Types of Truth

Perhaps if Leibniz's system has a single core principle, it is his distinction between necessary and contingent truths. His proofs of God's existence depend to some degree on this distinction, as well as his views of human freedom and defense of God in the face of evil. Although hinted at in Aristotle, Leibniz's distinction between necessity and contingency are explicated with clarity and at length, in seminal work that would lead Hume to his own distinction between "relations of ideas" and "matters of fact", and Kant to his distinction of "analytic" and "synthetic" propositions.

So let us start with the basics. "Necessary truths are those that can be demonstrated through an analysis of terms, so that in the end they become identities. . . . That is, necessary truths depend upon the principle of contradiction." (From, "On Contingency", p. 28.) Sound simple? Well, so far so good. "All squares have four sides" is a necessary truth, since anything with five sides is not a square. The word "square" is defined by its having four sides, such that "Five-sided square" is a contradiction in terms. In the same way, "If he is running, then he is running" is a necessary truth. The only thing that can make this proposition false is a contradictory (impossible) state of affairs: that he is running and not running.

Contingent truths, on the other hand, are truths such as "I am at home right now" or "This apple is red." There is no logical contradiction in my being elsewhere right now, nor in a non-red apple.

But this is where it gets complicated, and perhaps Leibniz is flirting with disaster here. Leibniz believes that each object is a bundle of properties; in the whole concept of "me" somewhere there is the property that I will be here at home right now. According to Leibniz, there is no truth that does not have some connection between subject and predicate. "This apple" contains, among other things "being red" as one of its properties.

So why is it, when I say, "This apple is red," that the truth of this does not rely on the principle of contradiction? Contained in the entire concept of the apple, redness is certainly among its properties; so when I say "This apple" I mean a red apple; the predicate adds nothing to the subject. We seem to be stuck with the idea that "This apple is red" is what Kant would call an analytic truth. If we analyze the subject "This apple" we will find "being red" among its properties, such that "This apple is not red" seems to imply a contradiction in terms.

By "This apple" is meant the entire content of the apple's properties, such that to enumerate any of them would mean merely analysis of the term "This apple" and not a synthesis with any idea not contained in "This apple". The entire distinction between necessary and contingent truths seems to have fallen apart.

The answer comes in the following way. Leibniz believed "All squares have four sides" was necessary in that its denial meant a contradiction in terms, while "This apple is red" issues from the nature of "This apple" and how it fits in with an infinity of other truths. According to Leibniz, the apple sits a certain distance from the sun; likewise in a certain location of the galaxy; the fact that I am contemplating it, or that you are, or that your grandparents' grandchildren would one day contemplate it, are all among its properties. Leibniz believed that could we know all the apple's properties, it would require omniscience (knowledge of everything). The apple sits in relation, in one way or another, to everything that is and ever was and ever will be. These relations are among its properties, such that the apple has infinite properties, and stands in relation to infinite things. To know them all I would have to literally know everything; to alter one property by biting the apple would require that I alter some properties of everything that is.

And this is where Leibniz's distinction between necessary and contingent truths becomes coherent. He believed that "This apple is red" might not have been true; and what it would mean for it not to be true is that every single truth in the cosmos would be altered were the apple not red. But, nonetheless, they could coherently be altered in such a manner, every single truth adjusted, every object having slightly different properties. In no coherent manner could God have created a five-sided square; but had he altered every object in the cosmos to account for it, he may have made this apple yellow. To make a necessary truth false God would need to create a true contradiction; to make a contingent truth false God would merely have had to alter the infinity of other contingent truths to allow for it.

An interesting consequence of this, one that is somewhat counterintuitive, is that "This apple is red, and possibly this apple is not red" is perfectly coherent. What does it mean for something to be possible? That it is not impossible (contradictory). Thus, "This apple is red" asserts a contingent truth; it is true, but its denial is not impossible, that is, it may have turned out not to be red. When I first heard in a lecture that "P and possibly not P" was coherent it sounded ridiculous. Given that I am named Jason, how is it possible that I'm not named Jason? It is possible in the sense that I may not have been named Jason; that is, it wouldn't have been impossible for me to have ended up with another name. Another little consequence of the distinction between contingency and necessity is that, "Necessarily, if I go out tomorrow, I will go out tomorrow," is true, but "If I go out tomorrow, necessarily I will go out tomorrow," is false. With the latter proposition we have merely misplaced the "necessarily" modifier. I may go out tomorrow, but by no means, when I do so, will it be impossible for me to have done otherwise (thus, while I do go out, I won't go out necessarily).

This is actually a common error. I once wrote a little argument that I thought proved once and for all that all truths are necessary truths. It rested on the rule in logic that one can switch names used to indicate the same object, always without a change in truth-value. Thus I imagined a wooden, green cube, and proceeded to argue it was a cube, it was made of wood, and it was green by necessity (it could not have been otherwise without a contradiction). I began by saying, "Necessarily, the cube has six sides." Then I switched names for it from "Cube" to "Wooden thing"—which I can do with no change in truth-value according to the rules of logic—and said, "Necessarily, the wooden thing has six sides." Again I switched terms and said, "Necessarily the green thing has six sides." Then I did the same thing with the other properties—being wooden, being green, and being a cube. "Necessarily, the green thing is green"; "Necessarily, the wooden thing is green" etc.

This seems too clever, and it is. In fact I made a very basic error. True, necessarily all cubes have six sides. But for any given cube, it does not have six sides by necessity, any more than it was a cube in the first place by necessity. That all cubes have six sides is necessary; but for any given cube, it has six sides by necessity if and only if it was a cube in the first place by necessity. Someone might have carved a pyramid out of it to begin with, so that it never was a cube at all. Necessarily all cubes are cubes, but not all cubes are by necessity cubes.

Just as "Necessarily if Fred is running, Fred is running" is true; but "If Fred is running, necessarily Fred is running" is false, due to a misplaced "Necessarily" indicator. Perhaps Fred is in fact running, but it could have turned out that he weren't. Just because he is running doesn't mean he's running by necessity; that is, it doesn't mean it was impossible for him to have not been running.

All in all, Leibniz's distinction between necessary and contingent truths proved to be his most seminal accomplishment. Right down to our own day work is being done with possible world theory and modality based on ideas that were first made explicit with Leibniz. It has consequences for whether we can prove the existence of God, and whether free will is possible.

Angelhaunt.net: Because earth's madness is heaven's sense.