentioned between = and =Def (equal for definition), i.e. the case in which the = makes part of the same definition. 34 G. Peano, op. cit. p. 208. 35 That does not mean, of course, that only the two calculi are reduced to one, neither that “translators” sign � and its inverse are abolished, which are on the contrary analysed in a deeper way, but simply that the two calculi, as said, are two faces of a same medal under all the points of view. 36 See G. Peano, op. cit. p. 215.


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About it he says : “Ma resta a far vedere come le varie definizioni particolari entrino in questa”,37 even if, his merit has been to discover a definite universal law for equality. Also the sign of the logic sum presents an innovation: in the previous notes its definition was a b = -[(-a) (-b)], adopting De Morgan’s Laws, while in this context the sign becomes a, b � K . . a

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∪ b = (c � K . a c . b c . c . x c). 38 The logic sum expresses the set up of single entities belonging to any class c containing the classes a and b : therefore the simplification obtained is remarkable, since laws (De Morgan’s) do not enter in avoiding the introduction of other signs as the negation. The negation itself , in fact, in this work results definite and no more primitive and this happens through the sign :

a � K . . ~a = (b � K . a

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∪ b = V . b . x � b).39 To the class ~a, belong all the elements of a class b, that added to a, produces as a result everything. In these cases, we have only to exchange the terms entering into a definition, such as to get several possibilities : the criterion of choice is purely arbitrary as the only reference lies in its simple use. Following this direction we can understand the choice of defining first the symbol “= ”, rather “ “ :

a � K . a = . = : b � K . b . a b.

37 Like the previous formula, G. Peano, op. cit. p. 215. 38 See G. Peano, op.cit. p. 211. With this, however, he does not intend to refuse the previous definition, only to say the greater simplicity of this. It, moreover, is set up in the calculus of the classes but, as in previous observations it is true also in the propositional calculus. 39 G. Peano, op. cit. p. 212. The sign V, representing all, or the universe of reference, is also defined, but it turns out useful, practically, only in this definition.


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If “ ” represents the absurd, a = expresses the empty class: in this illustration it appears as a class inside the others. There are a lot of advantages in the reduction of the calculation, like the possibility of considering the opposite sign as a term of comparison.40 In this work, the Author introduces new symbols that complete the theory more in depth : it is the case of “;” , “ ” and “

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∃”. The symbol “ “ represents the inversion of �, and it is defined :

�x = (y = x). as There appears the definition of its inversion:

a � K .

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∃ a : x, y � a . x, y . x = y : : x = a . = . a = �x.41 But the Author realizes a difficulty: he notices how impossible is to define simply a, in fact this is allowed only as equality x =

a. The problem can dealt with by means of a technical device, although it remains unsolved because it does not imply the need of a defining process. The group a, represents the elements (all equal) belonging to a, indeed a single element x = a.42 Through this symbol it is possible the definition of suspended before:

K (b � K . b a b).43

40 See G. Peano, op . cit. p. 211. 41 Op. cit. p. 215. 42 The same was true also for �. 43 Also in this case the definition is new, but it turns out, however, to be an analogous case to a –a, previous definition. See G. Peano, op. cit. p.216. With this symbol, moreover, the negation can also be defined: the Author, however, does not incline for any of the two procedures.


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In the definition of , appears also the new sign

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∃: it means “exists”, never adopted before: a � K . : a . = . a ~ = , a simplification of ~ = , to indicate a class which is not empty,44 a sign that will be used largely in the perceptive of a greater simplification of calculus. The last observation concerns the symbol “;” : (x ; y) = (a ; b) . = . x = a . y = b.45 It represents an ordered couple, whose elements belonging to different classes keep the same ordinal position: the concept has a fundamental importance, above all to demonstrate the valuable principles of importation and exportation of hypothesis, introduced for the first time in Notations, but in this case they receive a justification.46 The analysis of this paper,47 not included in the Formulario, becomes necessary in relation to the reduction of the ideas about

44 See Op. cit. p. 215. In the paper Delle proposizioni esistenziali he, explicitly, says “l’uso o non di simboli per indicare esiste, è questione di comodità pratica, non di necessità logica”, G. Peano, Delle proposizioni esistenziali, in International Congress of Mathematicians, Cambridge, August 1912, vol. 2, pp. 497-500, repr. Opere Scelte, cit. vol. 2, p. 388. 45 G. Peano, Studii di logica matematica, in “Atti della Reale Accademia delle Svienze di Torino”, vol. XXXII, 1896-97, pp. 565-583, repr. in Opere Scelte, cit. vol. 2, p. 214. The symbol is assumed as primitive, but the equality of two pairs is defined, which turns out, however, to be of remarkable importance in order to render the idea of what a ordered pair is 46 See G. Peano, op. cit. p. 214. The concept of ordered pair, moreover, will be the basis of the new theory of the relations exposed by Peano in some successive works. 47 The theory of the functions is also present here there are not, however, relevant innovations, but the substitution of symbol sim with rcp, so that it can be omitted. See op. cit. p. 216.


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logic:48 these ideas will be accepted and included in the edition (the second one) of the Formulario dated 1897. 1. 1. 4. The edition contains, actually, in its first part, a

presentation of the logical theory according to Peano, but this time starting from the calculation of classes not from propositions like in the first edition : however from the new calculation it can be drawn out the propositional one. The calculus derives from the primitive ideas, with the number nine described through a common language in fourteen propositions representing: the concept of class (K), belongings (�), the use of points (parenthesis), variables (a, b...x, y, z), with the convention to index all the apparent variables, the idea of , logic multiplication (

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∩), definition, couple (ordered, now symbolized by means x, y)

and negation (-), the use of for the substitution of variables.49 The theory is taken from previous Studii, with the only integration of the negation, considered as primitive and not specified in the work. The real calculation implies about two hundred and sixty propositions, divided in primitive and derivative (more than forty definitions). The primitive propositions (Pp), basis of the calculus are eleven: 1. a � K . . a a; 2. a, b � K . . ab � K; 3. a, b � K . . ab a; 4. a, b � K . . ab b; 5. a, b � K . a b . x � a . . x � b;

48 Issue, however, always opened: “Si può cercare di ridurre ulteriormente il numero delle idee ritenute primitive, ovvero tentare altre vie”, Op.cit. p. 217. 49 See G. Peano, Logique mathématique, in Formulaire de mathématiques, t. II, §1, Turin, Bocca Frères, repr. in Opere Scelte, vol 2, cit. pp. 218-281. See p. 221.


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6. a, b, c � K . a b . b c . . a c; 7. a, b, c � K . a b . a c . . a bc; 8. a, b, c � K : x � a . (x, y) � b . x, y (x, y) � c : x � a .

x : (x, y) � b . y . (x, y) � c; 9. (-a) � K; 10. -(-a) = a; 11. ab c . x � a . x -� c . . x -� b.50 The listed formulas set up a different organization of the calculus, in comparison with the other writings, in which it is evident the Author’s detachment from Boolean procedures. The Pp represent real laws, through their application it is possible to demonstrate all the other propositions: the 3 and the 4 convey the law of simplification (Simpl), the 6 the syllogism (Syll), the 7 the law of composition (Cmp), the 8 the law of exportation (Export) which modified, becomes the inversed law of importation(Import). Joined with these he presents and demonstrates: 1. ab c . = . a -c -b or the similar a b . = . -b -a;

2. (x, y) � b . x, y . x � a : = :

€

∃

€

yε (x, y) � b . x . x � a; representing the laws of transposition (Transp) and elimination (Elim).51 The laws listed are the principal to regulate the calculus: obviously they are not the only ones. Every proposition can be used as a law because of the lack of a formal procedure to get the demonstrations, or better, to make a distinction between the Pp (axioms) and derivates.

50 G. Peano, op. cit. p. 221 follow. To these formulas there should have been

added � K, but the Author does not consider it explicitly like Pp. 51 See G. Peano, op. cit. p. 221 follow.


19

The only laws (principles), considered external are the replacement of the apparent variables and the reversal of primitive propositions.52 As concerns the other symbols and ideas to be used in the calculus, they have been defined as equally as the Studii: it is given the generic definition of equality, proposed in the previous writing and here accepted completely. About the symbol , instead, is proposed the definition deriving from De Morgan’s laws, considered as primitive also the sign “–“. Also the symbols “= ”, “

€

∃”, ” ”, “� “, “ ”, are defined as similar as the Studii, even if, in relation to “ ”, it deals with a derivation.53 For the first time we have the definition of signs as K , , : u � K . . K u = K

€

xε [x u] u =

€

xε [

€

∃. u

€

yε (x � y) ] u =

€

xε [ y � u . y x � y ], that designate the class of u (subclasses of u), the sum and the logic product of the same subclasses belonging to u.54 The theory of functions expressed in the last group of propositions within the same writing, does not reveal any variations and it is presented through the notations already exposed in the other works, that shows the completeness assigned to this theory. 1. 1. 5. This does not mean that the changes about the calculation of classes or of propositions convey an incomplete theory; the changes brought have to be considered attempts to set the theory, making it use easier as the Author will say, in his writing Sulle formule di logica, appeared in the “Rivista di matematica” in 1899. 52 Op. cit. loc. cit. 53 Ibid. The discussion about this sign, has always remained suspended. 54 See Op. cit. p. 235. The symbols now introduced, have already been discussed, however they have not been defined, rather used as primitives.


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In this brief paper, in giving a justification to all the changes between the first and the second edition of the Formulario, he has pointed out how they must be considered a different conception of the calculus foundation: the propositions that can be demonstrated remain the same, are the primitive ones which changes, or better, the system of propositions that causes the order changes not so important. To this aim it is worthy to mention an important observation about the formula a . a b . . b 55 that right within the calculation of propositions, was impossible to interpret in the calculus of classes:56 the problem is solved through the Pp 25 of the second tome of the Formulaire de mathématique (1897): a, b � K . a b . x � a . . x � b, which is a syllogism, making possible such an interpretation.57 1. 1. 6. The possibility of obtaining different formulas of calculations, with a change in the order of Pp, is brought to the extreme consequences in the last work by Peano given up to logic: the third edition of Formulario, published in 1900. Above all the annotations: they get their definite form: “Cls” (class), “ ”(inversion of “�”), “

€

⊃” (deduction), “ ” (inversion

See G. Peano, Sulle formule di logica, in “Revue de mathématiques”, vol. VI, 1896-1899, pp. 48-52; repr. in Opere Scelte, vol. 2 cit. pp. 282-287, see p. 283. 56 Op. cit. loc. cit. It was this, in fact, the formula that prevented the complete translation, signalled, as said by Beppo Levi: however, there would be other “introducible” propositions, that are, transformed with different devices, so as to go around the problem. 57 G. Peano, Logique mathématique, repr. in Opere scelte, vol. 2, cit., p. 222.


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of �), and the symbols related to the definitions with Df. to indicate a possible definition.58 It is the explicit indication of possible definitions to grant exactly the possibility of obtaining equivalent theories: every possible definition in fact, could be a basic definition for another type of calculation, without causing a radical change in the theory itself.

The writing is divided in fourteen paragraphs, illustrating the theory concerning the different logic symbols: in the first there are the signs “Cls”, “�”, “ ”, “

€

⊃”, “;”, “=”, “ ”, similarly to what he had done in the previous edition of the Formulario.

The principal demonstrative laws are the same of the previous edition but among them we find distributive, associative, commutative and operative properties, that is Distrib, Assoc, Comm, Oper, considered as expressions of a reasoning.59

Paragraph two instead, contains two expressions of logic sum, the first defined in Studii, the other through De Morgan’s laws: both definitions equivalent bringing only changes in the order of propositions.60

The third paragraph presents the sign : its definition is x (a � Cls .

€

⊃a . x � a), which means null class.61

For the first time it is considered independent from its combination with “=” and is accepted as Pp � Cls; in this work Peano uses some propositions without saying if they are Pp, and this has caused doubts and uncertainties.

The next paragraph deals with the negation which is seen as primitive, like in the other edition of Formulario, but close to this interpretation the mathematician presents other four ones, (possible 58 See G. Peano, Formules de logique mathématique, in “Revue de Mathématique”, vol. VII, 1900-1901, pp. 1-41, repr. in Opere Scelte, cit. p. 304 follow. The other signs remain unchanged. 59 G. Peano, op. cit. p. 322. 60 See op. cit. p. 333, follow. 61 See op. cit. p. 338.


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definitions), where – becomes a derivate sign, without detailing the question.62 The paragraphs five, six and seven deal with the signs “ ”, “�”, “ ”, whose theory proves similar to the others.63

In the paragraph eight appears a new sign ” ”, symbolizing a couple where the elements belong to two different classes: u, v � Cls .

€

⊃ . (u v) = (x ; y) (x � u . y � v).64 through which it will be after to define the couple “;” of two classes, sketching a new possible theory.

The last paragraphs are given up to the theory of functions whose signs, presented in the paragraph nine, remain “ ” e “ ”, and the same happens with their use: however in the next paragraph is defined the symbol “|”, a sign of inversion, similar to the one adopted for connections, but meaning something different. In this presentation it takes on the meaning of inversion: a, b � Cls . u � b a .

€

⊃ . (ux) | x = u.65

This symbol differs from the classical sign of inversion since here we have to go back to the function which has produced a determined expression: if, for example, A represents an expression deriving from the application of a function, A | x indicates the function generating A.66

The sign of inversion is introduced in the last paragraph and takes the form -1.

62 See op. cit. p. 340 follow. For the first time from Operazioni della logica deduttiva the Author returns here on the issue of the development of one logical equation, a typical Boolean procedure, practically abandoned. 63 See op. cit. p. 347 follow. 64 Op. cit. p. 352. 65 Op. cit. p. 356. 66 Op. cit. p. 356. They are presign functions: that is true, however, also in the event of postsign functions.


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In the paragraph marked with eleven are presented the signs e , independently from K , , , but is a general case:

u a = y [

€

∃a x (ux = y)], used to indicate a function u of a determined class a.67

The theory of functions, in paragraphs twelve and thirteen, are included definitions of rcp, sim, idem, already contained in the previous editions of Formulario (including idem, that represents the identity idem x = x, functions, always similar, like+0, /1, ect., had the form id): the innovation is given by the signs Variab, F and Funct.

The sign F expresses a definite function: u, v

€

ε Cls . f

€

ε v u . x

€

ε u .

€

⊃ .

€

⊃ . vFu = g {

€

∃v u f [g = (f ; u)]},68 a couple formed by the mentioned function and the reference class, exactly Variab, a class of values belonging to a function.69

Similarly Funct indicates all the functions F whose extremes are both classes: uFv where u and v, are, exactly classes:70 it is obvious that these functions are peculiar cases applied to a determined class, so they have the same properties of general functions (sim, rcp, idem).71

The Formulario edition can be seen as the last systematic work by Peano about the logical theory: two more editions will be published after this one but the theory will not change any more, definitely stating the ideas expressed in 1900 edition.

The Peanian research turns to other arguments, the first is the creation of a universal language of communication able to knock 67 Op. cit. p. 357. The particular case mentioned above, in this paper is actually omitted, as it can be reduced, to the general case dealt with here. 68 Op. cit. p. 359. 69 Ibid. 70 Op. cit. loc. cit. 71 Op. cit. p. 360.


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down the linguistic barriers among people, making easier the exchange of information: the last edition of Formulario, in 1908, is written in a language which is impossible to understand. 1. 1. 7. This does not mean that the Italian mathematician abandons the logic research forever: in a series of papers, written between 1906 and 1916, he will study logic again , with no change of the theory contained in Formulario. In 1906 he publishes the note Super theorema de Cantor-Bernstein et additione, written in Latin sine flexione, where he deals with the question of proving the theorem by Cantor-Bernstein, about the possibility of deducing among cardinal numbers, from x ≤ y e x ≥ y, x = y.72 While we leave a part this demonstration aside which gains more importance for arithmetic than for logic, we will analyze two observations included in Additione, related to Zermelo principle (multiplicative axiom) and antinomy by Richard, two logical questions discussed for long time.

As concerns the axiom by Zermelo, the Peanian view is clear: the Italian logician refuses it as a reasoning though he admits its possible use in ambiguous cases.73

The Author considers the syllogism as a principal form of proof: in this case the reduction of the multiplicative axiom to

€

∃a (1) x�a .

€

⊃ . p (2) (1), (2).

€

⊃ . p 74

72 G. Peano, Super theorema de Cantor-Bernstein et additione, in “Revista de mathematica”, t. VIII, 1902-1906, pp. 136-157, repr. in Opere Scelte, cit .vol. 1 p. 337, follow. 73 Op. cit. p. 346 foll. The judgment on the said axiom had already been expressed in Demonstration de l’intégrabilité des equations differentielles ordinaries, in “Mathematische Annalen”, Bd. XXXVII (1890), pp. 182-228, repr. in Opere Scelte, cit. vol. 1 p.150 “on ne peut pas appliquer une infinitè de fois un loi arbitraire, avec laquelle à une classe a on fait correspondre un individue de cette classe”.


25

€

∃a (1) x�a .

€

⊃ . p (2) (1), (2).

€

⊃ . p 75

that is a wide form of syllogism, grants that confirms the practical usefulness of the axiom.

That does not mean deleting it a priori: it is useful in some cases, and he will accept it in a review about Principia Mathematica by Russell and Whitehead, giving also a definition k � Cls Cls .

€

⊃. Select k = (UkFk) f (a � k .

€

⊃a . f a � a). 76

As concerns the antinomy by Richard, about the possibility of defining numbers in an ordinary language , he will be the first, without receiving any attention, to recognize the pure linguistic characteristic of the opposition.77

The review about Principia, instead, retains a great importance because within it we can find a new theory of the connections, no more considered as functions, rather as classes of couples which have been considered a last integration to the Formulario.

To tell the truth, in Sulla definizione di funzione, published in 1911, he had defined a connection in the following way: Relatio = u [

€

∃ (a ; b) (a, b � Cls . u � Cls ‘ (a b))],

74 G. Peano, Super theorema de Cantor-Bernstein et additione, cit, in Opere Scelte, cit. p. 346. 75 G. Peano, Super theorema de Cantor-Bernstein et additione, cit, in Opere Scelte, cit. p. 346. 76 G. Peano, Recensione Whitehead-Russell, Principia Mathematica, in “Bollettino di bibliografia e storia delle scienze matematiche”, Loria, 1913, pp. 47-53, 75-81, repr. in Opere Scelte, cit. vol. 2, p. 400. 77 G. Peano, Super theorema de Cantor-Bernstein et additione, cit, in Opere Scelte, cit. p. 358.


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having in mind the idea of a connection as the set of elements in couples belonging to two classes specified with a and b.78

The prevalence of a connection becomes as follows: u � Relatio .

€

⊃ . Dominio u = x [

€

∃y (y ; x � u)].79

Through these definitions, it is possible also to get a new idea of function: Functio = Relatio u [ y ; x � u . z ; x � u .

€

⊃ x, y, z . y = z ]. 80

These definitions express the attempt to reduce the theory of connections, laid by Russel as the foundation of his mathematical theory, following the Peanian theory of classes, avoiding the adoption of another one.

All these observations will be studied again, till the extreme consequences in the review dated 1913.

He has the aim of interpreting the Russel theory and so introduces in this writing new symbols: Dyade, representing the pair of elements, which is the basic principle of the theory, E1 , representing the first element of Dyade itself, E2 , the second element, C, converse of a Dyade (if x ; y, means y ; x), to indicate the product of two connections (x ; y and y ; z = x ; z), and lastly, representing the sum and the product of the under classes belonging to a broken class.81

In this way he can explain the whole theory of connections without considering it independent from the class theory. Once analysed works that have set up the logical system, the base of mathematic foundations, we can deal with this topic: the 78 G. Peano, Sulla definizione di funzione, in “Rendiconti della Reale accademia dei Lincei”, Serie 5, Vol. XX, 1° sem. 1911, pp. 3-5. Repr. in Opere Scelte, cit. vol. 1 p. 365. 79 Op. cit. loc. cit. 80 Ibid. 81 G. Peano, Recensione Whitehead-Russell, Principia Mathematica, cit., in Opere Scelte, cit. vol. 2, p. 389 follow. The last two signs had been, in fact, omitted in the last editions of Formulario.


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mathematical sciences starting from arithmetic, are in Peano view some “extensions” of the logic.

He creates new symbols to express the mathematical principles and is able to found the mathematics itself through the logical rules, only by widening its area.

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