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:

A1. xPx

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.

T1. x=x

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:

T11. ~xpx

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:

........a__________v____z__y__x

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.