Ohad Nachtomy

In her stimulating article, Catherine Wilson considers the moment of worlds-making in Leibniz’s philosophy. She raises the following question: “How do possible substances give rise to possible worlds?“ and observes that the moment of world-making is as puzzling as it is interesting. In section 2 of her article, Wilson considers two approaches to the question. According to the first, possible individuals logically precede possible worlds and possible worlds are constituted either by combinations of possible individuals or by mechanically checking the compossibility relations among them. This approach assumes that possible individuals are given and complete and that worlds result from their various conjunctions. On this approach, a world is a set of compossible individuals, i.e., a set of complete individuals whose co-existence is possible.

This chapter seeks to highlight some of the main threads that Leibniz used in developing his views on infinity in his early years in Paris. In particular, I will be focusing on Leibniz’s encounters with Descartes, Galileo, and Spinoza. Through these encounters, some of the most significant features of Leibniz’s view of infinity will begin to emerge. Leibniz’s response to Descartes reveals his positive attitude to infinity. He rejects Descartes’s view that, since we are finite, we cannot comprehend the infinite and therefore should refrain from studying it. Likewise, Leibniz rejects Descartes’s view that the term ‘infinite’ should be reserved to God alone, as well as Descartes’s distinction between the infinite and the indefinite. Leibniz’s encounter with Galileo brings out his rejection of infinite number in response to Galileo’s paradox. This, in turn, leads him to face another formidable challenge, viz., to defend the claim that an infinite being is possible, while an infinite number is not. Leibniz’s encounter with Spinoza, I suggest, highlights the way he is approaching this problem by distinguishing between quantitative and non- quantitative senses of infinity. The strategy of employing different senses of infinity in different contexts will remain central in Leibniz’s approach to infinity for the rest of his career.

This work develops a Leibnizian approach to possibility by explicating the notions of possibility in general, in chapter 1; possible individuals in chapter 2; possible worlds in chapter 3; and actualization in chapter 4. ;A Leibnizian notion of possibility is characterized against the traditional view of an intelligible realm of thoughts in God's mind. It is understood in terms of self-consistent thoughts and is developed by explicating the notions of thought and of possibility in terms of the combinatorial structure common to both, that is, the construction of complex structures by progressive combinations of simple elements. According to this view, God mentally combines his simple forms, and complex concepts arise in his mind. Such complex and self-consistent concepts are deemed possible. ;Against this general view of possibility, a possible individual is identified with the combinatorial rule that produces a unique and maximally consistent structure of predicates. Such a rule is termed the "production rule" of the individual. The "production rule" unifies a series of predicates and constitutes the individual's logical subject and its principle of individuation. ;Given this view of a possible individual, the notion of a possible world is defined as a set of compossible individuals. It is argued that the compossiblity relation result from the consideration of two individuals in the same logical space. It is suggested that the inner structure of each individual entails its possible connections with other individuals, and that God's consideration of the individuals splits them into compossible sets or possible worlds. ;Finally, it is suggested that actualization involves endowing the "production rule" in God's mind with power of action. The union of a divine prescription of action with power of action yields an active and self-sufficient agent. Upon creation, the agent's activity generates the sequence of predicates prescribed to it by its production rule. ;The explication of the above notions accounts for the conceptual dependence of possible individuals and for the causal independence of created individuals--both of which are indispensable for Leibniz's notion of preestablished harmony.

La communication qui suit porte sur le concept de relation tel que le définit Leibniz dans sa correspondance avec Arnauld. La première partie présente trois des présupposés impliqués dans ce concept, à savoir 1) qu'il y a des relations entre des individus possibles, 2) que ces relations sont nécessaires à la notion de mondes possibles et 3) qu'elles sont également nécessaires pour compléter l'individuation des individus possibles. Dans la deuxième partie, on verra que le premier présupposé semble entrer en conflit avec le nominalisme de Leibniz et son refus d'attribuer une réalité aux relations. Un problème de circularité entrave la résolution de ce conflit: d'une part, Leibniz considère en effet que les relations interindividuelles présupposent des individus, mais d'autre part, le troisième présupposé indique que de telles relations sont nécessaires à l'individuation même des individus possibles. Dans la troisième partie, on proposera une solution à ce problème de circularité au moyen du concept d'espace logique de possibilités. Dans cet espace logique, la place qu'occupe un individu est à la fois 1) constitutive de son concept et 2) une notion relationnelle. Grâce au concept de relation que propose Leibniz, les concepts d'individus et l'espace logique en viennent ainsi à se constituer mutuellement, ce qui ouvre une avenue hors de la circularité