What is Proof?

[Brian Bosse]

What does it mean for someone to provide proof for a particular proposition? Intuitively, one might say that a proof for proposition Q is a list of propositions, P(1), P(2),..., P(n), when taken together is meant to lead one to accept the truth of Q. The idea might be expressed more formally as follows:

Definition of Proof: A set of propositions, {P(1), P(2),..., P(n)}, constitutes a proof of Q if and only if in some sense (P(1) ∧ P(2) ∧ ... ∧ P(n)) → Q.

The interesting thing about this definition is the phrase "in some sense." What are the different senses a set of propositions may entail another proposition? What does it mean for (P(1) ∧ P(2) ∧ ... ∧ P(n)) to entail Q? As was intimated above, this entailment is related to persuasion in that the idea is to convince someone of Q's truth given the truth of other propositions. The trouble with this is that persuasion is subjective and at times can be irrational. So, maybe we can sharpen the definition of proof as being a set of propositions that someone should (ought to) accept as entailing its conclusion? If this is the case, then how does one decide what the various senses of entailment are that place a moral demand on someone to accept the conclusion?

What has motivated this thread is the claim by some that circular argumentation in certain cases is valid, and in other cases is an informal fallacy. If this is the situation, then who gets to decide when it is appropriate and when it is not? One might ask if circular argumentation is an informal fallacy in the first place? Is so, why? If not, why not? These are all particular questions of the larger question I raised at the end of my last paragraph.

Sincerely,

Brian
P.S. I am hoping to take the feedback and discussion from this thread and start a series on my blog regarding proof.

 

[Brian Bosse]

Hello Paul,

This may be more of a side question, but why take the 'should' to be a moral should? Perhaps it can be construed in functionalist terms. For example, when my heart it properly functioning, it functions how it 'should' be functioning (in our case, according to God's design plan).

This is a good point. Another ‘should’ in terms of functionality could be in relation to rationality. Someone should accept proof X if they are rational. (This relates to Bahnsen’s Basketball court analogy.)

On the functionalist account, propositions P 1 ... n entail q for a person S iff S's cognitive faculties CF are properly functioning PF and S is persuaded (by any multitude of ways: they cohere with other beliefs he doesn't want to give up, they're logical, the empirical evidence is strong for him, etc., etc., etc.) that P1 ... n entail q.

This takes into account PF CF. This is very helpful.

Notice that PF is what is primary. P cannot be said to entail Q for someone who's Cfs are not PF. Cartesian demons could be playing natsy tricks on S and making S think that P entails Q. S could do his duty, it may seem logical, the demons could make Q fit with the rest of S's beliefs, etc. But, we would not want to say that P1 ... n was a proof (or that S was warranted in believing) for Q since S's CFs were not PF.

This is interesting. Couldn’t we define ‘proof’ in broad enough terms to encompass bad or misleading arguments? Isn’t it still a proof even if it is deceptive or wrong?

Lastly, regarding circularity. It may be that the warrant given by the circularity is not transitive, but is just a source of warrant (or pfoof). Only in the transitive sense would there be a fallacy.

Can you expand on this a little for me? It seems you are saying that in some cases a circular argument can provide warrant, but this does not imply that it does so in all cases.

Anyway, 'bout all I had time for, hope it helped somewhat (at least with some ideas of where to go).

This was great. I hope you can expand on some of these ideas for me.

Of course Clarkians and Cartesians will not like this since I obviously do not consider deductive proof to be the only kind of proof.

I, too, am thinking of ‘proof’ in broader terms than simply deductive arguments. Consider the courtroom where someone is not guilty until they have been “proven” guilty beyond a reasonable doubt.

Unfortunately, on their (Scripturalists) own terms, they can't prove that deductive proof is the only type of proof since they can't deduce that from the Bible.

I agree that Scripture does not teach us that deductive proofs are the only valid proofs.

Brian

 

[Pergamum]

I stink, therefore I am.






Actually, I am a solipsist. But I cannot get anyone else to believe me!

 

[Civbert]

...I agree that Scripture does not teach us that deductive proofs are the only valid proofs.

Brian,

Would that be "valid" in the logical sense or in the sense of being legitimate.

Certainly people use the term "proof" in many different ways. As you mentioned, there is the use of someone being "proven" innocent/guilty. But then all we have here is that there are strict and and loose definitions of proof. In our justice system, one can be considered "proven guilty" on a preponderance of circumstantial evidence that in no way proves the person is guilty in a logical/deductive sense. There are deductive proofs, and inductive proofs. But an inductive proof is deductively invalid (in the logical sense of valid).

So if someone asked what is a proof - all I could say is - "in what sense?"

Now what about philosophical proof? For instance, when someone claims to have a "valid proof of the existence of God". I would deny this is the case, and I can give a perfectly good explanation why the particular proof is invalid. Usually, the problem is simply a matter of question-begging. When one explicit assumes the premise one is attempting to conclude - then I would say they have not given a valid proof by any reasonable logical sense.

Looking forward to some interesting responses.