Leibniz' Doctrine of Necessary Truth

Margaret Dauler Wilson

• Publisher: Taylor & Francis
• Year: 2019
• Print ISBN: 9780367334628
• eBook ISBN: 9781000300871
• Edition: 1
• Topic: Philosophy
• Subtopic: Business General

Originally published in 1990. This study was first written in 1965 when interest in Leibniz was intensifying. The book looks in detail at the doctrine of necessity – that necessary truths are those derivable from the principle of identity by the substitution of definitions. It first considers views of philosophic predecessors, relating Leibniz' doctrine to Aristotle and Hobbes among others. The second section examines the conflict between his reductionistic and formalistic views and the opposing intuitionism and anti-reductionism of Descartes and Locke. The author critically examines the theory of necessity, including Leibniz's arguments against the views of Hobbes and Locke, concluding with distinctions between necessary and contingent truths.

CHAPTER 1

LEIBNIZ’ DOCTRINE AND SOME HISTORICAL ANTECEDENTS

In this chapter I shall approach Leibniz’ theory of necessity by first considering views elaborated by four of his predecessors—Aristotle, Aquinas, Nicholas of Autrecourt, and Hobbes. Each of these philosophers, as will become clear, anticipates Leibniz in respect of some of the elements of his doctrine. Except in the case of Hobbes, it will not be maintained that Leibniz was directly influenced, in arriving at his view of necessity, by study of the works of these writers. (Leibniz is known to have been a close student of Aristotle, and he was thoroughly educated in the scholastic tradition, but direct influence of particular philosophers on particular aspects of his doctrine of necessity cannot, in general, be established¹). Nor, of course, is the examination of the positions of these four philosophers in any way intended to constitute a comprehensive survey of the development of philosophic treatments of the concept of necessity up to the time of Leibniz. The aim is rather the far more modest one of (1) indicating what elements of Leibniz’ doctrine may assuredly be called “traditional,” and from what “traditions” they respectively derive; and (2) illuminating certain important but subtle differences that obtain between Leibniz’ doctrine and others similar to it, which have tended to pass unnoticed by commentators. (For example, it will be discovered that philosophers before Leibniz commonly held that necessity involves or consists in the “contradictoriness of the opposite,” without arriving at Leibniz’ view that necessary truths are those of which the denial can be reduced to a formal contradiction.)

I shall try to show that Leibniz’ theory proceeds from a sort of marriage between the Aristotelian-Scholastic tradition on the one hand and the tradition of the geometers and Hobbes on the other hand. From the former come the notions, (1) that necessary truth differs from contingent truth in resting strictly on a definitional relation between the subject and predicate terms; (2) (a) necessity/impossibility can be defined by reference to the principle of non-contradiction, which (b) is at the same time, in some sense, the first principle of knowledge; and (3) that there is an important distinction between a truth’s being necessary absolutely or in itself, and its being necessary only hypothetically or relative to certain premises. From the geometrical tradition derive, (1) emphasis on the importance of demonstration by means of definitions and axioms (with a clear indication of what is assumed in any proof); and (2) the view that the assumptions should be reduced to as small a number as possible—that it is valuable to attempt proofs even of apparently self-evident propositions.

1. ARISTOTLE

The conception of necessity.—Aristotle conceived of the problem of necessity as bound up with questions about the possibility and foundation of scientific knowledge. The latter questions, in turn, are approached from a metaphysical point of view. Aristotle asks not how we may come to know necessary truths, nor how we may recognize their necessity, but what connections in the objects of knowledge are necessary; what is therefore the appropriate subject matter for science. That which is not capable of being otherwise is necessarily as it is (τò μή ἐνδεχόμενον ἄλλως ἔχειν ἀναуκαῖόν ϕαμεν οὕτωϛ ἔχειν)². This is the primary sense of “necessary.”³ And only when we know that something is necessary in this sense do we say we have unqualified scientific knowledge of it (ἐπιστάσθαι οἰόμεθ ἓκαστον ἁπλῶϛ).⁴ “Consequently,” Aristotle continues, “that of which there is unqualified scientific knowledge is something which cannot be otherwise” (ὣστε οὗ ἁπλῶϛ ἔστιν ἐπιστήμη, τοῦτ’ ἀδύνατον ἄλλωϛ ἔχειν).⁵ Science,then,beginswith necessary premises. But we must consider what these are.

Aristotle finds that necessary premises are those involving predication “καθ αὑτό” or essential predication.⁶ Either what is predicated of the subject is part of the ousia of the subject (as line to triangle or point to line, for the ousia of the latter is in each case constituted by the former), or the subject is included in the explication of the nature (“τὸ τί ἐστιν”) of the predicate which is predicated of it. Thus to predicate straightness of a line is to predicate essentially, since straightness cannot be defined without reference to line (he gives several other examples).

It should be remarked that while Aristotle’s account of essential predication and hence of necessity involves reference to definition, this by no means reflects a desire on his part to represent the problem of necessity as one having to do with a strictly linguistic connection between subject and predicate terms of a proposition, as opposed to the real connection between a “being” and its attributes. This is clear from the fact that his concept of definition is inseparably bound up with his concept of ousia, and from the fact that he conceives the problem as one concerning the objects of scientific knowledge. Leibniz will be found to differ from Aristotle to the extent of speaking almost exclusively of necessity as a property of propositions, the subject and predicate terms of which are related in a particular way. On the other hand, we shall also find that Leibniz would not go beyond Aristotle to the point of accepting a view of necessity which divorced it from the level of “being”; nor even, in the last analysis, of altogether relinquishing the notion of necessary connections in rebus.

The principle of non-contradiction.—In the discussion of necessity in the Posterior Analytics there is no reference to a formal criterion of necessity, such as Leibniz was to seek in the principle of identity or non-contradiction. In the Metaphysics Aristotle does describe the principle of non-contradiction as the first principle of all demonstration, “for this is naturally the starting point even for all the other axioms.”⁷ Those who ask a proof even of this are simply uneducated:⁸ if this principle is denied there can be no reasons and a fortiori no proof of anything.⁹ But if we are to grasp the distance that separates Aristotle and Leibniz in their thoughts on this subject, it is necessary to observe the following points.

(1) As has been widely noticed, the principle of non-contradiction is not explicitly formulated by Aristotle as a formal or logical principle, but rather as a metaphysical principle.¹⁰ He characterizes it, along with the other axioms, as a principle of being qua being (τοῦ ὄντοϛ ἐστιν ἧ ὄν).¹¹ Aristotle does not usually say that if a given proposition is true, then its negation is false, and no proposition can be both true and false.¹² Instead, he is accustomed to formulate the principle as follows: “τὸ χὰρ αὐτὸ ἃμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό” (“The same thing cannot belong to (subsist in, be attributed to) and be denied of the same thing in the same respect”);¹³ or, even more strikingly: “οὐκ ἐνδέχετα τὸ αὐτὸ καθ ἕνα καὶ τὸν αὐτὸν χρόνον εἶναι καὶ μὴ εἶναι καὶ τἆλλα τὰ τοῦτου αὑτοῖϛ ἀντικείμευα τὸν τρόπον.” (“It is not possible for the same thing at one and the same time to be and not be, and similarly for all other such opposites.”)¹⁴ Leibniz, on the other hand, almost always expresses the principle in a way that would be more acceptable to modern logicians. (It will be shown, however, that he does not wholly abandon the ontological formulation.)

(2) Aristotle does not mention the principle of identity as being on a par with or in any way involved with the principle of non-contradiction. This relative disregard of the principle of identity appears to have been the rule in the Aristotelian and Scholastic tradition down to the time of Leibniz.¹⁵ (We shall see, however, that Nicholaus of Autrecourt, while not speaking of the principle of identity as such, emphasizes the point that propositions true by virtue of the principle of non-contradiction must be identities.)

(3) Most significantly, I think, Aristotle does not present the principle of contradiction as being in any way involved with the definition or criterion of necessity. He merely says that it is the most certain (βεβαιοτάτη) of all principles; the principle about which it is not possible to be mistaken (περὶ ἣν διαψευσθῆναι ἀδύνατον).¹⁶ As already noted, Aristotle defines the necessary as “that which cannot be otherwise.” Similarly, in De Interpretatione he equates ‘“necessary” with “opposite not possible” and “possible” with “opposite not necessary”¹⁷ This is of course perfectly in accord with ordinary language, but does not go much beyond it. (“What does it mean for something to be necessary?” “Well, that it has to be that way.”) There is an important step—a step, I would say, from tautology to theory as well as from ordinary language to logic—between the assertion that something is necessary if it cannot be otherwise, and the two-fold contention that necessity applies to propositions, and that a proposition is necessarily true if and only if its negation implies a formal contradiction (or that its being necessary means that its negation implies a formal contradiction). This point may partially be rephrased as follows: “self-contradictory” is not itself a modal expression; hence, the effort to explain necessity in terms of self-contradiction is potentially illuminating in a way that an explanation using other modal terms (“cannot be otherwise”) is not. (As we shall soon see, Leibniz was not the first to link necessity with self-contradiction. But I believe he was the first to attempt a systematic exploitation and development of this connection.)

It is true that Aristotle goes well beyond what would presumably be obvious to the ordinary speaker of Greek in his attempt to explain “cannot be otherwise” in terms of predication καθ’ αὑτό But this explication in turn does not seem to point to any general or self-sufficient criterion. It rather appears to rely on the arrival at intuitive understanding through the citation of examples.

The contrast between absolute and hypothetical necessity.—Finally, we should note that there already appears in Aristotle’s works a suggestion of the distinction between absolute and hypothetical necessity, which plays an important role in Leibniz’ thought. In one brief passage, that is, Aristotle explicitly recognizes a distinction between what is necessarily true ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Preface to the Garland Edition
• Acknowledgments
• Table of Contents
• Abbreviations
• Introduction
• Chapter I: Leibniz’ Doctrine and Some Historical Antecedents
• Chapter II: Seventeenth Century Opposition: The Intuitionism of Descartes and Locke
• Chapter III: Some Critical Challenges to Leibniz’ Doctrine of Necessity
• Chapter IV: Leibniz’ Opposition to Conventionalism and his Conception of Definition
• Conclusion
• Bibliography