A Timeless Genius: Leibniz’s Philosophy and Logic

The study of logic has long been one of the main areas of philosophical enquiry. Yet the theory of logic a particular philosopher adopts can only rarely be separated from other philosophical commitments they hold. Certainly, that could not be said of Leibniz. His philosophy of logic – logic here having a broad meaning, beyond the modern use which limits it primarily to a focus on propositional validity – is indeed inseparable from his metaphysics, his epistemology, and his philosophical system as a whole.

It is worth emphasizing that Leibniz is an especially systemic philosopher: he intends his work in one area to support his work in another. This underlying faith in the coherent knowability of the world is partly informed by Leibniz Christian faith, which informs many of his more specific philosophical views besides. This article begins with a discussion of what Leibniz understands ‘logic’ to mean. It then moves on to a discussion of the concepts of necessary and contingent truth in Leibniz’s work. This is followed by a discussion of Leibniz’s definition of contingent truths, before a conclusion focuses on the question of free will and its relation to Leibniz’s work.

The relationship between Leibniz’s logic and his philosophy at large is a longstanding issue. Bertrand Russell argued that Leibniz derived his views about substance from his view about logic. Logic here refers not to its modern meaning, i.e to do with the structure of valid reasoning, but rather to do with the nature of the proposition, the concept, and of truth itself.

Leibniz’s theory of the proposition and of truth must be understood synoptically, given that Leibniz’s theory of truth has much to do with his belief that true propositions can always be expressed in the form of a subject-predicate pairing. Leibniz’s conception of the subject and predicate are quite close to the modern one: the subject is that about which something is said, and predicate are things which are said about it. Leibniz’s use of “concept” is similarly still modern – a concept can roughly be understood as the meaning of a word. Leibniz applies “concept” broadly – not only is there a concept of, say, the church (with a small ‘c’, you’ll notice), there is also a concept of each individual church.

Leibniz develops a theory of inclusion/containment, which holds that in true propositions, the concept of predicates is included in that of a subject, meaning that it is included among those which constitute the concept of the subject. Leibniz himself contrasts this with the Scholastic approach: “for they consider, not concepts, but instances which are brought under universal concepts.” Leibniz’s preference for his own approach comes from his view that concepts “do not depend on the existence of individuals”.

To put the dichotomy here straightforwardly, Leibniz here wants to claim that things are defined in terms of descriptors we apply to them, rather than being actualized instances of those descriptive concepts.

Leibniz adds that a true proposition must be one which is either an identical proposition or reducible to one. This might seem as though Leibniz is only intending to leave us calling trivial propositions true. When he talks about identical propositions, Leibniz isn’t just speaking of propositions like ‘A is A’, but also ‘AB is A’. Not just ‘gray is gray’ but ‘the gray horse is gray’. Often that a proposition is identical is submerged.

There is a disagreement between Leibniz and Spinoza concerning possibility and necessity that clarifies this point. For Spinoza, all that is possible necessarily exists, because otherwise God could not be both necessary and absolutely infinite. Leibniz’s refutation of this is based on the fact that we can conceive of things which have not in fact existed, but for which there is no reason they might not have.

It seems to be a consequence of Leibniz’s views on predication that all truths are necessarily true, being that they are propositions of identity or reducible to them. Leibniz’s solution to this is to claim that contingent truths are those for which the reduction to a statement of identity proceeds to infinity, and so is only accessible to God. “If, at a given time, the concept of the predicate is in the concept of the subject, then how, without contradiction and impossibility, can the predicate not be in the subject at that time?”.

Leibniz’s solution is to claim that, whereas for necessary truths we can show that the predicate is included in the subject and how, this is not true for contingent truths. For contingent truths, to show this would take an infinite number of steps.

Leibniz is led in this direction by his work in mathematics and physics, in which he theorizes that rest is a special case of motion, where motion is infinitely little. This isn’t to say that contingent truths are really necessary, so much as a way of observing how the difference is articulated through our comprehension of it.

A question offers itself here – how is Leibniz so sure that human beings couldn’t make the necessary steps to explain the location of the predicate in the subject of contingent truths? Leibniz has another way of distinguishing necessary truths from contingent ones. Leibniz holds that necessary truths: “are based on the principles of contradiction and on the possibility or impossibility of essences themselves”. If P is a property of S, then to say that S having property P is a necessary truth is to say that it would be contradictory to claim otherwise. It is “based … on ideas pure and simple”.

Contingency and Necessity According to Leibniz

If that is the reason to believe in necessary truths, the reasons for believing in contingent truths are – as one would expect – less binding. They are “based only on … that which is or appears the best among several things which are equally possible”. In other words, they are based on the “free will of God or of creatures”. The reasons behind holding contingent beliefs “incline without necessitating”.

Given that most contingent truths are an effect of God’s will, we can broadly limit ourselves to a focus on truths of that kind. This distinction is based on the different reasons for necessary truths and contingent truths. At certain points Leibniz uses the phrase ‘principle of sufficient reason’ to refer to the acquisition only of contingent truths, and at other points in his work he seems to apply it to all truths: “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”.

Often, Leibniz refers to the principle of sufficient reason in the former sense (i.e. to refer only to contingent truths) as the ‘principle of the best’, indicating that whilst contingent beliefs do not necessitate our admitting them, we admit such contingent beliefs as appear to us to be the best (the best attempt to answer a certain question, or the best thing to do in the case of ‘normative’ beliefs). The principle of sufficient reason as it is used to refer to all truths is a logical principle, whereas its former sense it relies on metaphysical commitments – in particular, beliefs about God.

Many contingent truths are a matter of God’s will, and the very idea of God having a will was controversial. Descartes, for instance, effectively holds that God’s activity precedes any concept of the ‘best’ – what is best is what God wills, and what is best is the best because God wills it. Leibniz clearly believes Descartes’ view to leave God’s will as vacuous, a mere ‘fiction’. Moreover, Descartes’ view involves positing God’s understanding as pre-existing any truth which could be its object, which is (Leibniz thinks) patently absurd.

How do these two accounts of the contingent truth – the account based on the infinite analysis of identity between a proposition and its reduction to the expression of identity between a subject and what is predicated of it, and the account based on God’s willing-for-the-best – relate to one another? Leibniz isn’t explicit about this, but we can plausibly suggest the following answer. The reason why infinite analysis must be a component of contingent truths, and why this analysis cannot possibly be executed by a human being, is that the will behind a contingent truth is free, so no necessity – which in Leibniz’s view means logical necessity – can be brought to bear on an analysis of it.