First Truths

G. W. Leibniz

Copyright © Jonathan Bennett 2017. All rights reserved

[Brackets] enclose editorial explanations. Small ·dots· enclose material that has been added, but can be read as
though it were part of the original text. Occasional •bullets, and also indenting of passages that are not quotations,
are meant as aids to grasping the structure of a sentence or a thought. Bold type is used where Leibniz used
italics, apparently foir emphasis. aids to grasping the structure of a sentence or a thought. Bold type is used
where Leibniz used italics, apparently foir emphasis.
First launched: September 2004 Last amended: June 2006

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First truths are the ones that assert something of itself components·. Doing that is giving an a priori proof—a proof
or deny something of its opposite. For example, that doesn’t depend on experience. From among the axioms
•A is A that are accepted by mathematicians and by everyone else, I
•A is not not-A choose as an example this:
•If it is true that A is B, then it is false that A isn’t B A whole is bigger than its part, or
(i.e. false that A is not-B) A part is smaller than the whole.
•Everything is as it is This is easily demonstrated from the definition of ‘smaller’ or
•Everything is similar or equal to itself ‘bigger’ together with the basic axiom, that is, the axiom of
•Nothing is bigger or smaller than itself identity. Here is a definition of ‘smaller than’:
and others of this sort. Although they may have a rank- For x to be smaller than y is for x to be equal to a part
ordering among themselves, they can all be lumped together of y (which is bigger).
under the label ‘identities’. This is easy to grasp, and it fits with how people in general
Now, all other truths are reducible to first ones through go about comparing the sizes of things: they take away from
definitions, that is, by resolving notions ·into their simpler the bigger thing something equal to the smaller one, and find
First Truths G. W. Leibniz

something left over. With that definition in hand, here is an into identities, contrary to the nature of truth, which is
argument of the sort I have described: always an explicit or implicit identity. ·Thus, if the axiom
1. Everything is equal to itself (axiom of identity) were false, my account of truth would be false; which is why
2. A part is equal to itself (from 1) I say that (the truth of) the axiom follows from (the truth of)
3. A part is equal to a part of the whole (from 2) my account·.
4. A part is smaller than the whole (from 3 by the It also follows that when there is a perfect balance or
definition of ‘smaller than’). symmetry ·in a physical set-up· there will also be a balance
·Because all truths follow from first truths with the help of or symmetry in what follows from it. ·Stated more abstractly·:
definitions·, it follows that ·in any true proposition· the pred- when there is symmetry in •what is given, there will be sym-
icate or consequent is always in the subject or antecedent. metry in •what is unknown. This is because any reason for
It is just this—as Aristotle observes—that constitutes the an asymmetry in the unknown must derive from the givens,
nature of truth in general, or the ·true-making· connection and in the case as stated—·where we start from something
between the terms of a statement. In identities the con- symmetrical·—there is no such reason. An example of this is
nection of the predicate with the subject (its inclusion in Archimedes’ postulate at the beginning of his book on statics,
the subject) is explicit; in all other ·true· propositions it that if there are equal weights on both sides of a balance
is implicit, and has to be shown through the analysis of with equal arms, everything is in equilibrium.
notions; a priori demonstration rests on this. There is even a reason for eternal truths. Suppose that
This is true for every affirmative truth—universal or the world has existed from eternity, and that it contains
particular, necessary or contingent—and it holds when the nothing but little spheres; for such a world we would still
predicate is relational as well as when it isn’t. And a won- have to explain why it contained little spheres rather than
derful secret lies hidden in this, a secret that •contains the cubes. From these considerations it also follows that
nature of contingency, i.e. the essential difference between In nature there can’t be two individual things that
necessary and contingent truths, and •removes the difficul- differ in •number alone,
ties concerning the necessity—and thus the inevitability—of ·i.e. that don’t differ in any of their qualities, and differ
even those things that are free. only in being two things rather than one ·. For where there
These considerations have been regarded as too simple are two things it must be possible to explain why they are
and straightforward to merit much attention; but they do different—·why they are two, why it is that x is not y·—and
deserve attention because many things of great importance for that explanation we must look to ·qualitative· differences
follow from them. One of their direct consequences is the between the things. St. Thomas said that unembodied
received axiom minds never differ by number alone—·that is, no two of
Nothing is without a reason, or them are qualitatively exactly alike·; and the same must also
There is no effect without a cause. be said of other things, for we we never find two eggs or
If that axiom were false, there would be a truth that couldn’t two leaves or two blades of grass that are exactly alike. So
be proved a priori, that is, a truth that couldn’t be resolved exact likeness is found only in notions that are incomplete

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and abstract. In that context things are considered only The complete notion of an individual substance con-
•in a certain respect, not •in every way—as, for example, tains all its predicates—past, present, and future. If a
when we consider shapes alone, ignoring the matter that has substance will have a certain predicate, it is true now that
the shape. And so it is justifiable to consider two perfectly it will, and so that predicate is contained in the notion of
alike triangles in geometry, even though two perfectly alike the thing. Thus, everything that will happen to Peter or
triangular material things are not found anywhere. Gold and Judas—necessary events and also free ones—is contained in
other metals, also salts and many liquids, are taken to be the perfect individual notion of Peter or Judas,. . .
homogeneous, ·which implies that two portions of gold could how the sentence continues: . . . considered in the realm of
be qualitatively exactly alike·. This way of thinking and possibility by withdrawing the mind from the divine decree
talking is all right if it is understood as referring only to for creating him,. . .
differences that our senses can detect; but really none of the underlying line of thought: To grasp how the concept of
these substances is strictly homogeneous. ‘the complete notion of Judas’ is being used here, think of it
[Leibniz is about to use the phrase ‘purely extrinsic denomination’. This as the complete total utterly detailed specifications for Judas,
means ‘purely relational property’, meaning a relational property that
viewed as a possibility without any thought of whether God
isn’t grounded in any non-relational property. It might seem to us that
has chosen to make the possibility actual. That is the notion
a thing’s spatial relations to other things constitute such an extrinsic
that God employed when deciding to make Judas actual: he
denomination: the thing could be moved without being in anyway altered
pointed to the possibility Judas and said ‘Let him come into
in itself. That is what Leibniz is going to deny. The word ‘denomination’
existence’, which means that he pointed to that complete
(and Leibniz’s corresponding Latin) mark the fact that he wavers between
notion and said ‘Let that be actualized’.
making this a point about •the properties and relations a thing can have,
and •the linguistic expressions that can be used in talking about a thing.
. . . and is seen there by God. This makes it obvious that out
Although basically an external denomination is meant to be a relational
of infinitely many possible individuals God selected the ones
property, Leibniz sometimes writes as though it were a relational predi-
he thought would fit best with the supreme and hidden ends
cate.]
of his wisdom. Properly speaking, he didn’t decide that
It also follows that Peter would sin
There are no purely extrinsic denominations or that
—that is, denominations having absolutely no foundation in Judas would be damned.
the denominated thing. For the notion of the denominated All he decreed was that two possible notions should be
subject must contain the notion of the predicate; ·and, to actualized—the notion of
repeat what I said at the top of page 2, this applies to Peter, who would certainly sin (but freely, not neces-
relational predicates as well as qualitative ones, i.e. it sarily)
applies to seemingly extrinsic as well as to obviously intrinsic and the notion of
denominations·. So whenever ·any· denomination of a thing Judas, who would suffer damnation
is changed, there must be an alteration in the thing itself. —which is to decree that those two individuals, rather than

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other possible things, should come into existence. Don’t . . . has spatial relation R to a pebble that is at 2◦ C:
think that Peter’s eventual salvation occurs without the and, because there are no purely extrinsic denominations,
help of God’s grace, just because it is contained in the that change in your relational properties will be backed
eternal possible notion of Peter. For what that complete by a change in your intrinsic properties·. This fits with our
notion of Peter contains is the predicate achieves salvation experience of nature. In a bowl filled with liquid, a movement
with the help of God’s grace. [Leibniz says, puzzlingly, that the ·of the liquid· in the middle is passed on out to the edges,
complete notion contains this predicate sub notione possibilitatis = ‘under becoming harder and harder to detect the further it gets from
the notion of possibility’. That seems to say where in the complete notion the centre ·but never being wiped out altogether·. Well, the
the predicate will be found—‘Look it up in the file labelled Possibility’, as whole universe is just such a bowl!
it were—but that can’t be right.] Strictly speaking, one can say that no created substance
Every individual substance contains in its complete exercises a metaphysical action or influence on any-
notion the entire universe and everything that exists in thing else. [Leibniz is saying that no real causal force or energy passes
it—past, present, and future. [The next sentence is stronger than from one substance to another. ‘Influence’ here translates the Latin
what Leibniz wrote, but it seems to express what he meant.] That is influxus [= ‘in-flow’] which reflects one view about what would have to
because: for any given things x and y, there is a true propo- happen for one substance to act on another: according to this view, when
sition about how x relates to y, if only a comparison between the hot poker heats the water, some of its heat literally passes from one
them. And there is no purely extrinsic denomination, ·which to the other; when a man falls against a wall and knocks it down, some
implies that every relational truth reflects non-relational his motion passes to the wall. The basic idea is that of an accident—a
truths about the related things·. I have shown this in many property-instance—travelling from one substance to another. The poker’s
ways, all in harmony with one another. heat is an ‘accident’ in this sense; it is to be distinguished from the poker
Indeed, all individual created substances are differ- (an •individual substance) and from heat (a universal •property); it is
ent expressions of the same universe and of the same the-present-heat-of-this-particular-poker, an •individualized •property.
universal cause, namely God. But the expressions vary in Leibniz is sceptical about the transfer of accidents from one thing to
perfection, as do different pictures of the same town drawn another, but since he thinks that substances don’t act on one another,
or painted from different points of view. he doesn’t mind implying that if they did act on one another it would
Every individual created substance exercises physi- have to be by the transfer of accidents.] For one thing, there is no
cal action and passion on all the others. Any change explanation of how something—·an accident·—could pass
made in one substance leads to corresponding changes in all from one thing into the substance of another; but I’ll let
the others, because the change in the one makes a difference that pass. I have already shown that ·there is no work for
to the relational properties of the others. ·For example, a inter-substance causation to do, because· all a thing’s states
pebble on Mars becomes colder, so that you move from follow from its own ·complete· notion. What we call ‘causes’
having the property are, speaking with metaphysical strictness, only concurrent
. . . has spatial relation R to a pebble that is at 5◦ C •requirements. This too is illustrated by our experience of
to having the property nature. For bodies really rebound from others through the

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force of their own elasticity, and not through the force of There is no atom, which means that any body could
other things, even if a body other than x is •required in order be split·. In fact, every body, however small, is actually
for x’s elasticity to be able to act. subdivided. Because of that, each body, while it ·•constantly
Assuming that soul and body are distinct, from the changes because it· is acted on by everything else in the
foregoing we can explain their union, without appeal- universe in ways that make it alter, also •preserves all
ing to •the popular but unintelligible idea of something the states that have been impressed on it in the past and
in-flowing from one to the other, and without •the hypothesis contains in advance all that will be impressed on it in the
of ‘occasional causes’, which appeals to God as a kind of future. You might object:
puppet-master. [Leibniz says Deus ex machina—a God who comes ·Your view that every body is affected by every other
on-stage by being winched down from the ceiling of the theatre. The body, and that each body contains information about
phrase ‘occasional causes’ refers to the view that minds can’t literally act all its past and all its future states, could be true even
on bodies, and that when I will to raise my arm that act of my mind is the if there were atoms·. It could be that other bodies
For God’s wisdom and
prompt or ‘occasion’ for God to raise my arm.] affect an atom by •making it move in certain ways
workmanship enabled him to set up the soul and the body, ·and by •changing its shape·, and these are effects
at the outset, in such a way that from the first constitution that the atom can receive as a whole, without being
or notion of each of them everything that happens in it divided.
through itself corresponds perfectly to everything that hap- I reply that not only must there be effects produced in an
pens in the other through itself, just as if something—·some atom from all the impacts of the universe upon it, but
‘accident’·—passed from one to the other. This hypothesis also conversely the state of the whole universe must be
of mine (which I call the ‘hypothesis of concomitance’) is inferable from ·the states of· the atom—the cause must
true for all substances in the whole universe, but it can’t be be inferable from the effect. However, any given motion
sensed in all of them as it can in the case of the soul and the of an atom ·and any given shape· could have come about
body. through different impacts, so there is no way to infer from
There is no vacuum. For if there were empty space, two the present shape and motion of the atom what effects have
different parts of it could be perfectly similar and congruent been had upon it. ·And there is a different objection to
and indistinguishable from one another. Thus, they would atoms, independent of my metaphysics, namely the fact
differ in number alone—·differ in being two, but not in any that· one couldn’t explain why bodies of a certain smallness
other way·—which is absurd. One can also prove that time is couldn’t be further divided—·that is, there couldn’t be an
not a thing, in the same way as I just did for space, ·namely explanations of why there are any atoms·.
arguing that if time were a thing there could be stretches of From this it follows that every particle in the universe
empty time, i.e. time when nothing happens; and two parts contains a world of infinitely many creatures. However,
of such empty time would be exactly alike, differing only in the continuum is not divided into points, because points
number, which is absurd·. are not parts but boundaries; nor is it divided in all pos-

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sible ways, because the contained creatures are not all them is not one substance but many.
·separately· there. It’s just that a series of divisions could go Something unextended is required for the substance of
on ad infinitum separating some from others at each stage. bodies. Without that there would be no source for the
·But no such sequence separates out all the parts, all the •reality of phenomena or for •true unity. There is always
‘contained creatures’, because· every division leaves some a plurality of bodies, never just one (so that really there
of them clumped together—just as someone who bisects a isn’t a plurality either, ·because a many must consist of
line leaves clumped together some parts of it that would be many ones·). Cordemoy used a similar line of thought as
separated if the line were trisected. an argument for the existence of atoms. But since I have
There is no determinate shape in actual things, for no ruled out atoms, all that remains ·as a source of unity· is
determinate shape can be appropriate for infinitely many something unextended, analogous to the soul, which they
effects. So neither a circle, nor an ellipse, nor any other once called ‘form’ or ‘species’.
definable line exists except in the intellect; lines don’t exist Corporeal substance can’t come into existence ex-
until they are drawn, and parts don’t exist until they are cept through creation, or go out of existence except
separated off. through annihilation, because once a corporeal substance
•Extension and •motion, are not substances, but true exists it will last for ever, since there is no reason for it not
phenomena (like rainbows and reflections). The same holds to do so. Any body may come apart—its parts may come to
for •bodies, to the extent that there is nothing to them but be scattered—but this has nothing in common with its going
extension and motion. For there are no shapes in reality, out of existence. Therefore, animate things don’t come
and if we think about bodies purely as extended, each of into or go out of existence, but are only transformed.