The Principle of Sufficient Reason and Incompleteness

honestrosewater
Gold Member
2,071
5

Main Question or Discussion Point

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_the_Fourfold_Root_of_the_Principle_of_Sufficient_Reason#.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.)
 
Last edited by a moderator:

Answers and Replies

Gokul43201
Staff Emeritus
Science Advisor
Gold Member
6,987
14
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
Gold Member
2,071
5
Thanks. :biggrin:

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
Staff Emeritus
Gold Member
1,717
5
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.
 
89
0
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_the_Fourfold_Root_of_the_Principle_of_Sufficient_Reason#.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.
 
Last edited by a moderator:

Related Threads for: The Principle of Sufficient Reason and Incompleteness

  • Last Post
Replies
18
Views
2K
  • Last Post
2
Replies
30
Views
7K
  • Last Post
4
Replies
84
Views
6K
  • Last Post
3
Replies
56
Views
4K
Replies
12
Views
3K
  • Last Post
Replies
21
Views
4K
Replies
1
Views
1K
Top