The Leibnizian Background

Kant’s semantic (or, better, representation-theoretic) innovations are best illustrated by comparing them with Leibniz’s proto-semantics.

This seems to have been the reference-point which Kant had in mind

4 See Philip Kitcher (1983, Ch. 2), 1986.

when he explained his discoveries.⁵ According to Leibniz’s famous inesse-principle, “an affirmative truth is one whose predicate is in the subject”. According to this view of the semantics of propositions – we could call it the conceptual containment model – every proposi-tion is either explicitly or implicitly identical; confining attenproposi-tion to categorical and affirmative propositions, the contention is that every proposition of the form “A is B” (read: predicate B belongs to sub-ject A) reduces ultimately to one of the form “AB is B”, which shows that the predicate is a part of, or is contained in, the subject. Another way to put this is to say that, according to Leibniz, the truth of a true proposition is grounded in the principle of contradiction. One of his formulations of the principle is that identical statements are true and their contradictions are false.⁶ When we add to this the principle of sufficient reason (principium rationis), we get the two principles which for Leibniz underlie all knowledge. In its most general form the prin-ciple of sufficient reason can be expressed in some such form as

“there is nothing without reason” or “nothing happens without rea-son”, but it can also be formulated in a manner that is less metaphysi-cal and more logimetaphysi-cal in character. Formulated in this way, the princi-ple has to do with the demonstration of truths and it amounts to the requirement that every truth which is not identical (per se true) is provable from such identities. This formulation gives at the same time Leibniz’s definition of truth.⁷

The logical version of Leibniz’s principium rationis or the contention that every proposition must have an apriori proof (proof that is inde-pendent of experience) is likely to strike a less rationalistically inclined philosopher as far-fetched. Nevertheless, it gains some plausibility (or, at any rate, is less obviously false), if it is restricted to the sphere of mathematics.⁸

5 As is pointed out by Brittan (1978, p. 46).

6 See Kauppi (1960, Ch. II.4) for references and further discussion.

7 See Kauppi (1960, Ch. II.6).

8 Even though the principium rationis is supposed to apply to all truths, its content differs depending on whether the truths to which it is applied are necessary (like mathematical truths) or contingent. Firstly, since contingent truths are about infinitely complex individual substances, the possibility of

In the case of mathematics, the demand for a proof amounts to this. Since mathematical truths are among the truths of reason (vérités de raison), they are characterised by what Leibniz called “absolute ne-cessity”. And this means that proofs in mathematics rest on the prin-ciple of contradiction. That is to say, every mathematical truth – with the exception of explicitly identical truths, which “cannot be proved and have no need of proof”⁹ – are provable from explicitly identical truths, i.e., reducible to such truths by the substitution of definitional equivalences. That such a proof is capable of yielding a genuine dem-onstration (and hence knowledge of the demonstrated proposition) depends on two further claims. Firstly, identical truths are per se true, giving an a priori proof for them is beyond the reach of the finite human intellect: it is only God who grasps, with His infinite intellect, such truths.

Thus, Leibniz explains, in Discourse on Metaphysics, §8, that God, who sees the individual essence of Alexander the Great, sees in it “the foundation of and reason for all the predicates which can be truly predicated of him”. There-fore God knows apriori that Alexander defeated Darius and Porus. Similarly, he knows apriori whether Alexander died a natural death or whether he died by poison, truths which finite human beings can know only through history and hence aposteriori. Secondly, it follows from the absolute necessity of mathematical truths that their proofs can be grounded, in the manner Leib-niz explains, in the principle of contradiction. Even if there were an apriori proof for a contingent truth like “Alexander defeated Darius”, it could not be so grounded; such a truth is necessary only in the hypothetical sense that it is a consequence of God’s free decision and hence follows by “moral ne-cessity” from the principle that “God will always do the best” (id., §14).

What is less than perfect does not imply a contradiction in the logical sense (“Alexander who did not defeat Darius” is not a contradictory notion in itself it is only incompatible with what takes place in the best of all possi-ble worlds). The principle, therefore, that is operative in such would-be-proofs would have to be stronger than the principle of contradiction. Essen-tially, it would be a principle to the effect that the definition of, say, Alexan-der, on which the proof operates is a real definition. And what makes it a real definition is in the last instance the fact that the notion codifies God’s choice to create the actual Alexander from all possible Alexanders existing in his mind as different complete individual concepts. As Erik Stenius has put it, “the very definition of Alexander qua a real definition is an act of creation, the creation of Alexander” (1973, p. 332).

9Monadology, §35.

which makes their truth immediately evident. Secondly, the defini-tions that are made use of in the proof must be real definidefini-tions, and not merely nominal definitions; that is, they must be definitions of ideas that are really possible. In the case of mathematical ideas, how-ever, this possibility is said by Leibniz to be immediately evident.¹⁰ Leibniz’s views on this matter can be illustrated with the well-known example from his New Essays.¹¹ There Leibniz (or his alter ego Theophilus) gives a proof of the proposition “2 + 2 = 4”. In accor-dance with what was said above about proofs in mathematics, it can be described as follows. Leibniz assumes as definitions the following:

(1) 2 = 1 + 1; (2) 3 = 2 + 1; (3) 4 = 3 + 1. The substitutivity of iden-ticals is taken as an axiom, and the proof itself can be written as fol-lows:

1. 2 + 2 = 4 2. 2 + 1 + 1 = 4 3. 3 + 1 = 4 4. 4 = 4

Here the first line states the proposition to be proved. Each subse-quent line is obtained from the immediately preceding one by substi-tuting definitionally identical terms. The proof terminates with an explicitly identical proposition. This is the general schema that

ap-10New essays, bk. 4, Ch. ii, §1. Elsewhere in the Essays Leibniz illustrates the difference between real and nominal definitions with an example from geometry. To define parallel straight lines as “lines in the same plane which do not meet even if extended to infinity” is to give a merely nominal definition, since it can still be doubted whether such lines are possible. When it is ex-plained, however, that, given a straight line, a line can be drawn with the property that it is at each point equidistant from it, it is at once seen that the thing is possible and why the two lines never meet (id., 3.iii.18). As Kauppi (1960, pp. 106-8) points out, Leibniz distinguishes among real definitions a species, so-called “causal definitions”, which describe a method through which the thing defined is created. Geometrical definitions are precisely of this kind; as Leibniz says, geometers have recognised the need at least to postulate the possibility of things corresponding to their definitions.

11New Essays, bk. iv, Ch. vii, §10.

plies, according to Leibniz, to all (non-primitive) necessary truths or truths of reason (in particular, to mathematical truths).¹²