A Proof for the Principle of Sufficient Reason

Let Rx = there is a sufficient reason that explains x. (& represents conjunction; ~ represents negation.)

1. There exists an x such that x & ~Rx. [supposition]

2. p & ~Rp. [1, existential elimination]

3. Possibly, R(p&~Rp) [2, PPSR]

4. R(p&~Rp) [supposition]

5. Rp & R~Rp [4, Axiom 4 & Axiom 0]

6. Rp & ~Rp [5, application of Axiom 3 & Axiom 0 to right conjunct]

7. ~R(p&~Rp) [4-6, negation-intro; discharges assumption in #4]

8. Necessarily, ~R(p&~Rp) [7, Axiom 1 & Axiom 0]

9. It is not the case that possible, R(p&~Rp). [8, Axiom 2 & Axiom 0]

10. Possibly R(p&~Rp) and NOT possible R(p&~Rp) [3, 9, &intro]

11. It is not the case that there exists an x such that x & ~Rx. [1-10, negation-intro; discharges assumption in #1]

12. For any x, x only if Rx. [11, classical logic]

Supposing that, for every truth, there could be a sufficient reason that explains that truth, this argument demonstrates that, for every truth, there is a sufficient reason that explains that truth.

Commentary on the Axioms

Axiom 0. I cannot make sense of the possibility of a falsehood that is necessarily true.

Axiom 1. Understand by “proof” a deductively valid argument. Consider a set of premises {P} such that each member of the set is necessarily true. Then, according to the standard way of understanding necessity, these premises are true in all possible worlds. If the axiom is false, there is one such world in which, despite the premises being true, the conclusion C is false. But if all the premises of a deductively valid argument are true, there can be no such world. For if there is a deductively valid argument from a set of premises {P} to a conclusion C, then, according to the definition of “deductively valid,” C is true whenever each member of {P} is true.

Axiom 2. This follows from the standard definitions of necessity and possibility, according to “it is necessary that x” means “it is not possible that it is not the case that x.”

Axiom 3. Explanations provide answers to why or how questions: to explain x is to provide an answer to the question “Why is it the case that x?” or “How is it the case that x?” If x is false, then there is no answer to why or how x is the case (because x is not the case). So explanations are always explanations of truths. Now if there is a sufficient reason that explains x, then there is an explanation of x. The axiom follows.

Some might suppose that, even though the second law of thermodynamics is, strictly speaking, false, statistical mechanics nonetheless provides an explanation of the second law, and that the explanandum in such a case is the second law rather than the falsity of the second law. Fine. For those who think in this way, every occurrence of “sufficient reason that explains” in the axioms should be replaced with “sufficient reason that explains the truth of.” Then Axiom 3 states: Necessarily, for any claim x, there is a sufficient reason that explains the truth of x ONLY IF x is true. This axiom is obviously true. And it does not detract from the scope of the theorem, because we are interested only in explanations of truths.

Axiom 4. It is a logical truth that, given any conjunction, there is a deductively valid argument which has that conjunction as its sole premise and either of the conjuncts as its conclusion. Underlying this axiom is the idea that making deductively valid inferences is a legitimate part of any explanation. For if this is the case, then it is always possible to construct an explanation of a claim from an explanation of a conjunction containing that claim as a conjunct. For example, if {P} is a sufficient reason that explains C1&C2, then, since making deductively valid inferences is a legitimate part of any explanation, {P} and the deduction of C1 from C1&C2 is a sufficient reason that explains C1. It should be obvious that this remains so if we replace “sufficient reason that explains” with “sufficient reason that explains the truth of” in the statement of this axiom.

Commentary on PPSR

The only assumption in the preceding proof that does not seem to be necessarily true is PPSR. I take the history of scientific inquiry to provide a strong inductive argument for the truth of this postulate, because many truths that have thought to be unexplainable have turned out to have explanations and, therefore, turned out to be explainable.

People often object to the Principle of Sufficient Reason by claiming that some truths about the world are brute facts, where a brute fact is a truth for which there is no sufficient reason. And to provide evidence for the existence of brute facts, these people often point to truths for which we have no explanation.

The preceding proof demonstrates that those who make accept the existence of brute facts are, in fact, committed to a much stronger claim, namely, that there are necessarily brute facts, truths for which there could not be a sufficient reason. I do not see that pointing to the existence of unexplained truths provides any evidence whatsoever for supposing that there could not be an explanation of those truths.

Accordingly, our available evidence seems to favor PPSR over its denial. For the history of scientific inquiry provides evidence for PPSR, while there is no evidence for the denial of this postulate.

This inductive argument does not, of course, show that PPSR is necessarily true. For the argument does nothing to show that there could not be a possible world in which the history of scientific inquiry is littered with abject failures. Indeed, it strikes me that the history of scientific inquiry could have been different in this way. But the preceding argument in support of the postulate is consistent with this possibility. For what matters in that argument is the actual history of scientific inquiry. And the contingency of this history filters down (so to speak) to make PPSR merely contingently true. But that’s okay. The argument for the Principle of Sufficient Reason requires only the actual truth of PPSR; and this postulate, like other contingent truths, can be actually true despite not being necessarily true.

Observations

If Postulate 1 is true, then so is the Principle of Sufficient Reason. And if my argument for Postulate 1 is correct, the Principle of Sufficient Reason is true, not by virtue of its meaning, but instead by virtue of the way the world is. It is, to use Kantian terminology, a synthetic truth rather than an analytic one; and our justification for believing it is aposteriori (dependent on empirical evidence about the world) rather than apriori. To know whether it is true, we must engage in scientific inquiry; and to know that is true, we must have a track record of success in those inquiries.

Leibniz, recall, used the Principle of Sufficient Reason to argue that, because some truths about the world are contingent (namely, truths about contingent beings), a necessary being, God, exists. If that argument is valid, then my argument for Postulate 1 suggests that the contemporary felt relation between theism and scientific inquiry is a bit topsy-turvy. On the one hand, for the most part, contemporary atheists reject the existence of God and praise the success of scientific inquiry. The argument I have given suggests, however, that the success of scientific inquiry provides support for the existence of God (by virtue of the support it gives to Postulate 1, and provided that Leibniz’s argument is valid.) On the other hand, and for the most, some contemporary theists accept the existence of God and, to support this acceptance, maintain that there are certain truths about the world that scientific inquiry is not equipped to explain. The argument I have given suggests, however, that this kind of failure in scientific inquiry provides support for rejecting Postulate 1 and thereby undermines the Leibnizian argument for the existence of God.