Dr. Pushpa Rani Prasad |

**by Pushpa Rani Prasad**

Dr. Pushpa Rani Prasad, a Commissioned Principal, working at S.P. Mahila College, Dumka, Jharkhand (India) writes in both English and Hindi. She is a pioneer of modern vision and her voice for modern Indian women is miracle and outstanding.

**Abstract:**

**Leibniz’s theory of truth:**

Leibniz
has defined truth in terms of inclusion or containment, which is evident from
his following statement:

“In
every affirmative true proposition, necessary or contingent, universal or
singular, the concept of predicate is included in that of the subject,

*praedicatum inest subjecto*.”^{1}
Leibniz
means to say that in the case of a true proposition, the concept of predicate
is included in that of the subject.

He has
also defined truth in another way. He says that a true proposition is one which
is either an identical proposition or reducible to one.

Superficially, this appears quite
different from the previous definition, but the only difference is that in the
present definition, Leibniz is calling attention to the fact that the inclusion
of the concept of the predicate in that of the subject is not always obvious,
but often has to be shown. To understand what Leibniz is saying here, it is
necessary first of all to realize that when he speaks of an identical
proposition, he does not mean only propositions of the form ‘A is A’ - e.g. ‘A
man is a man’; he also means propositions of the form “AB is A” - e.g. ‘A white
man is white’. This is obviously related to what Leibniz has said about truth
in terms of inclusion; e.g., he could have said that in the case of the
proposition ‘A white man is white’ the concept of the predicate, whiteness is
included in that of the subject. In such a case, the inclusion is manifest or
evident; however, there are many cases in which the inclusion is concealed or
implicit. Such implicit inclusion is “shown by the analysis of terms, by
substituting for one another definitions and what is defined”.

^{2}
Leibniz means that in the case of true
proposition “Every man is rational” the inclusion of the concept of rationality
in that of man is only implicit; however, it can be made explicit by substituting
for “man” its definition “rational animal”, so that we get the identical
proposition “Every rational animal is rational”. It is in this way that a true
proposition is ‘reduced’ to an identical proposition.

**Necessary truth and contingent truth:**

According
to Leibniz in the case of necessary truths or truths of reasoning, its reason
or explanation can be discovered by analysis of the notions or concepts. Thus
truths of reason are necessary propositions in the sense that they are either themselves
self-evident propositions or reducible thereto. All truths of reason are
necessarily true, and their truth rests on the principle of contradiction. To
take an example given by Leibniz, we cannot deny the proposition that the
equilateral rectangle is a rectangle without being involved in contradiction.

Contingent truths or truths of fact, on
the other hand, are not necessary propositions. Their opposites are
conceivable, and they can be denied without logical contradiction. The
proposition, for example, that John Smith exists is not a necessary but a
contingent proposition. It is, indeed, logically and metaphysically
inconceivable that John Smith should not exist while he is existing. A true
existential statement that John Smith actually exists is a contingent
proposition, a truth of fact. We cannot deduce it from any

*a priori*self-evident truth; we know its truth*a posteriori*: But if John Smith actually exists, there must be a sufficient reason for his existence; that is, if it is true to say that John Smith exists, there must be a sufficient reason why it is true to say that he exists. Truths of fact, then, rest on the principle of sufficient reason.
Truths of reason embrace the sphere of
the possible, while truths of fact embrace the sphere of existential. When a
true proposition asserts existence of a subject, it is a truth of fact, a
contingent proposition, and not a truth of reason. However, there is one
exception to the rule that existential propositions are truths of fact. For the
proposition that God exists is a truth of reason or necessary proposition, and
denial of it involves for Leibniz a logical contradiction. However, apart from
this one exception, no truth of reason asserts existence of any subject.

We
find that Leibniz has employed the two principles – the Principle of
Contradiction and the Principle of Sufficient Reason in the service of his
distinction between the truths of reasoning and truths of fact, that is,
between necessary truths and contingent truths.

Before
a critical examination of the concept of truth and Leibniz’s distinction
between necessary truths and contingent truths, it would be pertinent to define
the Principle of Contradiction and the Principle of Sufficient Reason. The Principle
of Contradiction states simply that ‘a proposition cannot be true and false at
the same time, and that therefore A is A and cannot be not A’. The Principle of
Sufficient Reason in its classic form is simply that nothing is without a
reason

*(nihil est sine ratione)*or there is no effect without a cause. Leibniz suggests that the claim that nothing takes place without a sufficient reason means that nothing happens in such a way that it is impossible for someone with enough information to give a reason why it is so and not otherwise. According to Leibniz, the Principle of Sufficient Reason must actually follow from the predicate-in-notion principle, for if there were a truth that had no reason, then there would be a proposition whose subject did not contain the predicate, which is a violation of Leibniz’s conception of truth.
Now I will discuss a difficulty that is
presented by Leibniz's theory of truth. This theory seemed to imply that there
are no truths that are not necessary. It was pointed out earlier that Leibniz
defines a true proposition as one which is either an identical proposition or
is reducible to an identical proposition. The problem is, that he offers
exactly the same definition of a necessary truth. An alternative definition of
necessary truth - namely, that a necessary truth is one whose opposite implies
a contradiction - also causes difficulties, this time in connection with
Leibniz's theory of truth as it is stated in terms of containment. Parkinson
points put that, as Leibniz saw, “'If, at a given time the concept of predicate
is in the concept of subject, then how, without contradiction and
impossibility, can the predicate not be in the subject at that time?”

^{3}
Speaking of this problem and his
attempts to solve it, Leibniz says that “A new and unexpected light finally
arose in a quarter where I least hoped for it-namely, out of mathematical
considerations of the nature of infinite”.

^{4}The solution was this: in the case of necessary truths, the inclusion of the concept of predicate in that of the subject is something that we human beings can prove. That is, we can show in a finite number of steps that the concept of the predicate is included in that of the subject; or we can, in finite number of steps, reduce to an identical proposition the proposition whose truth is to be established. But in the case of contingent truths, we cannot do this. The concept of the predicate is indeed in that of the subject, but this can never be demonstrated, nor can the proposition ever be reduced to an equation or identity. Instead, the analysis proceeds to infinity. According to Leibniz it is only God who can see the connection of terms or the inclusion of the predicate in the subject, for he sees whatever is in the series.
Parkinson is of the view that this
solution raises problems of its own. First, what exactly was the light that
arose ‘out of mathematical considerations of the nature of the infinite’? It is
clear from Leibniz’s writings that he holds that ‘rest’ can be considered as a
special case of motion - motion which is infinitely little, or which vanishes into rest.
Similarly, a contingent truth can be regarded as a special case of the
inclusion of the concept of the predicate in that of the subject-namely, where the
analysis of the concepts that would be necessary to provide a proof is
infinite.

Leibniz says that mathematics suggested
the solution, but he also knew that it suggested an objection to it. The
difficulty in question involves the irrational numbers. Leibniz often compares
the distinction between necessary and contingent truths with that between
rational and irrational numbers, or, as he says, between numbers that are
commensurable and those that are incommensurable or “surd”. He explains his
difficulty by reference to incommensurable ratios or proportions. He says that
an incommensurable ratio is not expressible, which is to say that it cannot be
expressed by a finite series of numbers, the series required is infinite.
Correspondingly, the analysis of a contingent truth is infinite. Leibniz also
points out that in mathematics, we can establish demonstrations by showing that
the error involved is less than any assignable error. As this is so, it may
seem that human beings also will be able to comprehend contingent truths with
certainty. Leibniz, however, says that this is beyond our powers. We can indeed
establish proofs of the kind described - i.e. proofs in which the error involved
is less than any assignable error - in the case of incommensurable ratio. But
in the case of contingent truths, not even thus is conceded to created mind.

But what entitles Leibniz to be sure
about this? After all, there was a time when it was thought that human being
could not give mathematical proofs of the kind to which Leibniz refers; yet
such proofs were found. Why, then, should it be beyond human powers to find comparable
proofs of contingent truths?

Leibniz has also distinguished
necessary from contingent truths in a different way. Leibniz says that
necessary truths are based on principle of contradiction and on the possibility
and impossibility of essences themselves. But the reasons that one can bring
for a contingent truth are based on that which appears the best among several
things which are equally possible; such truths (unlike necessary truths) are
based on the free will of God or of creatures. Thus Leibniz has distinguished
between necessary and contingent truths by reference to the different reasons
that can be brought for them. In a paper for Samuel Clarke, Leibniz calls the
principles involved those of contradiction and sufficient reason respectively, and
says that “what is necessary is so by its essence because the opposite implies
a contradiction; but the contingent which exists owes its existence to the
principle of what is best, the sufficient reason for things.”

^{5}
Leibniz’s use of the terms here is confusing,
in that he often uses the term “principle of sufficient reason” in such a way
as to apply to absolutely all truths, necessary as well as contingent. In this
use, he says that the principle of sufficient reason is that by which we
consider that no fact can be real or existing and no proposition can be true
unless there is a sufficient reason, why it should be thus and not otherwise.
This version of the principle of sufficient reason, which applies both to
necessary and contingent truths, adds little to what has already been seen of
Leibniz's theory of truth. All that Leibniz has added to this is the statement
that to do so is to give a sufficient reason for the truth of the proposition.
However, we have seen that Leibniz calls the principle of sufficient reason to
distinguish contingent truth, (whose principle it is) from necessary truth (whose
principle is that of contradiction). In this sense of the principle of
sufficient reason, the reasons that one can bring for a contingent truth are
based on that which is or appears the best among several things which are
equally possible, and they are related to the free will of God or of creatures.
Or, as Leibniz says elsewhere the connection between the predicate and the
subject of a contingent truth is not a necessary one, but depends on an assumed
divine decree and on free will.

I pass now to a question which is no
less fundamental. This is the question of analytic and synthetic judgement and
their relation to necessity. As regards the range of analytic judgements,
Leibniz held that all the propositions of logic, Arithmetic and Geometry are of
this nature, while all existential propositions, except the existence of God,
are synthetic. As regards the meaning of analytic judgements, it will assist us
to have in our minds some of the instances, which Leibniz suggests. We shall
find that these instances suffer from one or other of two defects. Either the
instances can be easily seen to be not truly analytic - this is the case, for
example, in Arithmetic and Geometry - or they are tautologous, and so not
properly propositions at all. Thus, Leibniz says, on one occasion that
primitive truths of reason are identical, because they appear only to repeat the
same thing, without giving any information. Among the instances given by
Leibniz are ‘A’ is ‘A’, ‘The equilateral rectangle is a rectangle’ etc. Most of
these instances assert nothing; the remainder can hardly be considered the
foundations of any important truth. These propositions are clearly
tautological. The propositions of Arithmetic, as Kant discovered are one and
all synthetic. In the case of Geometry, which Leibniz also regards as analytic,
the opposite view is even more evident. Kant, by pointing out that mathematical
judgements are both necessary and synthetic, prepared the way for the view that
this is true of all judgements. It must be confessed that, if all propositions
are necessary, the notion of necessity is shorn of most of its importance.
Bertrand Russel in his book

*A Critical Expositions of the Philosophy of Leibniz*writes:
“Whatever view we adopt, however, as
regards the necessity of existential propositions, it must be admitted that arithmetical
propositions are both necessary and synthetic, and this is enough to destroy
the supposed connection of necessary and the analytic”

^{ 6}.**CONCLUSION:**

It is clear from the above discussions
that there is no consistent account of the principle of sufficient reason
because of differences of meaning which Leibniz gives to the expression
‘sufficient reason’.

In saying that contingent truths depend
on the free will of God and necessary truths do not, Leibniz was opposing
Descartes, who said that the will of God is involved even in necessary truths,
and also Spinoza, who took the view that God cannot be said to act from freedom
of will. In sum, Leibniz holds (against Spinoza) that there are contingent
truths which depend of the free will of God and of creatures. He also holds
(against Descartes) that Gods free will is not arbitrary, God acts for the sake
of the good which is independent of his will, and his actions are in accordance
with eternal truths which his will does not produce.

Regarding the question of analytic and
synthetic judgement and their relation to necessity Leibniz has held that all
propositions of Logic, Arithmetic and Geometry are analytic. It appear to me
that the view of Bertrand Russel that arithmetical propositions are both
necessary and synthetic is correct.

* * *

**Reference:**

2. Ibid : Page-202

3. Ibid : Page-203

4. Ibid : Page-203

5. Ibid : Page-207

6. Russel, Bertrand: A Critical Exposition of the Philospphy of Leibniz; Forgotten Books, 2013, p-24.

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