Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as (...) early as Galen, the importance of relational reasoning for mathematics had already been recognized. Further critical voices emerged in the Renaissance and the question of whether mathematical proofs could be recast syllogistically attracted more sustained attention over the following three centuries. Supported by more detailed analyses of Euclidean theorems, this led to attempts to extend logical theory to include relational reasoning, and to arguments purporting to reduce relational reasoning to a syllogistic form. Philosophical proposals to the effect that mathematical reasoning is heterogenous with respect to logical proofs were famously defended by Kant, and the implications of the debate about the adequacy of syllogistic logic for mathematics are at the very core of Kant's account of synthetic a priori judgments. While it is now widely accepted that syllogistic logic is not sufficient to account for the logic of mathematical proof, the history and the analysis of this debate, running from Aristotle to de Morgan and beyond, is a fascinating and crucial insight into the relationship between philosophy and mathematics. (shrink)

According to the received view (Bocheński, Kneale), from the end of the fourteenth to the second half of nineteenth century, logic enters a period of decadence. If one looks at this period, the richness of the topics and the complexity of the discussions that characterized medieval logic seem to belong to a completely different world: a simplified theory of the syllogism is the only surviving relic of a glorious past. Even though this negative appraisal is grounded on good reasons, it (...) overlooks, however, a remarkable innovation that imposes itself at the beginning of the sixteenth century: the attempt to connect the two previously separated disciplines of logic and mathematics. This happens along two opposite directions: the one aiming to base mathematical proofs on traditional (Aristotelian) logic; the other attempting to reduce logic to a mathematical (algebraical) calculus. This second trend was reinforced by the claim, mainly propagated by Hobbes, that the activity of thinking was the same as that of performing an arithmetical calculus. Thus, in the period of what Bocheński characterizes as ‘classical logic’, one may find the seeds of a process which was completed by Boole and Frege and opened the door to the contemporary, mathematical form of logic. (shrink)

This paper investigates the reason why, in the tradition of Western philosophy, a logic of relations was developed only in the second half of the nineteenth century. To this end, it moves along two different but interconnected paths: on the one hand, it attempts to reconstruct the main views concerning the ontology of relations during the middle ages; on the other, it focuses on the treatment of so-called oblique terms in the logical works of some preeminent authors belonging to the (...) scholastic and late-scholastic tradition. From the ontological point of view, realists and nominalist both denied that polyadic expressions of the language signify polyadic properties in the world extra. Some authors, such as Peter Auriol, claimed that polyadic expressions signify something merely mental, thus recognizing, even though in a limited ontological domain, the existence of full-fledged relations, that is, of ‘things’ simultaneousl... (shrink)

In a text written during his stay in Paris, Leibniz, to deny ontological reality to relations, employs an argument well known to the medieval thinkers and which later would be revived by Francis H. Bradley. If one assumes that relations are real and that a relation links any property to a subject – so runs the argument – then one falls prey to an infinite regress. Leibniz seems to be well aware of the consequences that this argument has for his (...) own metaphysical views, where the relation of inherence (‘inesse’) plays such a central role. Thus, he attempts first to interpret the relation of inherence as something ‘metaphoric’, originating from our ‘spatial way’ of looking at the surrounding world; and then he tries to reduce it to the part-whole relation which clearly he considers weaker, from the ontological point of view, than that of ‘being in’. (shrink)

Many of the problems traditionally related to the interpretation of Leibniz' theory of relations may be seen in a better light considering essentially two factors: 1) the different plans (ontological, metaphysical, psychological and logical-linguistic) implied by Leibniz reflections on the subject; 2) the reference to scholastic and late-scholastic texts read or consulted by Leibniz. Relations for Leibniz are, from a metaphysical point of view, denominations only seemingly external, they are in reality denominationes intrinsecae, and are founded on the general connection (...) of all things. From a psychological point of view they are abstract entities that our mind builds by resemblance. From an ontological point of view they are individual accidents inherent to the substances. From a logical-linguistic point of view they are abstract structures that connect the one to the other at least two subjects. The propositions in which they appear, as for example the proposition “Paris loves Helen” are transformed by Leibniz in equivalent propositions joined by operators, which in medieval logic were known as termini reduplicantes (terms which define mostly intensional contexts). This may be seen with sufficient clarity examining the last part, until now inedited, of the famous passage about Paris and Helen published by Couturat in his Opuscules. (shrink)

This book is a collection of essays published by the author in the long run of about 20 years and is centered on the reconstruction of Leibniz’s logical calculi. All the essays have been revised for the present edition and some of them constituted the background for Lenzen’s first monograph on Leibniz’s logic. A feature common to all these essays is the vindication of the relevance and originality of Leibniz’s logical achievements. Lenzen manifests strong dissatisfaction with the evaluations of Leibniz’s (...) logic previously offered by interpreters like Louis Couturat, Clarence I. Lewis, Karl Dürr, William and Martha Kneale, and states that till now Leibniz’s results in the field of logic have been widely underestimated. The book contains a careful and detailed examination of almost all Leibniz’s papers on the logical calculus and it is based on the knowledge of a wide range of texts unknown to previous interpreters. Lenzen’s acquaintance with the entire corpus of Leibniz’s logical texts is impressive. Some chapters of the book in particular contain very solid and useful logical analyses. Chapter 7, for instance, includes the most profound account of Leibniz’s theory of negation I ever read. Chapter 8 presents in a very clear way Leibniz’s attempt to reduce traditional syllogistic to a calculus based on logical inclusion between terms. Chapter 14 is devoted to Leibniz’s a priori proof of the existence of God and presents the first edition of an important manuscript on the proof. On chapters 3 and 5 a series of convincing reasons are given to argue that Leibniz’s concept of ens does not have to be considered a constant in the logical calculus. In brief: this work discusses a wide range of topics in such a clear and learned way that it will surely become a reference book for scholars interested in the study of Leibniz’s logical papers in the forthcoming years. (shrink)

This is an appreciation of Richard Arthur, assessing his contributions to Leibniz studies and recounting the nature of our friendship over the past 30 years.

In a text written during his stay in Paris, Leibniz, to deny ontological reality to relations, employs an argument well known to the medieval thinkers and which later would be revived by Francis H. Bradley. If one assumes that relations are real and that a relation links any property to a subject – so runs the argument – then one falls prey to an infinite regress. Leibniz seems to be well aware of the consequences that this argument has for his (...) own metaphysical views, where the relation of inherence (‘inesse’) plays such a central role. Thus, he attempts first to interpret the relation of inherence as something ‘metaphoric’, originating from our ‘spatial way’ of looking at the surrounding world; and then he tries to reduce it to the part-whole relation which clearly he considers weaker, from the ontological point of view, than that of ‘being in’. (shrink)

In a text written during his stay in Paris, Leibniz, to deny ontological reality to relations, employs an argument well known to the medieval thinkers and which later would be revived by Francis H. Bradley. If one assumes that relations are real and that a relation links any property to a subject – so runs the argument – then one falls prey to an infinite regress. Leibniz seems to be well aware of the consequences that this argument has for his (...) own metaphysical views, where the relation of inherence plays such a central role. Thus, he attempts first to interpret the relation of inherence as something ‘metaphoric’, originating from our ‘spatial way’ of looking at the surrounding world; and then he tries to reduce it to the part-whole relation which clearly he considers weaker, from the ontological point of view, than that of ‘being in’. (shrink)

This book offers a careful investigation of Leibniz’s theory of truth and, at the same time, an accurate analysis of many aspects of Leibniz’s ontology and philosophy of logic. Given its complexity and richness, I limit myself to illustrat ing certain aspects of Rauzy’s inquiry and to expressing some, albeit moderate, disagreement. Despite this disagreement, the author’s achievement is extraordinary. Anyone interested in Leibniz’s logic and metaphysics will read it with great profit.

At death's door, Leibniz firmly refused to see a priest. This seems to be in strong contrast with the central role that he attributes to God in his philosophy and with his explicit acceptance of the Christian religion. Thus, the circumstances surrounding Leibniz's death, as related by some direct and indirect witnesses, provide an opportunity to examine the sincerity of Leibniz's position on religion. The final conclusion is that Leibniz believed in a religion broadly inspired by Christian principles but that (...) he was not a particularly pious believer, which is fully in line with his own kind of ante litteram deism. (shrink)

The paper examines three essays that Leibniz wrote in 1688, immediately after the composition of the Discourse on Metaphysics, one of his most organic philosophical works. The main topics which emerge from these essays are: the relationship between substance and accidents; the nature of accidents; and, more generally, the nature of abstract entities. Given that accidents' nature is that of changing, Leibniz sees how hard it is to give an account of the relationship between substance and accidents that does not (...) imply any change in the substance, when its accidents undergo a change. To find a way out of this problem, Leibniz assumes that abstract terms (whiteness, wisdom, etc.) do not name real properties or real parts of substances, and that any sentence containing real abstract terms can be rephrased as a sentence containing logical abstract terms, which do no imply any commitment to the existence of real properties. Coherently with this nominalistic attitude, Leibniz ends up, in the Theodicy and in his correspondence with Des Bosses, considering accidents as modes or internal modifications of the substance. (shrink)

The paper examines three essays that Leibniz wrote in 1688, immediately after the composition of the Discourse on Metaphysics, one of his most organic philosophical works. The main topics which emerge from these essays are: the relationship between substance and accidents; the nature of accidents; and, more generally, the nature of abstract entities. Given that accidents’ nature is that of changing, Leibniz sees how hard it is to give an account of the relationship between substance and accidents that does not (...) imply any change in the substance, when its accidents undergo a change. To find a way out of this problem, Leibniz assumes that abstract terms do not name real properties or real parts of substances, and that any sentence containing real abstract terms can be rephrased as a sentence containing logical abstract terms, which do no imply any commitment to the existence of real properties. Coherently with this nominalistic attitude, Leibniz ends up, in the Theodicy and in his correspondence with Des Bosses, considering accidents as modes or internal modifications of the substance. (shrink)