Three lines of argument in the Correspondence, involving the PSR, are worth tracing through in detail. One concerns space; the other atoms; the third motion. The first and second are closely interwoven:
Clarke Against the PSR:
Why this
particular system of
matter, should be
created in one
particular place,
and that in another particular place; when, (all
place being absolutely indifferent to all matter,)
it would have been
exactly the
same thing vice versa, supposing the two systems (or the
particles) of matter to be alike; there could be no
other reason, but
the mere will of God.
(Clark, Second Reply, Alexander 1984, p.20-21).
Leibniz’s reply:
I say then, that if
space was an absolute being, there would something
happen for
which it would be impossible there should be a sufficient
reason. Which
is against my
axiom. And I prove it thus.
Space is
something absolutely
uniform; and, without the things
placed in it,
one point
of space does
not absolutely differ
in any respect
whatsoever from
another point of space. Now from hence it follows,
(supposing space
to be something in itself, besides the order of
bodies among
themselves,) that 'tis
impossible there should be a
reason, why
God, preserving the
same situations of bodies among
themselves, should
have placed them
in space after
one certain
particular manner,
and not otherwise; why
everything was not placed
the quite contrary way, for instance, by
changing East into West. But
if space
is nothing else, but that order or relation; and is nothing
at all without bodies,
but the possibility of placing them; then those
two states,
the one such as it now is, the
other supposed to be the
quite contrary
way, would not at all differ
from one another. Their
difference therefore
is only to be found in our chimerical supposition
of the
reality of the space in itself. But in truth the one would be
exactly the
same thing as
the other, they
being absolutely
indiscernible; and
consequently there is no room to
enquire after a
reason of
the preference of
the one to the other. (Leibniz, Third
Paper, Alexander,
p.26).
Why not refer the parts of space to matter?
Space being
uniform, there can be neither
any external nor internal
reason, by
which to distinguish its parts, and to make any choice
between them.
For, any external reason to
discern between them, can only be
grounded upon some
internal one. Otherwise we should discern
what is
indiscernible, or choose without discerning. (Leibniz, Forth
Paper, Alexander 1984, p.39).
Clarke, on why atomism is in trouble even if space is purely relational:
The case is the same, even though space were nothing real but only the mere order of bodies: for still it would be absolutely indifferent, and there could be no other reason mere will, why three equal particles should be placed or ranged in the order a, b, c, rather than in the contrary order. And therefore no argument can be drawn from this indifferency of all places, to prove that no space is real. For different places are really different or distinct one from another, though they be perfectly alike. (Clarke, 3rd Reply, p.30).
Leibniz, concluding that therefore atomism must be given up:
Tis a thing indifferent, to place three bodies, equal and perfectly alike, in any order whatsoever; and consequently they will never be placed in any order, by him who does nothing without wisdom. But then he being the author of things, no such things will be produced by him at all; and consequently there are no such things in nature. (Leibniz, 4th Paper, p.36).
Clarke insists that this is not enough:
This argument, if it was true, would prove that God neither has created, nor can possibly create any matter at all. For the perfectly solid parts of all matter, if you take them of equal figure and dimensions (which is always possible in supposition), are exactly alike; and therefore it would be perfectly indifferent if they were transposed in place; and consequently it were impossible (according to this learned author’s argument) for God to place them in those places wherein he did actually place them at the cretaion, for he might as easily have transposed their situation. (Clarke, 4th reply, p.45-46).
For the debate over the motion of the centre of mass of the universe, see:
Clarke, p.32; Leibniz, p.38; Clarke, p.48, Leibniz p. 63, Clarke p.101.
For next week read:
G. M. R. Parkinson, “Philosophy and Logic”, in The Cambridge Companion to Leibniz, N. Jolley, ed., C.U.P., 1995.
Readings on reserve:
Other writings by Leibniz:
Theodicy, (1710), trans. Huggard, Open Court: 1985.
New Essays on Human Understanding, (1704; first published 1765), trans. Remnant and Bennett, (abridged edition), Cambridge: 1982.
Philosophical Essays, trans. Ariew and Garber, Hackett.
(Principle essays are: “Discourse on Metaphysics”, 1686; “New System of the Nature and Communication of Substances”, 1695; “Mondadology”, 1714; “Principles of Nature and of Grace”, 1714.)
Philosophical Writings, trans. Morris and Parkinson, Everyman:1973.
Secondary Writings
E. Vailati, Leibniz and Clarke: A Study of their Correspondence, Oxford: 1997.
N. Rescher, G.W. Leibniz’s Monadology: An Edition for Students.
N. Jolley (ed.), The Cambridge Companion to Leibniz, Cambridge: 1995. (see especially G. Parksinson, “Philosophy and Logic”.
D. Rutherford, Leibniz and the Rational Order of Nature.
B. Mates, The Philosophy of Leibniz: Metaphysics and Language, Oxford: 1986.
H. Ishiguro, Leibniz’s Philosophy of Logic and Language, Cornell: 1972.
R. Adams, Leibniz: Determinist, Theist, Idealist, Oxford: 1994.
P. Edwards, The Encyclopedia of Philosophy, MacMillan: 1967.
Personal Notes
Brief biography of Leibniz (Intro to New Essays good for this).
How the start of paper 2 comes as a bit of surprise, given all the creeping about materialism.
Examples of PSR: The lever; Stevins and inclined plains; Galileo’s law of gravity?
Remark: synthetic a priori truth is what is at issue.
From PSR to PII: apply it to worlds.
Analyse the shift argument.
Probe: deduce something about “real” as opposed to nominal.
Consider again PII as Leibniz treats it. The leaves.
Permutations, as Clarke introduced it.
Goi back to shift argument: indexicals.
Also ask: shift argument and velocities, rotations, spinning frames of reference? Spacetime.
Choice in a State of Equilibrium
Leibniz argued in the Correspondence against the view that God could choose from a state of equilibrium. He first argued, against Clarke, that one could not choose A or B without choosing A or B in a particular manner (LV, 16-17); supplementing this, that if there were no reason to favour one particular manner for achieving A or B, then (because omniscient) God would not choose a world in which the choice of A or B was required (LV, 66); and finally that there could be no such perfectly symmetric situation, by the Identity of Indiscernibles (LIV, 4, LV, 25). (The latter will hardly do as an independent argument, if the PII is only motivated by the PSR.)
The strongest philosophical intuition in favour of Absolute Space is that matter could have been differently located – say three feet in this direction – where “this” may not have any descriptive significance, but nevertheless picks out a direction by virtue of demonstrative reference. To put the matter in terms of possible worlds, this amounts to the view that in the treatment of counterfactuals, certain entities can count as the same in two possible worlds. To understand Leibniz’s views on this matter, we need to understand his theory of counterfactuals (his treatment of possibility, or contingency). This too derived from the PSR, but here Leibniz offered a number of theories.
In 1771 it seemed to Leibniz that the PSR forced out-and-out necessitarianism (that everything that happens, happens necessarily).
For it is necessary to analyse everything into some reason, and not to stop until we arrive at a first reason – or else it must be admitted that something can exist without a sufficient reason for its existence, and this admission destroys the demonstration of the existence of God and of many Philosophical theorems. What then is the ultimate reason of the divine will? The divine intellect. For God wills those things that he understands to be best and most harmonious, and selects them, as it were, from an infinite number of all possibles.
Since God is the most perfect mind, however, it is impossible for him not to be affected by the most perfect harmony, and thus to be necessitated to the best by the very ideality of things…..Hence it follows that whatever has happened, is happening, or will happen is best and therefore necessary, but…with a necessity that takes nothing away from freedom because it takes nothing away from the will and the use of reason. (letter to Wedderkopf, 1671).
He subsequently rejected this conclusion. For a typical passage:
I reply that it is false that whatever follows from what is necessary through itself is necessary through itself. From truths, to be sure, nothing follows that is not true. Yet since a particular [conclusion] can follow from purely universal [premises], ...why may not something contingent, or necessary on the hypothesis of something else, follow from something that is necessary? (“The Philosopher’s Confession”, approx 1677)
further:
In this place we call necessary only that which is necessary through itself – that is, which has the reason of its existence and truth within itself. such are the Geometrical truths, and of existing things only God. The others, which follow from the supposition of this series of things – that is, from the harmony of things – or from the Existence of God, are contingent through themselves and only hypothetically necessary.
This is the line taken in the Correspondence (LV, 4-10).
On the other hand Leibniz was drawn to a way of incorporating the PSR quite generally into a definition of truth:
Since the individual concept of each person contains once and for all everything that will ever happen to him, one sees in it the proofs a priori or reasons for the truths of each event, or why one has occurred rather than another. (Letter to Arnauld, p.12).
Finally, I have given a decisive reason, which in my opinion ranks as a demonstration [for the PSR]: it is that always, in every true affirmative proposition, necessary or contingent, universal or singular, the concept of the predicate is included in some way or other in that of the subject:…or else I do not know what truth is. (Letter to Arnauld, p.56).
In this context Leibniz also spoke of the essence of a substance. We should distinguish this from the complete concept a substance; although there are conflicting views on this, we will take it (following Adams) that the complete concept of a substance is given by the individual concept or essence of that thing, considered together with the individual concepts or essences of other substances. Likewise we shall talk of the basic concept of a world, as derived from the complete concepts of all the substances in it (which must all be compossible with it), but not as involving the basic concepts of other worlds (whereas the complete concept of a world does). Only from the latter could one derive which world is the best, and hence, which world exists. By a world “possible in its own nature”, then, we understand a world as given by its basic concept.
Leibniz was also able to advance an independent account of contingency from this perspective:
And here is uncovered the secret distinction between Necessary and Contingent Truths, which no one will easily understand unless he has some tincture of Mathematics – namely, that in necessary propositions one arrives, by an analysis continued to some point, at an identical equation…but in contingent propositions the analysis proceeds to infinity by reasons of reasons, so that indeed one never has a full demonstration, although there is always, underneath, a reason for the truth, even if it is perfectly understood only by God, who alone goes through an infinite series in one act of the mind. (p.28, Philosophical Essays, ed. Ariew and Garber).
It is hard to make sense of the notion of “transworld identity”, an identity relation between worlds; it is not even clear whether it is qualitative or numerical identity that is at issue. But since Leibniz denied both kinds of identity, intraworld, we shall assume that the relation is in the first instance that of qualitative identity.
From Leibniz’s theory of truth (hereafter the “conceptual containment theory of truth” or CCT) it is a natural step to conclude that there can be no (qualitative) transworld identities; qualitatively identical things cannot lie in distinct possible worlds. For every predication of that thing is contained in the individual concept of that thing, so, certainly, if the thing in question is to have different predications in different possible worlds, it could not have the same individual concept.
It may be of course that qualitatively identical things lie in distinct possible worlds that differ not with respect to any predication of that thing, but with respect to predications of other things – that it may, in particular, be composed with other possible things. This too would be ruled out if by “individual concept” we mean the complete concept of a thing, which takes into account comparisons with all other things with which it is composed. On a “thin” reading of the CCT, this would not seem to be required; it would be enough if`, from the individual concept of a thing, one can deduce every true predication of that thing – comparative judgments with other things (including inter-relations with other things) need not be contained in its individual concept as well. Might one, on the basis of this, make sense of counterfactuals concerning the locations of objects in space, as required by Clarke? Surely not: for to that end, one would have to have a (thin notion of an) individual concept of a particular point in space, which would distinguish it from other points of space – precisely what is ruled out by the homogeneity of space.
It would seem that Clarke requires a “brute” identity across worlds that may be better made out in terms of numerical identities, independent of individuating concepts altogether. Why should not relations of identity be understood in this way? The reason can be traced once more to Leibniz’s conception of truth, and in part to his insistence that of all possible worlds only one is actual – this the unique preserve of God to decide. The two are in fact linked.
Evidently the CCT involves very considerable philosophical commitments: it is worth considering again why Leibniz endorsed it. Recall again his definition of truth:
[I]n every true affirmative proposition, necessary or contingent, universal or singular, the concept of the predicate is included in some way or other in that of the subject:…or else I do not know what truth is. (Leibniz -Arnauld Correspondence, H. T. Mason, ed., 1967, p.63).
This is an intensional theory, in that the truth or falsity of propositions depends on the relations among the concepts (“intensions”) of the terms occurring in the proposition; in contrast, in an extensional theory, the truth or falsity of propositions depends on the relations among the extensions of the terms (the class of things that fall under the terms).
The six traditional types of such propositions are:
universal affirmative: All men are married
universal negative: No man is married
particular affirmative: Some man is married
particular negative: Some man is not married
singular affirmative: Arnauld is married
singular negative: Arnauld is not married
Singular affirmatives and negatives, and universal affirmatives and negatives, appear to pose no problems to the CCT, but what of particular affirmatives? Here is Leibniz’s solution:
In a particular affirmative proposition it is enough that the thing should follow when something is added.....although metal does not by itself contain gold, nevertheless some metal, with an addition or specification (for example, that which makes up the greater part of a Hungarian ducat) is of such a nature as to involve the nature of gold. (Leibniz, Parkinson ed., p.51).
There is, however, a difficulty. Why not treat the singular affirmative and negative proposition in a similar way? There seem to be two readings of each available:
x is F x is not F
The concept of x The concept of x
contains the concept of F contains the concept of not-F.
x is F
x is not F
The concept of x, with The concept of x,
some consistent addition, contains with some consistent addition
the concept of F. contains the concept of not-F
This is undesirable in a number of respects, but in fact, given a simple assumption, the two readings collapse into one. All that is needed is that if there is no predicate not already contained in the concept of x which can consistently be added to it – that, in other words, the concept of x be its individual concept.
There is evidently a natural progression from an intensional logic to the CCT, but why did Leibniz insist on an intensional logic? One answer is surely that if concepts are understood in extensional terms, then truths about possible worlds would be settled by the extensions of terms – by whether or not possible worlds fall under some concept or not. But how can that be settled if possible worlds do not, in fact, exist? For normally, on an extensional approach, questions of truth or falsity are settled on what objects in fact exist. Yet possible worlds (at least all save one) do not exist.
This solves one problem – possible worlds do not have to exist for God to settle what is true concerning them (and to go on to decide which is the best) – but it raises another. For how do we, on Leibniz’s terms, do justice to the thought that what makes it true that “some pious human being is poor” is that, in fact, there is such a human being? Evidently Leibniz will say that the proposition is true if and only if the concept of existing in the actual world can be added to the concept of a poor pious human being without generating any conceptual inconsistency (whether or not the inconsistency can be displayed by a finite analysis); and that in turn will be true just if a poor pious person exists in the best of all possible worlds.
(Does there remain a problem for singular propositions, e.g. “Arnauld is (actually) celibate”? Can we suppose that the individual concept of Arnauld contains the concept of existing in the actual world? Can we suppose the complete concept of Arnauld contains the latter concept?)
Note that Leibniz was not alone in rejecting a “correspondence” theory of truth:
Frege, although a realist, did not believe in the correspondence theory of truth. ...The truth of a (complete) sentence or of the thought which it expresses is not relational: there is no question of our having first to discover the state of affairs which the sentence is intended to describe, and then to compare the sentence with it to see whether or not it corresponds; the sentence is simply true or false without qualification. Facts, in Frege’s ontology, are not further constituents of reality...alongside objects, truth-values, concepts, relations, and functions. They are, rather, to be identified with true thoughts. (Dummett, Frege, Philosopher of Language, p.442).
What of relational propositions? We shall take Leibniz’s view of relations, as stated in the Correspondence, at face-value: relations involve comparisons, and comparisons are always judgments, they are the work of the mind. Relations, it may be said, are “reducible”: relational truths are to follow from categorical truths concerning the things related. This view extends even to causal relations – indeed, this yields a stronger argument against the view that there are transworld identities, for it is plausible that there are causal relations among all the constituents of a possible world (in effect, it requires that the “thin” and “thick” sense of individual concept coincide – that the individual concept of a thing be its complete concept). Cf. “A Specimen of Discoveries About Marvellous Secrets”, in Philosophical Writings, Parkinson ed., p.84).
Readings for the final lecture:
“Monadology”, in Leibniz: Philosophical Writings , G. Parkinson, ed. p.179-94.