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.