Gottfried Leibniz: One of the Great renaissance men of Western thought - A Critical Estimate of Leibniz’s theory of Truth

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:
According to Leibniz’s theory of truth, the nature of truth consists of connection or inclusion of a predicate in a subject. Leibniz uses his apparently self-evident principles such as the principle of sufficient reason, the law of contradiction, the identity of indiscernible and predicate-in-notion theory of truth to develop a remarkable philosophical system that provides an intricate and through account of reality. Ultimately, Leibniz’s universe contains only God and non-composite immaterial, soul-like entities called “monads”. In this article, I have tried to critically examine his concept of truth, the principle of ‘sufficient reason’ and the principle of contradiction. I have tried to point out 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’.
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.
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Reference:
1. Parkinson, G.H.R.: Philosophy and Logic, In The Cambridge Companion to Leibniz; ed. By Nicholas Jolley, Cambridge University Press, 1995, p-200.
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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