The Principle of Sufficient Reason and Incompleteness

In summary, Schopenhauer argues that seeking a proof for the Principle of Sufficient Reason is an absurdity, as the very act of seeking proof assumes the principle to be true. He also suggests that the principle can be rephrased as "every true sentence is provable," but this too leads to a paradox. Schopenhauer's complaint may be misdirected, as he may be assuming a bivalent system and not acknowledging the four different "roots" of the PSR.
  • #1
honestrosewater
Gold Member
2,142
6
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.)
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Welcome back hrw!
honestrosewater said:
(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.
 
  • #3
Thanks. :biggrin:

Gokul43201 said:
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.
 
  • #4
Gokul43201 said:
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.
 
  • #5
honestrosewater said:
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.
 
Last edited by a moderator:

1. What is the Principle of Sufficient Reason (PSR)?

The PSR states that everything must have a reason or cause for its existence or occurrence. This means that nothing happens without a sufficient explanation or cause for why it happens.

2. How does the PSR relate to the Incompleteness Theorem?

The Incompleteness Theorem, proposed by mathematician Kurt Gödel, states that any formal system of axioms cannot prove all true statements within that system. This means that there will always be some statements that cannot be explained or proven within a given system, which aligns with the idea of the PSR that there are some things that cannot be fully explained or understood.

3. Can the PSR be proven?

No, the PSR cannot be proven as it is a philosophical principle rather than a scientific or mathematical theorem. It is a fundamental assumption that we make in order to understand the world and make sense of our experiences.

4. Are there any objections to the PSR?

Yes, there have been several objections raised against the PSR. Some argue that not everything needs a reason or cause, such as random events or actions of free will. Others argue that the PSR leads to an infinite regress of reasons, making it impossible to find a single ultimate cause or reason.

5. How does the PSR impact scientific research and inquiry?

The PSR can guide scientific research by encouraging scientists to seek out explanations and causes for phenomena rather than simply accepting them as they are. It also encourages scientists to continue searching for answers, even when they may seem elusive or incomplete. However, the PSR does not limit scientific inquiry and allows for the acceptance of unknown or unexplainable elements in the universe.

Similar threads

  • General Discussion
Replies
18
Views
3K
  • General Discussion
2
Replies
36
Views
4K
  • Programming and Computer Science
Replies
29
Views
2K
  • Cosmology
Replies
8
Views
2K
  • Quantum Interpretations and Foundations
Replies
2
Views
991
  • Sci-Fi Writing and World Building
Replies
7
Views
1K
  • Special and General Relativity
Replies
8
Views
2K
  • General Discussion
Replies
12
Views
3K
  • Precalculus Mathematics Homework Help
Replies
1
Views
735
Back
Top