This is the second edition of an important introduction to Leibniz's philosophy of logic and language first published in 1972. It takes issue with several traditional interpretations of Leibniz (by Russell amongst others) while revealing how Leibniz's thought is related to issues of great interest in current logical theory. For this new edition, the author has added new chapters on infinitesimals and conditionals as well as taking account of reviews of the first edition.

Mathematician and philosopher Hermann Weyl had our subject dead to rights.

"The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of Leibniz (...) to the writings of Hobbes."--BOOK JACKET. (shrink)

One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...) was made to defend Leibniz’s argument against my criticisms. In the present paper, I take up the matter again in an attempt to clarify the issues involved and to defend my original criticisms of the argument against the objections raised by Arthur. (shrink)

Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and (...) 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods. (shrink)

Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It (...) concerns the set of all numbers, N: since to every number there is a corresponding square, there are as many squares as numbers. But since there are non-squares between the squares, “all numbers, comprising the squares and the non-squares, are greater than the squares alone”, i.e. there must be fewer numbers in the set of all squares S than in N. Thus N is both equal to and greater than S. This is a contradiction, so one of the premises must be given up: the question is, which one? (shrink)

In last year’s Review Gregory Brown took issue with Laurence Carlin’s interpretation of Leibniz’s argument as to why there could be no world soul. Carlin’s contention, in Brown’s words, is that Leibniz denies a soul to the world but not to bodies on the grounds that “while both the world and [an] aggregate of limited spatial extent are infinite in multitude, the former, but not the latter, is infinite in respect of magnitude and hence cannot be considered a whole”. Brown (...) casts doubt on this interpretation—or rather, he begins by questioning its adequacy as an interpretation of a central passage, only to concede its essential correctness as an interpretation of Leibniz’s position, and to turn his attack on the latter itself. In this note I shall argue that Brown underestimates the subtlety of Leibniz’s views on the infinite, and that Carlin is basically correct in his suggestion that the infinite magnitude of the world is what precludes it from having even the phenomenal unity that a body does. (shrink)

Throughout his early writings, Leibniz was concerned with developing an acceptable account of God's relationship to the created world. In some of these early writings, he endorsed the idea that this relationship was similar to the human soul's relationship to the body. Though he eventually came to reject this idea, theanima mundi thesis remained the topic of several essays and correspondences during his career, culminating in the correspondence with Clarke. At first glance,Leibniz's discussions of this thesis may seem less important (...) in comparison to others, since it might seem like a topic which is far removed from what are regarded as his most important philosophical doctrines. I hope to show in what follows that such a view is mistaken. The large amount of attention Leibniz paid to this thesis is a sure indication of its importance to him. Further, as we shall see, his discussions of this thesis tum on some of his most interesting metaphysical topics, including the development of his thinking about the actual infinite, the structure of organic wholes, and the relationship between God and the created universe. In what follows, I examine these discussions chronologically, from the De Summa Rerum (1675-6), to the correspondence with Clarke (1715-6). (shrink)

Es hat sich als ausgesprochen schwierig erwiesen, für Leibniz' Auffassung, daß kontingente Wahrheiten unendlich komplex sind, eine Interpretation zu finden, die diese Auffassung kohärent erscheinen läßt. Dies liegt daran, daß seine Kommentatoren dazu neigen, sich für die unendliche Analyse einer kontingenten Wahrheit am Vorbild eines nicht endenden logischen Beweises zu orientieren. Ich versuche hingegen zu zeigen, daß unendliche Analysen als unendliche Reihe immer komplexerer und detaillierterer physischer Argumente aufgefaßt werden sollten. Ferner versuche ich zu zeigen, daß die Theorie der unendlichen (...) Analyse speziell für die Wahrung dessen entworfen wurde, was man ,innerweltliche' Kontingenz nennen könnte, d.h. einer Kontingenz, die aus internen Merkmalen einer gegebenen moglichen Welt hervorgeht, im Gegensatz zu einer Kontingenz, die aus Überlegungen dariiber entsteht, welche anderen Welten Gott geschaffen haben konnte. Um innerweltliche Köntingenz geht es, wenn Leibniz scheinbare Gefahren für die Freiheit betrachtet, die aus seiner Erklärung von Wahrheit als einem begrifflichen Enthaltensein entstehen. (shrink)

Im November 1672 schloss Leibniz, dass ein Kontinuum nicht aus Punkten besteht. Der Beweis, der als Diagonal-Paradox Bekanntheit erlangte, wurde von Leibniz vorgebracht, nachdem er die Existenz einer unendlichen Zahl verneint hatte. Vor kurzem haben mehrere Kommentatoren darzustellen versucht, dass der Leibniz'sche Beweis, unter dem Aspekt von Cantors Mengenlehre und seiner Lehre von den Kardinalzahlen gesehen, nicht stichhaltig sei. In diesem Artikel unternehme ich den Versuch, die philosophischen Annahmen, denen Leibniz' Gebrauch des Diagonal-Paradox unterliegt, offenzulegen, um zu zeigen, dass eine (...) solche Kritik unmöglich ist. Die Kritik gründet sich auf die Forderung, zwischen zwei Wegen, Größen miteinander zu vergleichen, zu unterscheiden; jedoch hatte Leibniz solch eine Unterscheidung schon im Sinn, die er 1672 aber vermeiden wollte. Gegen Ende 1670 dachte Leibniz, dass ein Weg existiere, ein Kontinuum aus Punkten durch eine Unterscheidung zwischen der Ausdehnung eines Körpers und seiner Größe zusammenzusetzen. Diese Unterscheidung erlaubte es Leibniz ebenfalls, verschiedene Größen gleichzusetzen und somit dem Diagonal-Paradox auszuweichen. Im November 1672 versuchte Leibniz jedoch, diese Unterscheidung zwischen Ausdehnung und Größe zu vermeiden, weil er davon überzeugt war, dass eine unendliche Zahl nicht möglich ist, was ihn dazu brachte, den Punkt als einen Bestandteil des Kontinuums zu verneinen. (shrink)