Monday, February 19, 2018

Leibniz on PSR and necessary truths

I just came across a quote from Leibniz that I must have read before but it never impressed itself on my mind: “no reason can be given for the ratio of 2 to 4 being the same as that of 4 to 8, not even in the divine will” (letter to Wedderkopf, 1671).

This makes me feel better for defending only a Principle of Sufficient Reason restricted to contingent truths. :-)


Philip Rand said...
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Red said...

Wouldn't such restricted psr entail that we might encounter odd brute necessities? I don't remember who but I recall a philosopher writing that he finds brute necessities intuitively even worse than brute contingencies.

Walter Van den Acker said...


I agree with this philosopher. To say that explanations end in a inexplicable fact that is brutely necessary does not seem to add anything interesting at all.
So, defending a PSR restricted to contingent truths to me only works if the necessary truth that is the "reason" for the contingent truth is demonstrated to be logically necessary, that is, if its non-existence violates the laws of logic in some way.
And the non-existence of God, e.g. doesn't seem to fall in that category.

Philip Rand said...
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Alexander R Pruss said...


That there are brute necessities does not imply that there are odd brute necessities. :-)

That said, I am a little disappointed by Leibniz's example. It seems to me that the the necessity of 2:4 :: 4:8 is not brute---it can be explained by more fundamental arithmetical necessities. But one has to remember that Leibniz is writing before the axiomatization of arithmetic.

Miguel said...

Professor Pruss,

I find this limitation on PSR to be problematic, because there are known examples of necessary facts/beings which, though necessary, would still cry out for explanation. And these cases can be particularly important for theistic philosophy in general. For instance, in view of problems with both platonism and standard aristotelianism, Saint Augustine and scholastic realists notably looked for explanations for how numbers and necessary truths could exist, leading them to conclude they existed in the divine mind. I'm sure you're familiar with this, especially given your work on actuality and possible worlds. But this is a case in which necessary beings clearly require an explanation (leading to the distinction between necessary beings which exist a se and those that exist ab alio), somethhing that would not be endorsed by the limited PSR.

This issue can be pressed further and show up even in cosmological arguments. The example of eternal truths may be taken to show that a being can be conditioned (having its existence dependent on another being) without having to be modally contingent. But then, for instance, the fact that quarks cannot exist alone but must always be joined with two or three others, would not imply that quarks are contingent; the atheist can insist that they are necessary and thus avoid explaining their existence. Of course suggesting quarks are necessary beings would be implausible, but the point is that even if, arguendo, they were, the fact that they are still dependent on other quarks to exist would seem to make them require a (non-circular) explanation, which in any case would not be endorsed by a limited PSR if quarks are necessary. An atheist could say that a (conditioned, dependent) cause in an infinite essentially ordered series of causes would require no explanation because it is necessary, but surely this would seem absurd and contradict what Aquinas and many other brilliant philosophers thought. But if we limit PSR to contingent facts, it seems it would not endorse such questions. You're probably aware of issues like these, since in your paper on Leibnizian cosmological arguments you mention that it is sensible to ask, if a necessary being had parts, what would be responsible for conjoining or keeping the parts together. Perhaps these issues can all be avoided by appeals to IBE, or by rejecting "odd brute necessities" (such as necessary beings who exist ab alio requiring no fuller explanation for their existence), but wouldn't it be better (and simpler), to just accept a full PSR and drop the limitation?

Excuse the long post (and any mistakes I might have made), but I'd be interested in hearing more of your thoughts on these issues. Looking forward to your upcoming book on necessary existence, too.

Sturer 111 said...

Hello Professor Pruss,
what would you now is the case? Do you agree with the conclusion of Leibnizs example, without agreeing with the example itself? What, in your opinion, would be a better example Leibniz should/could have given? If one cant think of a better one, then why believe the conclusion to be right in the first place?

Could Gods necessity not be of such a kind that its logical necessity eludes us only because our picture of God, so to say, is too simplistic? If God is indeed the necessary foundation of reality, thinking of God not existing would be like thinking of existence with existence not existing - this being seemingly a logical impossibility (not only the act of thinking about it, of course).

Does this sound plausible? Or would you reject taking this route?

Friendly regards,

Sturer 111 said...

*what would you now think is the case?

Alexander R Pruss said...

It seems plausible that 0=0 is necessary but not further explained.
But on the other hand it could be that 0=0 is self-explanatory.
I don't know which is the case.