The Principle of Sufficient Reason and Incompleteness

[honestrosewater]

The PSR is stated along the lines of "nothing is without a reason for its being". From Schopenhauer's http://en.wikisource.org/wiki/On_th...n#.C2.A7_14._On_the_Proofs_of_the_Principle.":

To seek a proof for the Principle of Sufficient Reason, is, moreover, an especially flagrant absurdity, which shows a want of reflection. Every proof is a demonstration of the reason for a judgment which has been pronounced, and which receives the predicate true in virtue precisely of that demonstration. This necessity for a reason is exactly what the Principle of Sufficient Reason expresses. Now if we require a proof of it, or, in other words, a demonstration of its reason, we thereby already assume it to be true, nay, we found our demand precisely upon that assumption, and thus we find ourselves involved in the circle of exacting a proof of our right to exact a proof.

His complaint seems misdirected to me. Does the negation of this sound like the assertion that there exist unprovable truths? Could we rephrase the PSR, or a version of it, as

PSR1) Every true sentence is provable.

If (PSR1) is true, it must have a proof. It says so itself. And it is really just a statement of completeness. Is he suggesting that we must take it as proof of itself or else assume that our system is incomplete?

Just a quick thought.

(Wow, philosophy is in GD now? Haha.)

 

[Gokul43201]

Welcome back hrw!

(Wow, philosophy is in GD now? Haha.)

Sad, but necessary, following some disappearances.

If your PSR1 statement is accurate, then it certainly looks like the complaint is misplaced. Also, isn't PSR1 at odds with Incompleteness.

 

[honestrosewater]

Thanks.

Also, isn't PSR1 at odds with Incompleteness.

Right, its negation, that there exists some true but unprovable sentence, is true of an incomplete system. His argument is not precise enough to me to be sure whether the PSR is analogous to completeness. The two options that I see are that (a) he is making a distinction between semantic implication and syntactic implication and the PSR is about completeness or (b) he isn't making such a distinction but rather taking the PSR to mean, in some sense, that, for every true sentence p, there exists some nonempty set of sentences that implies but does not include p. In other words, (b) means that there exist no logically true sentences (or tautologies), or that no sentence can act as proof of itself, which I can't really see him defending.

I don't often notice an acknowledgment in philosophical arguments of the system in which the argument is taking place, and I didn't see him say anything about it here. I assume he is assuming a bivalent system.

The point of the paper is that there are four different, often confused "roots" to the PSR, so perhaps I will have to read on for some clarity.

 

[loseyourname]

Welcome back hrw!
Sad, but necessary, following some disappearances.

Sorry about that, assuming it was my disappearance that necessitated it. Graduating proved to be rather hectic.

 

[Lord Ping]

The PSR is stated along the lines of "nothing is without a reason for its being". From Schopenhauer's http://en.wikisource.org/wiki/On_th...n#.C2.A7_14._On_the_Proofs_of_the_Principle.":

His complaint seems misdirected to me. Does the negation of this sound like the assertion that there exist unprovable truths? Could we rephrase the PSR, or a version of it, as

PSR1) Every true sentence is provable.

If (PSR1) is true, it must have a proof. It says so itself. And it is really just a statement of completeness. Is he suggesting that we must take it as proof of itself or else assume that our system is incomplete?

Just a quick thought.

(Wow, philosophy is in GD now? Haha.)

Thanks, the Schopenhauer passage is very interesting.

The argument seems to be:

____________

(1) Every true sentence is provable, and only true sentences are provable*.
Therefore (2) Demanding proof is a rational prerequisite to believing a sentence is true.

Does (2) apply to (1)?

For the rational answer to be yes, I must already rationally believe the truth of (1). Result: paradox.

Therefore (2) does not apply to (1)

____________

Let me know if you disagree with this formulation of the argument.

*= this clause seems to be an implicit assumption.