In contrast, claims Spinoza, the existence as well as the non-existence of a triangle or any other thing that is not a substance do no follow merely from the essence of the triangle:. The reason why a circle or triangle exists, or why it does not exist, does not follow from the nature of these things, but from the order of the whole of corporeal Nature. Thus, consider the following three propositions: i The triangle has three angles, ii The substance exists, and iii The triangle exists.
Proposition i is clearly self-explanatory, since the essence of the triangle which contains the nature of the number three and the nature of an angle is the sufficient reason for its having three angles adding any other information is of no explanatory value. Spinoza considers proposition ii as self-explanatory as well.
On the fourfold root of the principle of sufficient reason
Notice that Spinoza does not define substance in E1d3 as existing by virtue of its essence, but rather derives this claim from the definition of substance in E1p7. In contrast to i and ii , the existence of a triangle does not follow merely from its essence since it is caused by entities external to the triangle, and therefore has to be explained through these external causes.
Let us turn now to the question of the legitimacy of infinite regress. In E1p28, Spinoza openly states that within each attribute there is an infinite causal chain of finite modes:.
On the fourfold root of the principle of sufficient reason
Every singular thing, or any thing which is finite and has a determinate existence, can neither exist nor be determined to produce an effect unless it is determined to exist and produce an effect by another cause, which is also finite and has a determinate existence; and again, this cause also can neither exist nor be determined to produce an effect unless it is determined to exist and produce an effect by another, which is also finite and has a determinate existence, and so on, to infinity. Each link in this causal chain is preceded by infinitely many causes and is followed by infinitely many effects.
Each link also provides at least part of the explanation for the existence of the following link. Is the existence of such an infinite regress consistent with the PSR? Spinoza has no qualms about answering this question in the positive. We may legitimately ask what is the cause of each link in the chain, and the answer would be: the preceding link. Spinoza also thinks that we may—indeed, should—ask what is the cause, or reason, for the existence of the entire infinite chain? He simply thinks that he can provide a clear answer to the latter question as well: God or the being which is the ground of its own existence.
For as I find it in a certain Jew, called Rab Chasdai, it runs as follows: if there is an infinite regress of causes , then all things that are will also have been caused; but it does not pertain to anything which has been caused, to exist necessarily by the force of its own nature; therefore, there is nothing in Nature to whose essence it pertains to exist necessarily; but the latter is absurd; therefore, the former is also. Hence the force of this argument does not lie in the impossibility of there being an actual infinite or an infinite regress of causes, but only in the supposition that things which do not exist necessarily by their own nature are not determined to exist by a thing which does necessarily exist by its own nature.
In this passage, Spinoza follows the late medieval Jewish philosopher Hasdai Crescas in rejecting the Aristotelian ban on actual infinity see Melamed For Spinoza and Crescas the existence of an infinite regress of causes is perfectly legitimate. Yet, if all the items in this infinite chain are contingent beings i.
The ultimate reason for the instantiation of such an infinite chain of contingent beings must, claims Spinoza, be a being whose existence is not contingent for otherwise, the chain will remain merely contingent and its instantiation in reality would not be sufficiently explained. Thus, Spinoza allows for an infinite regress of causes or explanations as long as the entire infinite chain is grounded in a being which exists by virtue of its mere essence. He was the first to call it by name and, arguably, the first to formulate it with full generality. His treatment of the PSR is also noteworthy for its systematicity and the centrality that he accords it.
For example, in the Monadology he writes:. Our reasonings are based on two great principles, that of contradiction, in virtue of which we judge that which involves a contradiction to be false, and that which is opposed or contradictory to the false to be true. And that of sufficient reason, by virtue of which we consider that we can find no true or existent fact, no true assertion, without there being a sufficient reason why it is thus and not otherwise, although most of the time these reasons cannot be known to us. These principles are characterized in what appears to be epistemic terms.
And yet it is clear that Leibniz intends them to have metaphysical as well as epistemic import. In the case of the PSR, this will become more evident when we discuss how Leibniz understands the notion of a sufficient reason but it is already indicated in the passage quoted above by the fact that Leibniz explicitly states that there are sufficient reasons for every truth or fact even if such reasons are unknowable by us. The scope of the PSR, as stated above, includes facts and truths. Leibniz sometimes, however, characterizes the scope of the principle in different terms.
For example, he writes:. This suggests a version of the PSR that applies not to truths or facts but rather events: Every event has a sufficient reason. Rather they are usually understood as indicating that Leibniz views the scope of the PSR to be very wide, perhaps even absolutely general, but at least wide enough to encompass facts, truths, and events see Rodriguez-Pererya forthcoming. Leibniz associates the Principle of Contradiction and the PSR with a variety of domains where each is especially important. For example, there are domains where the truths of the domain depend on one of the two principles.
There are also domains that are characterized in terms of subject matter or areas of inquiry. It is far from obvious that these two ways of assigning domains to the Principles are equivalent. For example, it would be natural to assume that metaphysics and natural theology would include necessary propositions and thus the PSR would encroach upon the territory assigned to the Principle of Contradiction according to the modal characterization.
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We must therefore distinguish distinct ways of associating the principles with various domains. The first way is to specify the domain to which each principle applies.
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Leibniz appears to believe that, according to this approach, there is a single universal domain and it is associated equally with each principle. There are no contradictory contingent facts or truths and so the Principle of Noncontradiction applies to all contingent truths as well as all necessary truths. Likewise, it is usually assumed that, for Leibniz, every necessary truth has a sufficient reason see Broad 12 and 34 and Rodriguez-Pererya forthcoming.
For example, mathematical truths, might have sufficient reasons in the form of proofs that rest on statements of identity. Thus the PSR applies to all necessary truths as well as all contingent truths. The second way of associating the principles with specific domains is to specify the domain of truths that is grounded on or depends on each principle. According to Leibniz, only contingent truths depend on and are grounded by the PSR.
Likewise, Leibniz believes that only necessary truths depend on and are grounded by the Principle of Contradiction. What does it mean for some truths to depend on a principle? Leibniz is not explicit on this point but we will get a better idea in the next section when we consider the question of what counts as a sufficient reason.
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The third way is to specify a domain of truths that can be investigated on the basis of each principle. For Leibniz, we can know mathematical truths only on the basis of the Principle of Contradiction and only metaphysical, theological, and physical truths require the PSR in order to be known. We will see in more detail in what way the PSR allows us to investigate these domains the section on applications.
An a priori proof is a proof that reflects the causal order. Thus a sufficient reason would be a proof that is both a demonstration and an explanation see Adams Let us begin with truth. To keep things simple, we will focus only on categorical propositions of subject-predicate form. A proposition is true, according to Leibniz, just in case the concept of the predicate is contained in the concept of the subject.
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Uncontroversially, the concept of the predicate is unmarried is contained in the concept bachelor and it is this conceptual containment which explains the truth of the statement bachelors are unmarried. But Leibniz makes the further highly controversial claim that all true statements are true for this reason, even statements like Caesar crossed the Rubicon. That is, this statement is true because the concept crossed the Rubicon is contained in the concept of Caesar. This theory of truth is sometimes called the conceptual containment theory of truth.
It has as a consequence that all truths are analytic. After all, how could an analytic truth be contingent? In response to this worry, Leibniz develops an account of contingency in terms of infinite analysis. A demonstration results when an identity is obtained through the process of analysis in a finite number of steps. Leibniz claims that all and only necessary truths have a finite demonstration by analysis and all and only contingent truths do not have a demonstration by analysis in a finite number of steps.
In this way, he preserves the distinction between necessary and contingent truths while also maintaining that all truths are analytic, that is, true in virtue of the meaning of the concepts involved. This has led some commenters to think that Leibniz gave up the account of sufficient reason as an a priori proof.
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