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Logica dimostrativaS. Bozzi aa Department of Philosophy , Università degli Studi , via Festa delPerdono, 7 – 20122, Milano , ItalyPublished online: 15 Mar 2013.

To cite this article: S. Bozzi (2013) Logica dimostrativa, History and Philosophy of Logic, 34:2,183-187, DOI: 10.1080/01445340.2012.760432

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HISTORY AND PHILOSOPHY OF LOGIC, 2013

Book Reviews

Gerolamo Saccheri, Logica dimostrativa. Edited and translated by Massimo Mugnai andMassimo Girondino. Pisa: Edizioni della Normale, 2012. Vol. I. LVII +295 pp. Vol. II.II + 168 pp. ¤40.00. ISBN 978-88-7642-437-3.

Reviewed by S. Bozzi, Department of Philosophy, Università degli Studi, via Festa delPerdono, 7 – 20122 Milano, Italy.

© 2013 S. Bozzi

The two volumes under review contain, respectively, an Italian translation by MassimoGirondino of the 1701 edition of Saccheri’s Logica Demonstrativa and a photoreproductionof the Latin text. The translation is prefaced by an ample introduction by Massimo Mugnai,dedicated to the context of Saccheri’s book and an evaluation of its place in the historyof logic. The translation, which generally is very good, is followed by a large set of notes(due to Mugnai) illustrating the results and notions of traditional logic mentioned in the textwith many references and explanations concerning the most difficult points. The latter partof the volume contains appendices relative to Saccheri’s life and some texts (in Latin) bySaccheri himself on logic and metaphysics. This is not the first modern Italian translation ofthe Logica Demonstrativa; in 2011 another edition of the Latin text with a translation and arich introduction had already appeared, due to the late Corrado Mangione and Paolo Pagli(Saccheri 2011). The criteria adopted in the translation, the notes and introduction of the twoworks are very different. The aim of the edition by Pagli and Mangione is the underlining ofthe novelties in the Logica Demonstrativa and its connection with the geometrical resultsby Saccheri on the foundation of geometry. In particular, large space in the Introductionby Mangione and Pagli is dedicated to the role of the celebrated consequentia mirabilis, if(if not-p then p) then p, in the Logica Demostrativa. While in his Euclides ab omni naevovindicatus (1733), this thesis is the logical cornerstone of the purported demonstrations ofinconsistency for the non-Euclidean hypotheses on parallels, in the Logica Demonstrativathe role is very different and Mangione and Pagli emphasize the interest of its combinationwith the recourse to self-reference in the various demonstrations of non-conclusivity forsyllogistic pairs of premises presented by Saccheri.

Although he concurs with Mangione and Pagli in the valuation of the relevance of thiscombination of self-reference and Saccheri’s thesis, Mugnai’s approach, in the Introduction,in a sense is diametrically opposed to theirs. Mugnai puts a great emphasis on Saccheri’sdebts to the medieval tradition of Summulae logicae, and in the notes the connection withscholastic logic, medieval and contemporary, is amply documented. In particular, Mugnaiadvances the thesis that a very plausible source for the Logica Demonstrativa is a workby the Jesuit Honoré Fabri published in 1646 (Fabri 1646). According to Mugnai (andGirondino, whose dissertation published in 2009 is amply cited by Mugnai) the peculiarapproach adopted by Saccheri in his exposition, with its explicit reference to the euclideanmodel (definitions, axioms, hypotheses and demonstrations) follows the method adoptedby Fabri in his lessons. This fact is particularly interesting since Saccheri underlines thisaspect of his work already in the title – Logica Demonstrativa – where the qualification‘demonstrative’ refers to the method of justification adopted in the exposition, not to the

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logic exposed. This is a clear indication that Saccheri saw in this style of exposition thechief novelty of his work.

Professedly, the Logica Demonstrativa is an introduction to logic covering the usualthemes present in this literature during the seventeenth century, particularly in the Jesuitscholastic tradition: term theory, supposition, the variety of propositions and their mutualrelations (large space is devoted to opposition and equipollence), consequence theory, andcategorical syllogisms with an ample treatment of the rules regarding them. Finally, ananalysis of modals. All this material is presented in the first part of the book, titled Analyticaprior, in which the peculiarity of the approach adopted by Saccheri is more apparent. In thesecond part, Analytica posterior, the themes are science and scientific demonstration withthe classical partition of subjects: definition, hypothesis, axiom and division. Strangely, inview of Saccheri’s mathematical interests, the treatment is rather scanty and traditional andoffers no surprises save for some interesting observations about real definitions and theiruselessness in science which recall similar statements in the contemporary Logique de PortRoyal. Novelties appears again in the third and fourth parts devoted to Topical and Sophisticargumentation where we can recognize the able defender of Jesuit theses in controversialistdebates that Saccheri was. It is a pity that the editors instead of texts of purely biographicalinterest have not given us some specimens of Saccheri’s controversialist exploits since thestyle of Logica Demonstrativa shows clearly that for him this dialectical perspective wasapplicable not only to these themes but also to the defense of logical principles developed inthe book, whose primary aim seems to be the search for arguments suitable to put traditionallogical lore above any reasonable doubt.

In his Introduction, Mugnai sees instead Saccheri’s book chiefly as an essay in the direc-tion of a presentation of logic in a mathematical style, as it was not uncommon in largesectors of logical research during the seventeenth century. His general historical perspective(as amply illustrated in a paper published in 2010 in this journal (Mugnai 2010)) is thatduring this period a great effort was spent by logicians interested in the new developmentsin mathematics in clarifying the relations among the two disciplines. This happened in twoways: either by offering a logical perspective on mathematics or by furnishing an expositionof the materia logica in the euclidean synthetic mode or in an algebraic style. From thispoint of view, Mugnai’s general conclusion seems to be that Saccheri’s accomplishmentwas not completely satisfying and does not justify the enthusiastic valuation of his logicalwork exemplified for instance by Giovanni Vailati, one of the first modern logicians to writeon Saccheri’s work. A large part of Mugnai’s Introduction is dedicated to proving this factillustrating how Saccheri, even if deeply interested in the general trend toward a mathe-matically inspired exposition on logic, substantially was very conservative in his outlook,reproducing traditional views and limiting himself to marginal and sometimes confusedmodifications, the only real novelty in his book being the above mentioned use of self-reference and consequentia mirabilis. In a sense, for Mugnai, Logica Demonstrativa is notso advanced as to be a truly interesting step toward a more mathematical vision of logicand too distant from the proper horizon of traditional logic to offer a suitable presentationof its themes and methods. (See for instance the concluding remarks in the Introduction onpage LXVII.)

This valuation has an indubitable plausibility. Saccheri is often confused, in many caseshe adopts conflicting definitions and perspectives and oscillates between new insights andstandard traditional exposition. In part this is certainly due to the didactical character ofthe work but, in the opinion of the present reviewer, Mugnai is right in insisting on thenecessity to put Saccheri’s logical work in the proper historical context, underlining hisdebts to traditional logic and his eventual misunderstandings of it. So acting, we canhave a more articulated view of the different perspectives in the logical debates during

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this period and a sort of counterbalance to the valuations of the Logica Demonstrativawhose chief interest is in the possible contiguity of Saccheri’s ideas with modern con-ceptions rather than in their historical background. What is not completely convincing inMugnai’s analysis is the specification of what this tradition is and what are its relationsto the subsequent developments of logic. Even if, regarding logic in the seventeenth cen-tury, Mugnai claims different views from theirs, in the valuation of the differences betweentraditional and mathematical logic his general perspective is not too distant from the his-toriographic trend initiated by Łukasiewicz, the Kneales, etc. which he criticizes in thepaper we cited above. In many places in the Introduction and the notes (for instance seepp. XVI and XXIV), he seems to understand that progress toward a mathematical view oflogic means progress toward a vision underpinning the centrality of propositional logicagainst syllogistic logic and toward a Fregean rather than a Boolean perspective, thislast being too consonant with the traditional view and not directed toward an axiomati-zation of logic as a deductive basis for mathematical theories. The strict adoption of thisstandpoint leads him sometimes to valuations that in my opinion are too rigid and hideinteresting aspects of Saccheri’s approach. This is the case, for instance, of his treatmentof syllogistic rules. Mugnai underrates the role played by considerations of monotonic-ity in their demonstration. In this manner, he leaves indeterminate the meaning of thePostulatum presented by Saccheri in Chapter 4 notwithstanding the fact that its role isprecisely that of assuring – in conformity to Distribution Theory – the existence of theterms necessary to obtain the countermodels needed to disprove conclusions of syllo-gistic pairs not conforming to the rules. This underrating is an effect of Mugnai’s beliefthat it is the Theory of Consequentiae (and therefore a propositionally oriented approach)which plays the major role in the demonstration. A perusal of the medieval literature showsinstead that the methods employed by Saccheri were rooted in the Theory of Terms andthat the formulation in the language of consequentiae is only a translation from resultsobtained without recourse to them. A style of presentation traditional at least from thefourteenth century.

In my opinion, a similar rigidity appears in the treatment Mugnai gives of opposition ina section of the Introduction not casually titled ‘Saccheri and the logic of propositions’.From the first lines, Mugnai states that ‘Saccheri seems not to have a clear concept ofpropositionality’ and that this is shown in his treatment of opposition between propositions.The first example Mugnai gives to prove his contention is opposition between conditionalstatements. The question had a long history before Saccheri and we know for instance fromthe Commentary on Prior Analytics by Pseudo-Scotus that the problem was debated in thefourteenth century. Pseudo-Scotus takes the position that for Mugnai is the only possible,that is, that the proposition contradictory to a conditional ‘If A then B’ is the conjunction ‘Aand not B’; not so Saccheri who – in Mugnai’s opinion – is forced to a false solution by hisdefinition of conditional as a conditioned affirmation and must see the contradictory of ‘IfA then B’ in ‘If A then not B’. Mugnai himself admits that in other contexts Saccheri adoptsthe correct solution, but the problem, in my opinion, is another. Why not? If we regard anhypothetical as a conditioned assertion it is plausible to say – as Saccheri does – that thequality of ‘If A then B’ is the quality of B, the conditioned, and not that of the conditioning.The modern debates on the possible forms of conditionals, starting at least from the BarberShop Paradox by Lewis Carroll, show ad abundantiam that there are many reasons to see ‘IfA then not B’ as a contrary to ‘If A then B’. Obviously not as the contradictory, but nowheredoes Saccheri nor traditional logic assert that Positive is the contradictory of Negative.This is necessary only if we see in propositional negation (which gives the contradictoryof a proposition) the only way to express opposition. Mugnai seems to undervalue thefact that for traditional logic opposition is not explained by propositional negation, as is

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apparent from the fact that it duly discriminates contrariety from contradiction, a distinctionthat is impossible on the pure propositional plane without quantifiers or modalities. In thetraditional perspective, the key to opposition is not negation. How the two perspectivesare divergent we can see in Mugnai himself when he corrects Saccheri (see p. XXXI) andsays that the contradictory of ‘Every horse of some man runs’ (in Latin, ‘Omnis equusalicuius hominis currit’) is ‘not: Every horse of some man runs’ that is ‘Some horse ofsome man doesn’t run’, which is obviously untrue. The contradictory statement is ‘Thereis no man all of whose horses run’, that is ‘Every man has a horse who doesn’t run’, asSaccheri justly states. The error – in my opinion – is due to an unjustified privilege assignedto the purely propositional aspect of negation; in the case of natural language, this can bemisleading – as our example shows – because the linguistic form to which negation appliesdoes not give a correct logical analysis of its structure, hiding as it does the nesting ofquantifiers.

The same problem reappears more explicitly, when Mugnai analyzes Saccheri’s treatmentof the relations between contradiction and contrariety. Saccheri gives new definitions ofcontrary and contradictory which are based on mutual refutation but does not assume, asAristotle did, the presence in the propositions of an indication of quantity (or modality).For Saccheri, B is contrary to A when the affirmation of B not only implies the falsity of Abut implies more, and is contradictory when it affirms only what is strictly necessary for thefalsity ofA. For instance, a contrary of ‘Peter sleeps’can be ‘Peter doesn’t sleep and is drunk’,while the contradictory is ‘Peter doesn’t sleep’. In this perspective, a proposition can havemany contraries and only one contradictory. Curiously enough, Girondino in his translationuses always the determinate article ‘the’ and says ‘the contrary of …’ contradicting in thismanner what Saccheri explicitly affirms. What is important is that from Saccheri’s definitionit follows that the contradictory of a proposition is the weakest proposition implied by everycontrary; the fact is interesting because in this manner we can see the analogy with whattoday we do in lattice theory when we define pseudo-complement as the maximum amongcontraries, that is, the weakest proposition (if it exists) implied by every contrary, just asSaccheri says. In the case of Boolean algebras this coincides with the complement andvalidates not only the principle of contradiction but also the excluded middle. The problemof contradiction as the maximal form of opposition among propositions was deeply rooted inmedieval logic as we can see for instance from the pseudo-thomistic opusculum De quattuoroppositis or the debates among Walter Burley and Geraldus Odonis in the fourteenth century.In this perspective, the definition of contrariety given by Saccheri (I don’t know if he waspreceded by others) is interesting for his generality and can be applied to any proposition.Another of its merits is that it shows clearly why it is possible that we cannot have acontradictory of a prescribed form and why in some case contradiction and contrariety canamount to the same. The last is the case with singular propositions and in the seventeenthcentury John Wallis had given a complete analysis of the situation in the context of ageneral inquiry concerning singular propositions and their relations with universals andparticulars. Mugnai underlines the connection with singulars but he conceals the generalmeaning of the definition presenting it merely as an astute ‘trick’ (cf. XXXIV) employedby Saccheri to circumvent the ‘anomalies’ offered by singular propositions in contrast tocategorical with quantitative determinations. What anomalies? It seems to me that Mugnaisees as anomalies what are the distinctive properties of singulars and so misses the generalmeaning of Saccheri’s definitions.

Much more could be said, but I hope I have given the reader an idea of the richness ofproblems and new insights that this edition of Logica Demonstrativa can offer. Even if thiswork by Saccheri is not a major contribution to logic – as Mugnai justly asserts – ponderingon its position in the history of logic can be very rewarding.

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ReferencesSaccheri, G. 2011. Logica dimostrativa, a cura di P. Pagli, C. Mangione, Milano: Bompiani.Fabri, H. 1646. Philosophiae Tomus primus, qui complectitur scientiarum Methodum sex Libris esplicatam: Logi-

cam Analyticam duodecim libris demonstratam, et aliquot controversias logicas, breviter disputatas,Auctore Petro Mosnerio Doctore Medico,cuncta excerpta ex praelectionibus R.P.Hon.Fabri, Soc.Iesu,Lugduni.

Mugnai, M. 2010. ‘Logic and mathematics in the seventeenth century’, History and Philosophy of Logic, 31,297–314.

http://dx.doi.org/10.1080/01445340.2012.760432

Burt C. Hopkins, The Origin of the Logic of Symbolic Mathematics. Edmund Husserl andJacob Klein. Bloomington and Indianapolis: Indiana University Press. 2011. 592 pp.$49.95. ISBN 978-0-253-35671-0.

Reviewed by Stefania Centrone, Philosophisches Seminar, Universität Hamburg, Von-Melle-Park 6, 20146, Hamburg, Germany.

© 2013 Stefania Centrone

This book sets out to understand the contributions of Edmund Husserl and Jacob Klein toa correct understanding of the origin of the ‘logic of symbolic mathematics’. The ‘logic’whose origin is in question is understood by the author, following Jacob Klein, not as whatis meant today, i.e. a formal language and a calculus, but rather as the epistemological expla-nation of what is behind the transformation of mathematics from the science of magnitudesand quantities to an abstract formal science – or algebra. At stake in this transformation forthe author is the ‘blind’ use of signs typical of modern mathematical praxis that stands atthe origin of the movement towards ‘algebraization’, ‘formalization’ of modern mathemat-ics,1 which begins with Vieta and is developed in the mathematics of Simon Stevin, RenéDescartes, and John Wallis. ‘Logic’ in this sense is inseparable from the cogitatio caecaof the Leibnizian heritage – a theme, by the way, that plays an important role in Husserl’sreflections from the very beginning onwards.At the basis of logic, thus understood, turns outto be ‘formalization’, by which Burt Hopkins intends the concept-formation by abstractionresponsible for generating concepts, such as ‘quantity in general’ and ‘anything whatever’(Etwas Überhaupt) – in contrast to Aristotelian abstraction (aphairesis), which generates‘formal’ objects that remain bound to determinate objects and to concept-definition byabstraction in Frege’s sense. The author concentrates, on the one hand, on the text of theRussian-American philosopher and former associate of Husserl, Jacob Klein, entitled Diegriechische Logistik und die Entstehung der Algebra, which was originally published in twoparts in 1934 and 1936 and then in the English translation as Greek Mathematical Thoughtand the Origin of Algebra2 in 1968. On the other hand, he concentrates on three textsby Husserl, Philosophy of Arithmetic (1891), The Crisis of European Sciences and Tran-scendental Phenomenology (1935), The Origin of Geometry as an Intentional-HistoricalProblem (1939) and on Klein’s discussion of the latter two texts in his Phenomenology andthe History of Science (1940).

Hopkins observes that Husserl’s reflections in his youthful Philosophy of Arithmetic3

as well as in his Logical Investigations4 seem, prima facie, to move in the same direction

1 Cp. Giusti 2000, 105.2 Henceforth cited as Klein 1968.3 Henceforth cited as PA.4 Henceforth cited as LI.

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