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Leibniz was a philosopher of principles: the Principles of Contradiction, of Sufficient Reason, of Identity of Indiscernibles, of Plenitude, of the Best, and of Continuity are among the most famous Leibnizian principles.

[Footnote: According to Ortega y Gasset, Leibniz is both the philosopher who used the greatest number of maximally general principles and the philosopher who introduced the greatest number of new principles — Reference: Ortega y Gasset, J. 1979. La Idea de Principio en Leibniz y la Evolución de la Teoría Deductiva. Madrid]

… Leibniz gives several different formulations of the Principle of Contradiction, among them the following ones:

• PC1: For any two contradictory propositions p and q, one is true and the other is false.

• PC2: For any proposition p, p is either true or false.

• PC3: For any proposition p, p is not both true and false.

• PC4: For any proposition p, if p implies a contradiction, then p is false.

• PC5: For any proposition p, if p is false, then not-p is true.

• PC6: For any proposition p, if p is an identical proposition, then p is true.

… One might think that the fact that Leibniz referred to such a variety of principles as the Principle of Contradiction does not necessarily suggest confusion on his part, since in at least one text he formulates several principles having to do with truth and falsity, including PC2 and PC3, and says that all of them are usually included in one designation, “Principle of Contradiction”.

Thus, one might think that he used the phrase “Principle of Contradiction” as a collective noun. … But, overall, there is little evidence for this collective noun hypothesis. The texts suggest that on different occasions he used “Principle of Contradiction” to refer to different principles. This is puzzling, given the subtlety and power of Leibniz’s mind, for it suggests that he did not really distinguish between the different versions of the principle.

A hypothesis that would save Leibniz’s clarity of mind with respect to the distinction between these principles is that he thought of “Principle of Contradiction” as a name of whatever principle played a certain function in his theory—roughly, a principle that, in his view, excluded true contradictions and served to ground mathematical and necessary truths in general.

According to this hypothesis, the meaning of “Principle of Contradiction” is relatively constant throughout Leibniz’s work, but in different texts he proposes different principles as playing that role. Thus calling these different principles “Principle of Contradiction” is consistent with him having noticed that they are different.

I think this hypothesis is more likely than the collective noun one …

In any case, Leibniz did see some of these principles as more fundamental than others. … Leibniz seems to equate PC3 and PC6:

The great foundation of mathematics is the principle of contradiction, or identity, that is, that a proposition cannot be true and false at the same time; and in this way A is A, and cannot be not A.

Here Leibniz seems to equate PC3 with the proposition that A is A. Now, Leibniz thinks that L and L is true are coincident propositions, that is, that they can be substituted for one another without loss of truth. Therefore, A is A is coincident with A is A is true, and in general, every identity is coincident with a proposition saying that it is true. But then, by universal generalization, one obtains PC6. However, I am not aware that Leibniz ever proposed such a derivation of PC6.

Furthermore, it is not clear that PC3 and PC6 are equivalent, and Leibniz does not explain why this is supposed to be so.

Let us discuss the formulation of the Principle of Sufficient Reason. The essence of this principle is that there are no brute facts or truths, that is, there are no facts or truths for which no explanation can be given. Leibniz also gives various formulations of this principle, formulations that are not equivalent—at least not without presupposing further Leibnizian ideas. Here are three such formulations:

• PSR1: Nothing occurs without a sufficient reason why it is so and not otherwise.

• PSR2: Nothing occurs without a cause.

• PSR3: Every truth has an a priori proof.

… The Principle of Identity of Indiscernibles is also formulated in different ways. Typically, Leibniz formulates it in these ways:

… It is important to note that the principle [Identity of Indiscernibles] means, according to Leibniz, that things must differ intrinsically. A mere relational difference is not sufficient. Furthermore, for Leibniz things that differ must differ qualitatively. That is, numerically different things must differ with respect to how they are intrinsically, and not with respect to which ones they are.

Thus, if a and b are different but their difference is simply due to the fact that one of them is a and the other is b, a and b will differ solo numero, and this would be a violation of the Identity of Indiscernibles.

For a and b to satisfy the Identity of Indiscernibles, they must differ qualitatively—they must differ more than numerically, in the sense that their difference must be grounded in a difference in their intrinsic qualities.

… according to Leibniz, what can be known of a thing by inspecting it on its own are its qualities. Thus, all the denominations of a thing must be founded on its qualities, and therefore the difference between two things must be a qualitative difference. ….

… according to Leibniz, similarity is sharing of qualities, and qualities are what can be known about a thing by inspecting it by itself, without comparing it to other things.

… Thus, Leibniz claims to derive the Identity of Indiscernibles from his thesis that substances have complete individual concepts. But he can only establish the Identity of Indiscernibles if he assumes a thesis (that identity reduces to intrinsic qualitative char- acter) that establishes by itself the Identity of Indiscernibles. ….

…. Leibniz’s argument for the Identity of Indiscernibles is a simple one.

The argument is that individual substances have complete concepts that permit to deduce everything that is true of them.

Since they permit to deduce everything about them, they permit to deduce facts about the identity of substances. But those complete concepts are purely qualitative. Therefore, there cannot be two substances that resemble each other perfectly. This is, I think, a valid argument, but with very controversial premises. …

[You have now reached the end of this Social Science Files exhibit.]

Gonzalo Rodriguez-Pereyra has written a book on this topic and is a fairly recent Social Science Files subscriber. He is at University of Oxford, but his first job was at Universidad Nacional de Córdoba in Argentina [founded 1613] , which is an excellent university. I have visited often because for 8 years I sent Australian students to study there, and also because I worked in Mendoza in 1995. But my first trip to Córdoba was in 1963 when my father drove us across La Pampa from Buenos Aires in VW van!

Note our #1 Google hit when searching for the source of Kenny’s quote, which I think may have been Kenny’s own translation as I cannot find it verbatim.

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