Algebraische Betrachtungen zu den Aristotelischen Syllogismen by H. GerickeReview by: Klaus HärtigThe Journal of Symbolic Logic, Vol. 22, No. 3 (Sep., 1957), pp. 308-309Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2963626 .

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308 REVIEWS

held in Paris in 1900, Vacca acted as an intermediary between Peano and Russell, and at the same time he brought his own contributions to the development of logic and mathematics, especially in the field of historical research.

Vacca's philosophical ideas on mathematics were not very modern or consistent. He was a Platonist, in a metaphysically engaging and somewhat naive sense of the word. This position would appear not to be in accord with his researches on the tech- nical sources of many mathematical concepts.

Some of these researches are seen in the second of the present essays, "Mathe- matics and technique: The origin and development of mathematical concepts." This is a collection of historical notes, which is indeed far from systematic; there are, however, some interesting remarks. On the same level is the first essay, "Why did science not develop in China?" - on which theme there is also some discussion by a number of Italian scientists at the end of the volume.

The third paper ("Mathematical logic and logistic - On the postulates of arithmetic and their compatibility") is the text of a lecture held in Rome in April, 1945, and is of a more direct concern to this JOURNAL. The first part has a prevailingly historical interest, but it is based on a distinction between logistic and mathematical logic which the author wishes to trace back to Peano. Logistic is "a theory of numbered things and not of numbers in themselves"; it considers "any numbered or numerable entity as a number, and deals with these entities as arithmetic does with numbers." Mathematical logic, on the other hand, "sets out essentially a minimum problem, namely the problem of formulating and enunciating the smallest and simplest system of logical notions and propositions necessary and sufficient to represent symbolically mathematical truths and their demonstrations, that is their connections and laws of derivation."

The second part of the paper is devoted to some modifications that can make the postulates of elementary arithmetic more obvious and acceptable. By means of two new definitions, Vacca succeeds in reducing Peano's six postulates to three. He defines a "chain" as "any class that contains zero among its members, and, if it contains a member, contains also its successor"; then, he enunciates the following two postu- lates: 1. Number is a chain. 2. Any chain contains all the numbers. The author claims that they correspond to Peano's first four postulates. As a second step, he defines "the class of numbers between zero and any number a" as such that "if a = 0, then the class comprises the unique individual zero; while the class of numbers between zero and the successor of a comprises besides the numbers between zero and a also the successor of a." The third postulate is enunciated as follows: 3. If a is a number, its successor does not belong to the class of numbers between zero and a. This postulate includes Peano's fifth and sixth ones.

A major value of this new set of postulate is, according to the author, their greater simplicity and the obviousness of their compatibility. Only two primitive ideas are necessary: zero and successor. Number is the class defined by the three postulates, and precisely as an unlimited (post. 1), unique (post. 2), and open (post. 3) chain. 'The existence of this chain appears evident to our topological intuition of the real chains that we can conceive in space. So arithmetic presents itself as the simplest and most elementary abstraction of the notion of unlimited, unique, and open chain" (p. 40).

The last part of the paper gives a translation of Vacca's definitions and postulates into Peano's symbolic notation. DOMENICO PARISI

H. GERICKE. A lgebraische Betrachtungen zu den A ristotelischen Syllogismen. Archiv der Mathematik (Basel und Stuttgart), Bd. 3, Heft 6 (fur 1952, erschienen 1953), S. 421-433.

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REVIEWS 309

Die Studie ist hauptsachlich gedacht als "Beispiel - vielleicht eines der nachst- liegenden - einer Relationenalgebra."

Verf. charakterisiert eine Menge 9 von "Begriffen" S. P, M, (mit einer reflexiven und transitiven Relation ?) als Booleschen Verband, wobei er Vereinigung, Durch- schnitt und Komplement von Begriffen, die Elemente 0 ("nichts") und 1 ("alles") sowie = und < einfiuhrt. Im Booleschen Verband der "Urteile," d.h. der zweistelligen Relationen zwischen Begriffen, werden die (Halb-)Ordnung (x ? y), das Produkt (die Verkettung) xy, der Durchschnitt x A y etc., das Einselement r (= 93 x 93) und das Null element / wie ublich erklart, auch die "Konversion": SxP +-* PxS. Die klassischen vier Urteile a, e, i, o werden in bekannter Weise definiert, z.B. SaP als 0 < S ? P < 1. Sie werden durch a und 6 zu dem System X. erganzt, dieses zu dem System X. = [a, e, i, o, a, 6, u, v], dieses durch r, t, g, k, g, k zu E Definitionen:

SuP+-+S< 1AP< 1ASUP= 1, SvP+-O < SAO < PASUP < 1,

SgP4-*0 < S = P < 1 oder: aAd =g

SkP4-.0 < S = P < 1 oder: eAu = k.

Der Syllogismus "der ersten Figur" VSVMVP(SxM A MyP -* SzP) ist mit xy ? z equivalent; die andern drei SchluBfiguren sind bei Vorhandensein von a und 6 wegen

= e und i = i entbehrlich. Hauptinhalt der Arbeit ist das Aufstellen der Multipli- kationstafel fur 114, das sich wesentlich auf die Umformung E8 = [a, ak, i, ik, kak, ki, ka, kik] stutzt. Gericke schlieBt durch eine Zusatzforderung (Z) die Existenz von "Individualbegriffen" aus - VM(O < M -* 3L 0 < L < M) - und erzwingt dadurch die Abgeschlossenheit von 1,4 bzgl. Multiplikation. (Aquivalent mit (Z): ai = r.) In der Multiplikationstafel fur B - sie nennt in 32 Fallen das Ergebnis r - liest man z.B. an ia = i die Guiltigkeit der Syllogismen Darii, Datisi, Dimatis und Disamis ab. "Ex mere negatives nil sequitur" ist in E8 falsch, denn ee = v. In der Teil-Tafel fur E. treten, aul3er diesen sechs Relationen selbst, v und r auf. Erst aus der (nur einem

Diagramm zu entnehmenden) Falschheit der Satze

v ? a, e, i, o, d, 6; a, d, i ? e, o, 6; e, o, d6 ? a, a, i usw.

ergibt sich die Ungultigkeit aller 232 als ungultig bekannten Syllogismen. Ref. bemerkt, daB beispielsweise die Bedingung / < eAv notwendig und hinreichend

dafur ist, daB ein Boolescher Verband die Struktur der klassischen Syllogistik hat (namlich genau die bekannten 24 gultigen Syllogismen als gultige besitzt). So erscheint Forderung (Z), die die Existenz unendlich vieler disjunkter Begriffe impliziert, als erhebliche Beschrankung der Allgemeinheit.

Zumindest A. De Morgan hatte wohl zitiert werden mussen. (In 202 sowohl E als auch E8 . Am Schlu3 von 203 eine iibersichtliche Multiplikationstafel fur das Relationen- system [g, aA6, aAO, k, eAv, UAi, ZAOAVA6]. In 206: "... the ordinary syllogism being one case ... of the composition of relations.") KLAUS HARTIG

LEWIS CARROLL. Symbolic logic. Part I. Elementary. Reprint of the fourth edition (674). Berkeley Enterprises, Inc., New York 1955, xxxi + 203 pp.

The method of diagrams which Carroll develops in this book, and his ingenious examples of sorites (or as he calls them, "soriteses"), are well known and require no comment.

The propositional forms with which he deals are (i) negative and affirmative "existen- tials," asserting in effect that some class is or is not null, and (ii) "propositions of relation," i.e., propositions of the forms "Some S is P," "No S is P," and "All S is P." The latter are respectively equated with "SP's exist," "SFP's do not exist," and "SP's exist, but SP's do not," and Carroll works out amusingly and skilfully the consequences of these decisions and of some alternatives. He has no separate partic-

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