Abstract
On the Logical Interpretation of compossibility two substances are compossible if and only if the supposition of their co-existence is a logically consistent one, they are incompossible if the supposition of their co-existence is logically inconsistent. Most scholars of Leibniz’s work have abandoned the Logical Interpretation in the face of considerations that they think render the Logical Interpretation untenable. In this chapter I maintain that these scholars are mistaken. In my view, abandonment of the Logical Interpretation is premature because the arguments reared against it fall short of a successful refutation of the Logical Interpretation. I argue that these arguments against the Logical Interpretation are successful against only two strains of the Logical Interpretation, but not to a third logical interpretation that I shall present. I call the two strains LS1 and LS2. I call the new interpretation “The Reformed Logical Interpretation.” That there are two distinct strains of the Logical Interpretation is rarely discussed in the literature. I correct this shortcoming by demonstrating that the strongest arguments against it bear on the Logical Interpretation in two quite different ways. Ways that correspond to the two original strains of the Logical Interpretation. I then move to show that the Reformed Logical Interpretation is different from LS1 and LS2 and is not susceptible to the criticisms raised against LS1 and LS2. By virtue of that fact the Reformed Logical Interpretation succeeds where LS1 and LS2 fail, I conclude that the Reformed Logical Interpretation should be taken as the default Logical Interpretation thus keeping the Logical Interpretation a viable solution to the problem of incompossibility.
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Notes
- 1.
I shall use the phrases ‘doctrine of incompossibility’ and ‘doctrine of compossibility’ interchangeably to refer to the general topic. Naturally, I will use the terms ‘compossibility’ or ‘incompossibility’ when referring to whether substances can coexist, or cannot coexist respectively.
- 2.
- 3.
- 4.
- 5.
It is generally accepted that Leibniz held that these two principles are equivalent: see Leibniz’s second letter to Clarke (AG.321) and On Contingency (AG.28/Grua.302-6).
- 6.
I think this begins to unfold an intimate relationship between the doctrine of incompossibility, and the doctrine of universal expression. This relationship would be one of, at least, strong correspondence. If entities are compossible, then they are connected, they mutually express one another. If they express one another, then they are compossible. By parity of reasoning, collections of incompossible entities will not exhibit universal expression. The question is whether or not this relationship is as strong as logical entailment. To my mind this relationship should be as strong as logical entailment; the rest of the paper gives reasons (mostly indirect) that demonstrate why this must be so. If the relation were not a logical one, it is doubtful that we could provide a satisfying reply to why it isn’t just an accident that compossible collections always exhibit universal expression.
- 7.
There is some textual evidence for this claim. For instance, Leibniz advances the dictum that “there are no purely extrinsic denominations” (A.VI.vi.227/C.9, 250, 251/GP.VII.311/Grua.387).
- 8.
A page later, he writes: “It needs to be stressed here that what is at issue is not a strictly logical doctrine, but a metaphysical one. The theory [the reducibility doctrine] turns on the identification of substances through complete individual notions and the operation of a grounding relationship that is able to build rational connections into the defining conceptions of substances (thus making their defining descriptions more than a matter of merely conjoining predications, though not more than a matter of connecting them” (Rescher 2013, 79).
- 9.
“Extrinsic denominations, with which I am inclined to identify what recent authors are calling ‘relational properties,’ are reducible to intrinsic denominations, that is, to those concepts that are themselves simple or are compositions of such simples. [. . .] Relational propositions are similarly reducible to nonrelational propositions, the predicate concepts of which are intrinsic denominations of the individual substances that fall under their subject concepts” (Mates 1986, 222).
- 10.
See also A.VI.iv.1541⁄AG.42.
- 11.
Leibniz writes: “For we know on other grounds that the harmony of phenomena in souls does not arise from the influence of bodies but is pre-established” (A.VI.iv.1550/AG.47).
- 12.
I shall also take the phrase ‘relational concept’ to be interchangeable with ‘relational predicate’ when speaking of relations in terms of complete concepts. The presence of the term ‘concept’ in this phrase is connected to our understanding of Leibniz’s talk of extrinsic and intrinsic denominations. Extrinsic denominations correspond to what we now call relational concepts and intrinsic denominations to monadic/one-place predicates. It is safe to make this inference judging from Leibniz’s thoughts in the following passages. “It also follows that there are no purely extrinsic denominations, denominations which have absolutely no foundation in the very thing denominated. For it is necessary that the notion [concept] of the subject denominated contain the notion [concept] of the predicate. And consequently, whenever the denomination of a thing is changed, there must be a variation in the thing itself” (AG.32). “Every individual substance contains in its perfect notion the entire universe and everything that exists in it, past, present, and future. For there is nothing upon which one cannot impose some true denomination from another thing, at the very least a denomination of comparison and relation. Moreover, there is no purely extrinsic denomination. I have shown the same thing in many other ways, all in harmony with one another” (AG.32-3). More texts could be summoned, but the point is relayed well enough by these two. Since the notion/concept of the subject denominated contains the notion/concept of the predicate, it follows that the denomination is a sort of concept under which we understand the subject, something true of it. This comes out even more clearly in the second quote. There Leibniz connects the presence of denominations with universal expression. If comparison and relation can be made by means of denominations, then the plural term “denominations” stands for the concepts under which relations are subsumed. In this way the interconnectedness of all things is made possible. Mates describes the matter well: “In the so called region of ideas, the counterparts of declarative sentences are propositions or thoughts; correspondingly, the counterparts of definite or indefinite descriptions (or abbreviations of such) are denominations. Thus the ontological status of denominations is that of concepts; in short, a denomination is a kind of concept” (Mates 1986, 218).
- 13.
This is not, of course, to say that the world-apart doctrine is not important. I think it might actually be accorded the status of necessity in Leibniz’s system given that it seems to be a necessary upshot of the containment notion of predication and truth. And since the predicate-in-subject notion of truth appears to be necessary, it is probable that the world-apart doctrine is also necessary.
- 14.
Messina and Rutherford think that this criticism applies to the Logical Interpretation as a whole, but as I’ve already noted, it is more accurate to say that it applies only to LS1.
- 15.
“God, in designing the world, purposed solely to manifest and communicate his perfections in the way that was most efficacious, and most worthy of his greatness, his wisdom and his goodness.” See also: “There is no doubt that when God resolved to act outside himself, he made choice of a manner of action which should be worthy of the sovereignly perfect Being, that is, which should be infinitely simple and uniform, and yet of an infinite fecundity” (GP.V1.238/H.254-5).
- 16.
Leibniz is committed to both these propositions as we can see from the following passages: “After due consideration I take as a principle the harmony of things: that is that the greatest amount of essence that can exist, does exist.” (A.VI.iii.472); “[God is] the harmony of things” (A.VI.i.499), and “the divine mind consists of the ideas of all things [. . .]. In God there are infinite really diverse substances, yet God is indivisible” (A.VI.i.511-12/L.118).
- 17.
We are already familiar with identity/non-contradiction (Identity: A = B) as one of these relations; one that determines the intelligibility of a concept. There are, of course, more: Containment: A ∈ B; Converse Containment: A inest ipsi B; Conjunction: AB; Negation: ~ A. These last four do not determine the intelligibility of a concept as identity does but they do give us the basic stock of fundamental logical operations that can be performed on concepts.
- 18.
In the February 5, 1712 notes for a letter to Des Bosses he writes: “God not only sees individual monads and the modifications of every monad whatsoever, but he also sees their relations, and in this consists the reality of relations and of truth” (AG.199). That is to say, the reality of relations and truth consists in their being seen by God.
- 19.
Mark Kulstad argues for the same position. See Kulstad (1980), 417–32. According to Kulstad, Leibniz accepts subject-predicate logic in the “wide sense.” This means that for Leibniz relational propositions are reducible to either relational, or non-relational subject-predicate propositions. I’ve rendered this thought by saying that Leibniz accepts two kinds of accidents: non-relational concepts (monadic predicates), and relational concepts (two-place predicates).
- 20.
- 21.
A very good reason for having the doctrine of universal expression carry the weight here is that this doctrine is conceptually prior to compossibility. In fact, the doctrine of universal expression follows from the predicate in subject notion.
- 22.
Monadology sections 56–7, AG.220. This is reiterated in the Discourse on Metaphysics: “Moreover, every substance is like a complete world and like a mirror of God or of the whole universe, which each one expresses in its own way, somewhat as the same city is variously represented depending upon the different positions from which it is viewed. Thus the universe is in some way multiplied as many times as there are substances, and the glory of God is likewise multiplied by as many entirely different representations of his work” (AG.42).
- 23.
There are deep issues here about substance individuation and its connection to the doctrine of incompossibility. While it is true that I shall have to deal with this in the rest of the chapter, I will not be giving it the attention that it deserves. Instead, I will only give a general account of how substance individuation that could undergird an account of compossibility that is in keeping with all the strictures that I have given above.
- 24.
They say: “The lawful approach faces the following objection: By using ‘lawfulness’ rather than ‘no contradiction’ as the operative notion, one violates Leibniz’s own close association between possibility claims and the idea of contradiction” (Cover and O’Leary-Hawthorne 1999, 132).
- 25.
Formally the difference can be expressed in the following manner. Co-possibility: ◊β & ◊α; Compossibility: ◊(β & α), where ‘β’ and ‘α’ are complete concepts. This expression is different from Hintikka, who—like others—incorporates the notion of existence: Possibility: ◊ ∃x(Ax) & ◊ ∃x(Bx); Compossibility: ◊[ ∃ × (A×) & ∃ × (B×)] (Hintikka 1972, 159–60).
- 26.
Mogens Laerke defends the position that before the Paris years Leibniz held a modal philosophy in which possibilia were mere abstractions, and are never actually conceived by God. According to Laerke, God only conceives and brings about the actual beings. But he argues that all of this changes with the advent of Leibniz’s mature modal metaphysics after 1677 (Laerke 2007). In the mature metaphysics possibilia have their being in the mind of God; so God does conceive of them, but wills to bring about only the ones comprising the best set. Thus the principle of the best takes on a new importance in the mature picture. Laerke maintains that Leibniz abandoned the old view because he thought it brought him too close to the view of Spinoza according to whom only the actually existing things were possible. The bearing this has on my interest here is that for me God’s perfect rationality and his willing the best are one and the same. That is to say, God wills the best because he is perfectly rational, and he is perfectly rational because he is perfectly good. This convertibility of rationality with goodness—if anything—intensifies as Leibniz’s career progresses; and judging from its strong connections with his philosophy of possibility, it is likely that he thought that he had discovered a way of holding off Spinoza without giving up divine choice. For instance he writes in Principles of Nature and Grace Based on Reason, that “everything is ordered in things once and for all, with as much order and agreement as possible, since supreme wisdom and goodness can only act with perfect harmony” (AG.13).
- 27.
See Nachtomy (2007). Nachtomy presents the combinatorial approach to the production of possibilities in the divine mind by elaborating the idea that God introspects to produce the possibles.
- 28.
“I also find completely strange the expression of some other philosophers” who say that the eternal truths of metaphysics and geometry and consequently also the rules of goodness, justice, and perfection are merely the effects of the will of God; instead, it seems to me, they are only the consequences of his understanding, which, assuredly, does not depend on his will, any more than does his essence” (AG.36).
- 29.
I use the word ‘mutual’ to express the sense in which my version of logicality recognizes that the complete concept of a substance shares in the intelligibility of its world-mates to the extent that no substance’s complete concept can be fully understood in isolation from the complete concepts of the other members of its world.
- 30.
A similar account is offered by Gregory Brown (1987). Brown argues that we need to make a distinction between what he calls “monadically complete concepts” (let’s call them MCCs for short) and complete individual concepts. MCCs contain only primitive (monadic predicates) concepts, or their complements. For both Brown and me, GCs (what are MCCs for Brown) cannot exist in more than one world, only complete individual concepts can be candidates for instantiation at more than one world, but even they are essentially tethered to their world. Indeed, the closest Leibniz comes to a positive definition of a world is to say that it is set of related complete concepts.
- 31.
It might appear that appeal to God’s plans in an account of compossibility is question-begging especially since I’ve pointed out the danger in appeals to God’s aesthetic sensibilities. The difficulty is quickly removed, however, if we see that there is only a problem with appealing to God’s plans, or other extra-logical features of Leibniz’s system to account for compossibility. There is no problem with appeal to God’s intentions to explain why there is a world, so long as the metaphysical machinery used for world-creation does not depend on divine intentions. Compossibility is this mechanism.
- 32.
What is more, because relations between GCs are required to individuate the substances we are looking at, there is reason to think of Universal Expression (UE) as a primitive notion.
- 33.
“A contains B is a true proposition if A non-B entails a contradiction. This applies both to categorical and to hypothetical propositions, e.g., ‘If A contains B, C contains D’ can be formulated as follows: ‘That A contains B contains that C contains D’; therefore ‘A containing B and at the same time C not containing D’ entails a contradiction.” The translation belongs to Wolfgang Lenzen. See Lenzen 2004, 36.
- 34.
This principle supports the doctrine of pre-established harmony. Leibniz’s understanding of the union between soul and body, for instance, would not be possible without it. One can see the principle at work in the following passage:
“[E]very individual substance contains the whole series of things in its complete concept notion, and harmonizes with everything else, and to that extent contains something of the infinite. Because this has not been understood, the union of the soul and the body has also been taken to the inexplicable. For, in metaphysical rigor, they do not flow into one another, nor indeed, does God move the one on the occasion of the other and divert it from its own proper course. But following its own laws from the time they were instituted with and admirable but infallible constancy, each agrees with the other as exactly as they would if there were a true influx. And there is something similar in all substances, even those that are most distant from one another, although in them the agreement does not appear so distinctly” (AG.100).
In another place he writes that “one substance does not influence another, and therefore, the mind derives all of its operations from within itself, even though its nature is so ordered from the beginning that its operations harmonize with the operations of all other things” (AG.102–3).
- 35.
For instance, in a September 29, 1698 letter to Andre Morell he writes, “I am effectively of the opinion that God could not do better than he does, and that all the imperfections we think we find in the world only originate from our ignorance” (GP.III.589/SLT.198).
- 36.
A question that has bearing on this problem is what the nature of laws is for Leibniz. I am inclined to say that laws for Leibniz are generalized statements regarding the behaviour of the entire set of individuals that compose a world. If we apply this conception of laws to the expression of one substance by another, it is apparent that laws of expression have to be generalized statements about the predicational facts of the expressing individual.
- 37.
This is importantly different from a view defended by James Messina and Donald Rutherford called the “Cosmological Interpretation” (Messina and Rutherford 2009). According to the Cosmological Interpretation two substances are compossible only if God can conceive of them as being part of the same world, where ‘world’ is understood as a unified spatio-temporal order. On this reading, in conceiving of an individual substance, God conceives of that substance as it would be spatiotemporally related to other members of its world. Information as to how it would be related to other substances is contained in the complete concept of the substance in question. This might lead one to think that since information as to how the substance is to relate to other members of its world is contained within its complete individual concept, we can read-off the world the substance is a part of from the complete individual concept. And so it may seem that the Reformed Logical Interpretation is identical to the Cosmological Interpretation. This is mistaken because the Cosmological Interpretation requires the information within the complete individual concept of the substance and the notion of a world understood as a unified spatiotemporal order in order to account for compossibility. According to the Cosmological Interpretation, “the states of some substances are understood to explain the states of other substances, in accordance with the laws of their world. The general form of such laws will involve appeal to how dependence rins among the states of substances vary with respect to their relative spatiotemporal positions” (Messina and Rutherford 2009, 971; my emphasis). I take this to mean that—even including the internal states of substances—the account of compossibility offered by the Cosmological Interpretation bottoms out at complete concepts sharing a common spatiotemporal order. The Reformed Logical Interpretation, on the other hand, requires only the predicational structure of the substance’s complete individual concept to render a verdict on compossibility; it has no need for reference to the common spatiotemporal order of substances. According to the Reformed Logical Interpretation it is central to compossibility that the complete concept of substances express each other. It is, therefore, universal expression that determines whether or not substances are members of the same world, whether they share the same spatiotemporal order. Hence compossibility is conceptually prior to the common spatiotemporal order of substances. So long as God conceives of complete concepts as expressing one another, then he sees them as compossible. Thus the Reformed Logical Interpretation bottoms out at the universal expression of complete concepts.
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Chiek, Y. (2016). Compossibility and Co-possibility. In: Brown, G., Chiek, Y. (eds) Leibniz on Compossibility and Possible Worlds. The New Synthese Historical Library, vol 75. Springer, Cham. https://doi.org/10.1007/978-3-319-42695-2_5
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