Ohad Nachtomy

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.

This paper develops some important observations from a recent article by Maria Rosa Antognazza published in The Leibniz Review 2015 under the title “The Hypercategorematic Infinite”, from which I take up the characterization of God, the most perfect Being, as infinite in a hypercategorematic sense, i.e., as a being beyond any determination. By contrast, creatures are determinate beings, and are thus limited and particular expressions of the divine essence. But since Leibniz takes both God and creatures to be infinite, creatures are simultaneously infinite and limited. This leads to seeing creatures as infinite in kind, in distinction from the absolute and hypercategorematic infinity of God. I present three lines of argument to substantiate this point: (1) seeing creatures as entailing a particular sequence of perfections and imperfections; (2) seeing creatures under the rubric of an intermediate degree of infinity and perfection that Leibniz, in 1676, calls “maximum in kind”; and (3) observing that primitive force, a defining feature of created substance, may be seen as infinite in a metaphysical sense. This leads to viewing Leibniz’s use of infinity within a Neoplatonic framework of descending degrees of Being: from the hypercategorematic infinite, identified with the most perfect Being; to the intermediate degree of maximum in kind, identified with creatures; to the lowest degree of entia rationis (or beings of reason), identified with mathematical and abstract entities.

Während Leibniz' Vorstellung eines vollständigen Begriffs viel Beachtung fand, blieb die Frage seiner Begründung im Verstand Gottes eher unbeachtet. In diesem Aufsatz versuche ich auf diese Frage einzugehen, indem ich den Zeitraum (ungefähr 1672-1679), in dem Leibniz die Vorstellung eines vollständigen Begriffs als eine explizite Definition eines Individuums entwickelte, näher untersuche. Meine Darstellung über die Begründung des individuellen Begriffs im Verstand Gottes beinhaltet drei Thesen: (1) Leibniz sieht einen inneren Zusammenhang zwischen der Bildung einfacher Begriffe zu zusammengesetzten Begriffen und der Individuation von Begriffen, so daß die Zusammengesetztheit der Begriffe zu ihrer Individualität und Einzigartigkeit beiträgt. (2) Ein individueller Begriff sollte nicht als eine Menge von Prädikaten betrachtet werden, sondern vielmehr als eine einzigartige Struktur von Prädikaten, in der die Reihenfolge der Prädikate eine essentielle Rolle spielt. (3) Ein individueller Begriff sollte nicht nur mit einer einzigartigen Struktur in Verbindung gebracht werden, sondern auch mit einer kombinatorischen Regel, die eine solche einzigartige Struktur von Prädikaten im Verstand Gottes erzeugt

This paper examines the Leibnizian background to Kant's critique of the ontological argument. I present Kant's claim that existence is not a real predicate, already formulated in his pre-critical essay of 1673, as a generalization of Leibniz's reasoning regarding the existence of created things. The first section studies Leibniz's equivocations on the notion of existence and shows that he employs two distinct notions of existence ? one for God and another for created substances. The second section examines Kant's position in his early paper of 1763. My claim is that Kant's view of existence in 1763, namely that it is not a predicate, is strongly related to the logical notion of possibility, formulated by Leibniz and accepted by Kant

This paper argues that God's immanent causation and Spinoza's account of activity as adequate causation (of finite modes) do not always go together in Spinoza's thought. We show that there is good reason to doubt that this is the case in Spinoza's early Short Treatise on God, Man and His Well‐being. In the Short Treatise, Spinoza defends an account of God's immanent causation without fully endorsing the account of activity as adequate causation that he will later introduce in the Ethics (E3def2). We turn to an examination of how God's immanent causation relates to the activity of finite things in the Ethics. We consider two ways to think about the link between God, seen as immanent cause, and the activity of finite things: namely, in terms of entailment and in terms of production. We argue that the productive model is most promising for understanding the way in which the activity of finite things and God's immanent causality are connected in Spinoza's (mature) philosophy.

We argue in this paper that Leibniz’s characterization of a substance as “un être” in his correspondence with Arnauld stresses the per se unity of substance rather than oneness in number. We employ two central lines of reasoning. The first is a response to Mogens Lærke’s claim that one can mark the difference between Spinoza and Leibniz by observing that, while Spinoza’s notion of substance is essentially non-numerical, Leibniz’s view of substance is numerical. We argue that Leibniz, like Spinoza, qualifies the substance as “one” primarily in a non-numerical sense, where non-numerical means per se unity or qualitative uniqueness. The second line of reasoning suggests that the term “one” should be understood as a-unity-presupposed-by-multiplicity in two senses: a) externally, in the sense of being presupposed by higher complex structures, such as aggregates, and, b) internally, in the sense of having itself a complex structure. We develop an analogy along these lines between the role the notion of a fundamental unity plays in Leibniz’s view of numbers and his view of substance. In other words, we suggest that looking at the role units play in Leibniz’s view of mathematics can shed some light on the role they play in his metaphysics.

We argue in this paper that Leibniz’s characterization of a substance as “un être” in his correspondence with Arnauld stresses the per se unity of substance rather than oneness in number. We employ two central lines of reasoning. The first is a response to Mogens Lærke’s claim that one can mark the difference between Spinoza and Leibniz by observing that, while Spinoza’s notion of substance is essentially non-numerical, Leibniz’s view of substance is numerical. We argue that Leibniz, like Spinoza, qualifies the substance as “one” primarily in a non-numerical sense, where non-numerical means per se unity or qualitative uniqueness. The second line of reasoning suggests that the term “one” should be understood as a-unity-presupposed-by-multiplicity in two senses: a) externally, in the sense of being presupposed by higher complex structures, such as aggregates, and, b) internally, in the sense of having itself a complex structure. We develop an analogy along these lines between the role the notion of a fundamental unity plays in Leibniz’s view of numbers and his view of substance. In other words, we suggest that looking at the role units play in Leibniz’s view of mathematics can shed some light on the role they play in his metaphysics.

In notes from 1675-76 Leibniz is using the notion of an infinite number as an illustration of an impossible notion. In the same notes, he is also using this notion in contrast to the possibility of the ‘Ens perfectissumum’ (A.6.3 572; Pk 91; A.6.3 325). I suggest that Leibniz’s concern about the possibility of the notion of ‘the greatest or the most perfect being’ is partly motivated by his observation that similar notions, such as ‘the greatest number’, are impossible. This leads to the question how Leibniz convinced himself that the notion of the greatest number is self-contradictory and that of the greatest being is not. I consider two suggestions, one that stress the difference between beings and numbers and one that stress the difference between two notions of infinity, and conclude that neither of them provides a satisfactory solution to this question.

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.