The reader who wants to know more about mereology = the logic of parts is referred to Peter Simons's "Parts", which is an excellent survey and presentation of many mereological principles and systems, including how these principles may be used philosophically.
In what follows I give a very simple non-standard logic for parts (that is very close to the ordinary algebraic principles for "<" and "="), mainly because it makes intuitive sense, is clear, and permits the raising of a few points of interpretation of Leibniz's "Monadology", to which the present text is an appendix.
1. A very simple system. I presume some basic logical knowledge, essentially some familiarity with first order predicate logic, and its notations and modes of presentation. Taking this for granted, and not formalizing everything, I start with two simple axioms and two simple definitions:
A2. xPy & yPz ==> xPz
D=. x=y iff xPy & yPx
Dp. xpy iff xPy & ~yPx
The axioms say that 1. any thing is part of itself and 2. every part of something that is part of something is part of the last thing. Taken together, these two axioms say that the relation of being a part of is reflexive and transitive.
The definitions say that a thing x is the same as a thing y if and only x it is part of y and y is part of x, while x is a proper part of a thing y if and only if x is a part of y and y is not a part of x.
These seem to me very plausible axioms and definitions, that are very close to how the term "is a part of" is used in English (without exhausting this use). I have used a small "p" in the definition of "proper part" to remain close to Leibniz's terminology.
Since identity was defined, I start with proving that it has the usual attributes of reflexivity, symmetry and transitivity.
By A1 xPx which is by logic equivalent to xPx&xPx which by D= amounts to x=x.
T2. x=y ==> y=x
By D= x=y iff xPy&yPx but since & commutes xPy&yPx iff yPx&xPy which by D= amounts to y=x.
T3. x=y & y=z ==> x=z
Suppose x=y & y=z. By D= xPy & yPx & yPz & zPy, so xPz & zPx by A2, whence x=z by D=.
So we have proved the defined '=' is reflexive, symmetric and transitive. It also has the usual substitution properties for statements involving = by T3, since given x=y & y=z one may infer x=z i.e. substitute x for y in y=z, and likewise given y=z infer x=z from x=y. So in either case, making the substitution abbreviates the reasoning that can be carried out also without making the substitutions, but to the same effect. The same claim holds for statements involving P and p:
T4. x=y & yPz ==> xPz
T5. x=y & zpy ==> zpx
T4 is proved thus: Suppose x=y. By D= xPy & yPx. So if yPz, xPz by A2. And T5 thus: Suppose x=y and zpy. By D= and Dp xPy & yPx & zPy & ~yPz whence yPx & zPy whence zPx by A2. Now suppose xPz. Since yPx we have by A2 that yPz and a contradiction, so ~xPz whence since zPx we have zpx.
Similar theorems hold for the inverses of the conclusions, and thus indeed one may make these substitutions directly.
Next, we come to a theorem that characterises the proper part relation: if x is a proper part of y, y is not a proper part of x. This contrasts with A1 concerning mere parts:
T6. xpy ==> ~ypx
Suppose xpy. By Dp xPy & ~yPx. Now suppose ypx. By Dp again yPx & ~xPy. Contradiction, so ~ypx.
We chose to define proper parts using only the notion of part, but might have proceeded otherwise, as shown by the next theorem: to be a proper part of y is to be a part of y while not being the same as y:
T7. xpy iff xPy & ~(x=y)
Suppose xpy. By Dp xPy & ~yPx. Now suppose x=y. Then xPy & yPx by D=. Contradiction, so xpy ==> xPy & ~(x=y). Suppose xPy & ~(x=y). Then by D= xPy & (~xPy V ~yPx) whence xPy&~yPx i.e. xpy. Hence T7.
And now we have an equivalence that characterises parts: x is part of y precisely if x is a proper part of y or the same as y:
T8. xPy iff xpy V x=y
First RL. Suppose xpy V x=y. Now suppose ~xPy. By D= ~(x=y) and so xpy, whence by Dp xPy. Contradiction, so xPy. Next LR. Suppose xPy. Now suppose ~(xpy V x=y) i.e. ~xpy & ~(x=y). By A4 from ~xpy we have ~xPy V x=y, and so by ~(x=y) we have ~xPy, contradicting our supposition. So (xpy V x=y). Hence T8.
To conclude this section, we state and prove three related theorems about proper parts. These theorems will turn out to be important below.
First: proper parts of proper parts of any thing are proper parts of that thing. and mirrors A2 for mere parts.
T9. xpy & ypz ==> xpz
Suppose xpy & ypz. From Dp xPy&~(x=y) & yPz&~(y=z). Since xPy&yPz we have xPz by A2. Now suppose zPx. So yPz&zPx whence yPx by A2. Since we have xPy it follows x=y by D=. Contradiction, and so ~zPx whence since xPz it follows xpz by Dp.
Next, by T9 proper parts of proper parts of any thing z are not identical to z:
T10. xpy & ypz ==> ~(x=z)
This follows immediately from T9 and Dp. To conclude this introductory section, there is the related thesis that nothing is a proper part of itself:
For suppose xpx. Then xPx & ~(x=x) by Dp. But ~(x=x) ==> ~xPx by D=, and so T11.
The advantage of this very simple system is that it perfectly mimics part of ordinary algebra for the notions "<" and "=". The reason not to use the algebraic notions nor the algebraic notations is that the intended senses of "part" is meant to apply to things of any kinds that have parts.
2. Leibniz's arguments about parts. The above system needs supplementation to do justice to what Leibniz may have meant. We start with
1. The Monad, of which we shall here speak, is nothing but a simple substance, which enters into compounds. By 'simple' is meant 'without parts.' (Theod. 10.)
When considering (1) in the text, I took this as a definition and remarked that Leibniz seemed to have meant by 'simple' 'without proper parts', though he did not use the term 'proper'.
My reasons are A1 and T11: every thing is part of itself, but no thing is a proper part of itself. These reasons might have been considered not very serious, if it were not the case that the above system perfectly mimics algebra, and so has a very secure and well-known interpretation, while Leibniz might have arrived at the same results as section 1 by following a similar course that uses the parallels between 'being a (proper) part of' and 'being smaller than'.
Given that correspondence, that requires A1 i.e. every thing is a part of itself, it follows that in that sense of 'part' there are no simple things, and therefore I concluded Leibniz meant and should have said that what is simple is without proper parts rather than without parts.
I do think it is probable this is indeed what Leibniz meant, and this will have some important consequences in what follows, e.g. about Leibniz's motivation for the existence of simple substances:
2. And there must be simple substances, since there are compounds; for a compound is nothing but a collection or aggregatum of simple things.
We have at this point the machinery to state and consider some theorems. First, there is the truth that every thing has a part or has no parts:
T12. (x)[ (Ey)(yPx) V ~(Ey)(yPx) ]
This must be a theorem by the logic for quantification (I presume here), but since we have from A1 that ~(Ey)(yPx) is false T12 is not of much help in clarifying Leibniz's argument in (2). A1 has the consequence that, in the defined sense of part, every thing is part of itself, and that, therefore, in the defined sense of part, there is no thing which has no parts, and so there is no thing that can play the role of nothing for parts, and also there is no thing which can play the role of a simple thing if a simple thing is defined as a thing without parts.
But we noticed Leibniz seemed to confuse parts and proper parts, and we have on the same lines as T12 by quantification-theory the theorem
T13. (x)[ (Ey)(ypx) V ~(Ey)(ypx) ]
i.e. every thing has a proper part or has no proper part. We have seen that, on our reconstruction, Leibniz's sense of simple thing should be reconstructed as a thing without proper parts, and we can accordingly define the left inner component of T13 as follows
DC. Cx iff (Ey)(ypx)
A compound thing is a thing with some proper part. From DC we get ~Cx iff (y)~(ypx) iff (y)(yPx ==> x=y) by Dp. So a thing is not a compound iff it has no proper parts iff all its parts are the same as it, and therefore we can define
DS. Sx iff ~Cx iff ~(Ey)(ypx)
A simple thing is a thing that is not compound. Now Leibniz's assumption in his (2) may be written as (Ex)(Cx) i.e. there are compound things, and his conclusion as (Ex)Sx i.e. there are simple things. But in the present set-up it clearly doesn't follow from this assumption that there are simple things as defined, as Leibniz says in (2), so for the moment we shall neither assume (Ex)Cx nor (Ex)Sx, and instead consider what can be done within the context of our assumptions, and turn a little later to an assumption that does link composite and simple things, as defined in this appendix.
What Leibniz says in his (3) to (9) I take to belong to the - Leibnizian - interpretation of a logic of parts rather than as belonging to the formal logic of parts, so I will skip it in this appendix. In (10) we get Leibniz's next assumption
10. I assume also as admitted that every created being, and consequently the created Monad, is subject to change, and further that this change is continuous in each.
Having the apparatus of quantification, we have on the same line as T12 and T13
T14. (x) [ (Ey)(ypx & Sy) V ~(Ey)(ypx & Sy)) ]
i.e. every thing has a simple proper part or lacks a simple proper part. Since this appendix is motivated by the assumption that Leibniz somewhat confused parts and proper parts, we should expect, if this assumption is correct, that T14 does express something close to what Leibniz might have had in mind, and indeed it seems to do, as we shall see in a moment.
First, as before, we can introduce definitions concerning the disjuncts in T14:
DT. Tx iff (Ey)(ypx & Sy)
DI. Nx iff ~Tx iff ~(Ey)(ypx & Sy)
The readings I suggest are respectively 'x is terminal' and 'x is non-terminal': x is terminal if it has some simple proper part, and nonterminal if not. These are mere abbreviations for the disjuncts in T14, but they are in aid of the following definitions, that will be essential in what follows, and involve earlier definitions:
DI. Ix iff Cx & Nx
DR. Rx iff ~Ix iff Sx V Tx
The readings I suggest are respectively ‘x is ideal’ and ‘x is real’: x is ideal if it is compound and non-terminating, and real if not, in which case, by earlier definitions, it is simple or terminating. (Note that the pattern of definition used here differs somewhat subtly from that used in the previous pair, since Ix iff (Ey)(ypx) & Nx.)
Obviously, every thing is either real or ideal, as defined, just as every thing is also either terminal or non-terminal, and either simple or compound.
The notion of x satisfying ~(Ey)(ypx & Sy) - x is non-terminal - is interesting, for we have the following theorem about ideals:
T15. Ix iff (y)(ypx ==> Cy) & ~Sx
For Ix iff ~(Ey)(ypx & Sy)) & ~Sx iff ~(Ey)(ypx & ~(Ez)(zpy)) & ~Sx iff (y)(ypx ==> Cy) & ~Sx by quantification logic and DI, DC and DN.
Since by T10 proper parts of proper parts of x are distinct from x while also by T9 each new proper part of a proper part of x is a proper part of x, it follows this proper part must have again a proper part, and so on - so we have here a kind of infinity, which is another reason for the letter "I" in DI.
There is an easy picture, that relates to the closeness of proper part and the notion of smaller than:
each element being a proper part of all elements to the right of it, and the dots to the left of a indicating an infinite chain like the chain to the right of a that is the beginning of such a chain starting at x.
The reader should also be aware how close being an ideal, which amounts to having proper parts which have proper parts without end, is to being divisible without end.
Now, since being continuous generally is assumed to involve infinity, the definition DI with its consequence T15 seem to have some justification as an explication for what Leibniz might have had in mind. Also, an infinite chain as pictured can be seen as resulting from the notion that between any two things there is a third, or as from the notion that some things have proper parts that have proper parts without end.
Indeed, considering Leibniz next statements
11. It follows from what has just been said, that the natural changes of the Monads come from an internal principle, since an external cause can have no influence upon their inner being. (Theod. 396, 400.)
12. But, besides the principle of the change, there must be a particular series of changes [un detail de ce qui change], which constitutes, so to speak, the specific nature and variety of the simple substances.
13. This particular series of changes should involve a multiplicity in the unit [unite] or in that which is simple. For, as every natural change takes place gradually, something changes and something remains unchanged; and consequently a simple substance must be affected and related in many ways, although it has no parts.
it seems to follows that DI does some justice to what Leibniz might have meant.
However, the conclusion of (13), when considered in the light of this appendix, which presumes that Leibniz confused parts and proper parts, and consequently simple things and ideal things, should be read as "and consequently an ideal substance must be affected and related in many ways, although it has no proper parts with simple parts."
One reason for my presumption is that the whole machinery of talk about the parts of things seems pointless if when it comes to what really matters there suddenly are no parts; another reason is that there simply is a confusion of 'part' and 'proper part' in natural language: 'part' tends to be used ambiguously; a third reason is that the axioms and definitions used in my reconstruction are elementary and conform completely to similar ones for the notions of <=, < and =; a fourth reason is that my reconstruction remains close to what Leibniz says; and a fifth reason is that Leibniz seems rather inconsistent in his (13), since he requires that a simple substance contains a multiplicity, which does not seem possible on his given definition of 'simple’, but which does seem possible on our reconstruction of Leibniz's 'part' as 'proper part', and our consequent reconstruction of Leibniz's 'simple' as 'ideal' i.e. as having no proper part that lacks a proper part.
To provide further support for this, we shall formulate an axiom about proper parts, for which we get the motivation from the following points of Leibniz:
16. We have in ourselves experience of a multiplicity in simple substance, when we find that the least thought of which we are conscious involves variety in its object. (...)
36. (...) There is an infinity of present and past forms and motions which go to make up the efficient cause of my present writing; and there is an infinity of minute tendencies and dispositions of my soul, which go to make its final cause.
56. Now this connexion or adaptation of all created things to each and of each to all, means that each simple substance has relations which express all the others, and, consequently, that it is a perpetual living mirror of the universe. (Theod. 130, 360.)
62. Thus, although each created Monad represents the whole universe, it represents more distinctly the body which specially pertains to it, and of which it is the entelechy; and as this body expresses the whole universe through the connexion of all matter in the plenum, the soul also represents the whole universe in representing this body, which belongs to it in a special way. (Theod. 400.)
64. Thus the organic body of each living being is a kind of divine machine or natural automaton, which infinitely surpasses all artificial automata. For a machine made by the skill of man is not a machine in each of its parts. (...) But the machines of nature, namely, living bodies, are still machines in their smallest parts ad infinitum. It is this that constitutes the difference between nature and art, that is to say, between the divine art and ours. (Theod. 134, 146, 194, 403.)
65. And the Author of nature has been able to employ this divine and infinitely wonderful power of art, because each portion of matter is not only infinitely divisible, as the ancients observed, but is also actually subdivided without end, each part into further parts, of which each has some motion of its own; otherwise it would be impossible for each portion of matter to express the whole universe. (Theod. Prelim., Disc. de la Conform. 70, and 195.)
Lets first note that our reconstruction, that insists there are no proper parts without proper parts where Leibniz says there are no 'parts' is consistent in itself, while motivated by Leibniz's own talk of parts, and usage of terms such as "in" when speaking of "a multiplicity in simple substance" in (16) and especially of "the machines of nature, namely, living bodies, are still machines in their smallest parts ad infinitum" in (64), and (65) "each portion of matter is not only infinitely divisible, as the ancients observed, but is also actually subdivided without end, each part into further parts".
Given this it seems to me what may be hinted at in these points can be expressed, in the context of our assumptions and theorems, as the assumption that everything that has a proper part has an ideal part, which ideal part by what was said around T15 indeed is "subdivided without end (..) into further parts". Thus our third axiom is:
A3. (Ey)(ypx) ==> (Ey)(ypx & Iy)
As Leibniz seems to have confused proper parts and parts, in the present reconstruction he also confused simple things and ideal things, but if indeed he did make the first confusion, it seems sensible to conclude A3 is what Leibniz might have in mind when writing down in his (2) "there must be simple substances, since there are compounds", for - having undone the confusions - this is what A3 says. (Say: there are ideal things, if there are compound things - and obviously the hypothesis of A3 by DC simply is "x is a compound thing"). Also, A3 is very close in sense to what Leibniz claims in (64) and (65), which I have just cited.
Hence, I also think Pierre Bayle - mentioned in (16) - was right in seeing problems with Leibniz's argument, and Leibniz himself might have noticed there is a palpable difficulty if, on the on hand, in (1) he claims Monads are simple things that have no parts, and, on the other hand, in (13), he claims simple things should involve a multiplicity, while, in (64), he claims his simple things (Monads) have "smallest parts ad infinitum".
Although A3 may not yield all that was said by Leibniz in his above points, it has some interesting consequences. First, by quantification theory, there is from A3
T16. (x) [ (Ey)(ypx) iff (Ey)(ypx & Iy) ]
This has the consequence that Sx iff (y)(ypx ==> Rx) i.e. x is simple iff all proper parts of x are real, but since a simple thing has no proper parts this doesn't matter (while we also know by DR that simple things are real).
And we have a neat characterisation of being simple: x is simple iff x is real and not compound:
T17. Sx iff Rx & ~Cx
First, RL is immediate since ~Cx iff Sx by DC and DS. And for LR suppose Sx: By DS and DC ~Cx. And by DI and DR Rx, so we're done (and we have also shown simple things are real).
Next, by standard logic, T16 is equivalent to
T18. (x) [ (Ey)(ypx) & (Ey)(ypx & Iy) V ~(Ey)(ypx) & ~(Ey)(ypx & Iy) ]
and so we have it that every thing either is compound and has an ideal proper part or is not compound and has no ideal proper part, and so by DS everything is either compound with an ideal proper part or simple.
So every thing has an ideal proper part if it has a proper part, and some things have some simple parts as well, and the latter are real. And according to (64), one of the ideal parts that are part of any thing is "the divine machine or natural automaton".
And instead of the fairly long and somewhat complicated T18 we may use our defined terms to formulate the simple but comprehensive
T19. (x) [ Ix V Tx V Sx ]
which is an immediate consequence of the given definitions: everything is either ideal or terminating or simple i.e. everything either has proper parts without end or some proper part with an end (besides proper parts without end as is required by A3) or no proper parts at all.
Also, we have some further textual evidence for our A3, for we read
72. (..) nor are there souls entirely separate [from bodies] nor unembodied spirits [genies sans corps]. God alone is completely without body. (Theod. 90, 124.)
The first part of this quotation conforms to our T16, and we can shed some light on God by what we have achieved formally.
For like Leibniz we have by way of A3 and T18 taken as the mark of physical compounds that these contain a simple proper part besides containing a proper part that makes them a compound thing. The real is also defined with an existential quantor, and the ideal as its denial, and accordingly without an existential quantor: something is ideal iff it has no proper part that is without some proper part. But since the ideal is defined by denying it is real, the ideal itself involves no existential hypothesis. Accordingly, we may frame a definition of being divine:
DD. Dx iff (Ey)(ypx & (z)(zpx ==> Iz))
I.e. x is divine iff x is a compound that has no real proper part. So God - if He conforms to definition - is wholly ideal, which works out in the present system of assumptions that He has proper parts but no simple parts at all. By contrast, from DD, ~(Ex)Dx iff (x)(y)(ypx ==> (Ez)(zpx & Rz)) i.e. nothing is divine iff everything that is compound has some real proper part (besides some ideal part, according to A3).
Thus, to banish the Lord - as defined - from this mereological schema of things one may introduce a fourth axiom that parallels A3
A4. (Ey)(ypx) ==> (Ey)(ypx & Ry)
3. Concluding remarks. I have no firm beliefs about how close all of the above is to Leibniz's intuitions. All I do claim is that this appendix gives some simple formal logic of parts that does some justice to what Leibniz might have had in mind when writing his Monadology, and that is quite close to formal treatments Leibniz might have given if he had had the tools of modern logic, and had decided, like we did, to use axioms for parts similar to well-known theorems for <, <= and = in elementary mathematics.
One thing that does follow if my formalities are more adequate than not, is that on Leibniz's idealism human beings indeed are not finite machines, but may be infinite machines, and are so, according to Leibniz, on the present reconstruction, because they contain, besides whatever finite physical parts they have, an infinitely small infinite part (that does their feeling, desiring and believing, and is figuratively a divine spark). And here I use the term "infinite machine" in the modern sense, i.e. an entity that can be specified by primitive recursive rules, and that differs from finite machines or computers essentially in having infinitely many parts on which it can record the results of its computations.
How plausible the reader thinks this is he should decide for himself. Apart from God and theology, it should be pointed out that there is good evidence for infinities in nature (since between every two real points there are supposed to be infinitely many other points) and for infinitely small particles in nature (such as differentials, especially if explained as in non-standard calculus texts), while the mathematics of infinite machines is simple and well-known.
Also, the simple logic of parts I presented is very similar to a subset of the system Peter Simons in his 'Parts' considers 'is the minimum of a relation if it is to be one of proper part to a whole' (p. 362). And the basic axiom A3 can be taken as claiming that every compound is divisible without end, and thus may require no more than the infinite divisibility of space (and perhaps the notion of a field - of electro-magnetism or gravity - between any two points or parts of space).
Hence those who have no great faith in the existence of a divinity on that basis alone have no good basis for denying the mind may be like Leibniz conceived it to be, even if - like me - they wholly reject Leibniz's theology or theological inspirations. Also, given DD they may simply deny that there is anything real that has no simple parts (and thus in effect assume A4), and they may do so because this is simpler and more consistent (since God, if he exists, differs from all other things in having no real parts).
This leaves the formal results in this appendix unaffected. (Incidentally, McTaggart likewise thought human beings are immortal souls without believing in a real divine soul or a creator. I have not read his own arguments, but the present appendix at least gives some possible justification for such a position - say, there are infinities in each living creature, but there are no infinities outside a living creature, and no unembodied infinities.)
It may amuse some readers if I provide a little tale Leibniz might have liked - apart from its levity - that gives a brief schematic synopsis of a Leibnizian metaphysics (politically rectified for the benefit of the majority of my academic readers, who, like good scientists, prefer pleasing euphemisms over unpalatable facts):
There is, was and will be an infinite class that comprises all classes, finite and infinite, and all possibilities whatsoever. Being all-comprehensive, It is unique; being infinite It thinks, for infinite things can think since they can represent themselves, which they can do because, unlike finite things, they have subsets as numerous as they are themselves. At one point this infinite class decided to make some possible things real, which It did by lopping off an infinity from some of Its many possible infinite parts, which made these parts finite, unthinking and rudderless, for which reason It put an infinite particle in each of them to enable them to steer themselves and express their finite parts as good as they can. Also, It took care that this infinite part adequately mirrors all It created, where 'adequately' is to be understood from the finite part the infinite part guides: the less complex that is, the more confused its mirroring, and It took care all finite parts It made fitted together in what It considered the best and most pleasing way. Finally, It took care that the most complex of the finite parts It created were capable of confusedly mirroring the infinite class that created all from himself. Then It sat back, considered Its work and said it was good, even though in fact - as It could see - nothing of Its creation was capable enough to prove what It could see It was, nor capable enough to see Its creation in more than a very confused way.
Whatever its other merits, the previous paragraph, in the context of the present appendix, is at least a little clearer than a lot of speculation I have read concerning the divinity and infinity. It should also be observed that there seems to me hardly any evidence for such a hypothesis that gives it more than a very slender weight.
To first part: Monadology - part A
To second part: Monadology - part B
To Leibniz's Preface to the Nouveaux Essays
Amsterdam, July 10-14, 1998.