A friend of mine showed me this logical “proof”, and I’ve had little luck finding where exactly an illegal step is made. Of course, reasoning like this can be used to prove any proposition whatsoever. Proving the existence of God is just for dramatic effect.
Let S be the sentence
[ul]If S is true, then God exists.[/ul]
Obviously, if we can show that S is true, then we will have shown that God exists. Thus we proceed to prove S. Since S is a conditional statement, we do this by assuming the antecedent, and then deriving the consequent.
Assume the antecedent; that is, suppose that
[ul]S is true[/ul]
is true. Then,
[ul]If S is true, then God exists[/ul]
is true. Thus we have that both
[ul]If S is true, then God exists[/ul]
and
[ul]S is true[/ul]
are true, so, by modus ponens,
[ul]God exists[/ul]
is true. This completes the proof that S is true. As already mentioned, this proves that God exists.
I think you left out part of the “proof,” Tyrrell. You have considered the case where S is true. Now suppose S were false.
Then, the sentence “If S is true, then God exists” is TRUE, because a syllogism with a false antecedent is always true. So S is false implies S is true, a contradiction. Therefore the first case, where S is true, must obtain.
I believe the problem is in the fact that the symbol S is a part of statement S. This produces Russell’s Paradox or something related to it.
The sentence S is a conditional sentence of the form “If A then B”. In sentences of this form, A is called the antecedent, and B is called the consequent. The way one proves such sentences is to assume the antecedent A and derive the consequent B. Successfully doing this doesn’t mean that I’ve proved that B is true. It only means that I’ve proved that if A is true, then B is true. But this is precisely what S claims. So (it seems) I’ve proved S.
No. I can truthfully assert that “if 4 is even, then it is divisible by 2”, but that doesn’t mean that I’m entertaining the notion that 4 is not even.
december, actually, I think what you’ve done is provide a clever alternative proof of S, but my proof was complete. Regarding this paradox’s similarity to the Russell’s paradox, and its even closer (in my opinion) similarity to the liar paradox, this one is different in a way which intrigues me. The other paradoxes dealt with sentences that could not be true or false, since assuming either implied the other, if you follow. However, in the case of this paradox, assuming “S is true” does not imply that S is false (so far as I can see). Though, as you showed, assuming “S is false” does imply that S is true.
The “illegal” step is the conditional–“if S is true, then God exists” (call it T), which remains unproven (or unaccepted) in the argument. Your friend’s argument is valid (it violates no laws of logic), but not sound (the premises are not true, or accepted as true). Prove the conditional, and the argument might have some merit, but then you’re back to the same set of arguments as before, about God’s existence.
Your friend may be thinking about the case where S is a contradiction, and therefore false. Since all conditionals are necessarily true if the precedent term is false, the conditional is tautological. I doubt that will convince anyone, though.
This is, where it goes wrong already. You mentioned the liar paradox yourself, which is essentially “Let S be the sentence ‘S is false’”.
With this kind of self-reference you can easily build sentences that are neither true nor false. Logic - at least any logic I know - does not apply to them.
This is the classic “begging the question” – you’re assuming what you set out to prove.
You are attempting to prove that God exists.
“If S is true, then God exists” is logically equivalent to saying that “Proof of the truth of S requires that God exist” – which means that S is logically equivalent (not necessarily definitionally equivalent) to the conclusion, i.e. “God exists.”
Now, your next step is to assume S. Hence you assume a point that is logically equivalent to what you set out to prove.
Might I point out to your friend that if the God in question is the Christian one, His assignment to those who believe in Him was not to “prove His existence” but to show Him to others through the quality of the life you live?
re
Pooh would say: “…If I climb that tree to get to the top…then I’ll climb that tree to get to the top…but, if I climb that tree to get the honey I’ll fall off that tree and hit every branch on the way down…”
If ‘S’ is true then God exists…
If ‘T’ is true then God does not exist… In the natural progression of the SDMB we will debate this all afternoon…
Or course not, but that doesn’t explain why the logic is wrong.
I don’t think that you can so easily banish self-reference from logical discourse. Consider, for example, the apparently true sentence
[ul]This sentence contains thirty-six letters.[/ul]
(Hope I counted that right). And many books contain a statement along the lines of “this is a book about…”? I don’t think that, by doing so, such a book has destroyed any claim to logical correctness.
I don’t think that my next step was to assume S. Rather, my next step was to prove S. Since S is a conditional statement, I did this in the way one proves conditional statements. That is, I assumed the antecedent of S, and, on the basis of that assumption, derived the consequent of S.
I see now that I probably should have mentioned at the beginning that my friend and I are both atheists. Our interest in this “proof” is purely as a logical puzzle or paradox. I probably should have appended “or, a logical paradox” to the title of this thread. I did not mean to give the impression that there is any attempt at convincing anyone of any theological truths here. As I said in the OP, making “God exists” the conclusion of the “proof” is for dramatic effect only.
I did not claim that there are no true self-referential sentences.
But what is the goal, here?
Do we want maximal freedom in constructing sentences?
Or are we more concerned about consistency in our logic?
I could easily dispense with sentences stating their own length, if I could get rid of inconsistencies that way.
With natural language there’s no chance, to find a maximal set of sentences that allows for a consistent logic.
(That’s why formal logic starts with the definition of a formal language.)
Anyway the OP asked, where the “prove” was flawed. I stick to my point: It’s the self-reference.
The argument is like this: if A, then B, then B. Leave God out of it, since that just confuses the issue.
This isn’t begging the question, it’s a circular argument. To prove the antecedent B, you have to prove the conditional. To prove the conditional, you’re assuming the conditional. You can go on like this infinitely, just because it’s circular: if A, then B, then B, then B…
Maybe that is the best way to think of this. The “proof” uses self-reference in a tricky way to make what looks like valid reasoning be, in fact, circular.
A long car ride helped me clear my head about this problem.
Let me reformulate it, just to be clear:
A = if “A” is true, then B. (note that A is a token in the conditional–this clears up any confusion about self-reference, which isn’t a problem here).
Assume A. Then, A is true; so, by A, B is true. If that’s not an accurate restatement, please say so.
What you’ve got there isn’t an argument, it’s an argument form. You’re correct that modus ponens allows you to assert B in this case–as I said before, this violates no laws of logic.
The problem is in assuming A. In natural deduction, one is justified in hypothesising a precedent term only if that assumption can be later discharged as part of the proof–either by showing it to be necessary, irrelevant, or a contradiction. The conditional isn’t necessarily true, and it’s not contradictory. In the case of B being “God exists”, the only way to discharge the hypothesis is to prove the consequent–that God exists–which makes the conditional necessarily true, and the hypothesis is demonstrated to be irrelevant, discharging it. Without an independent proof of God’s existence, the argument as you stated it makes no claims about the existence of God because you’re not justified in asserting the truth of A by hypothesis.
Since the argument is an incomplete proof, there’s no logical flaw–it’s simply not a sound argument yet.
I’m afraid you’re missing the point of the proof.
Easy to get confused when things refer to themselves, so I’d like to start with the ‘normal’ case:
Let’s assume we want to proof a sentence S = “A -> B”.
This can be done by the following steps:
Assume A
Show that B is true under this assumption.
If you can do that, you know that S is true.
Now the “proof” in the OP has exactly the form above. (It’s proving S first!)
The only suspicious thing (except proving any arbitrary statement is, that “A” is replaced with “S”.
Step 1) Assume S, exactly following the schema for the proof for S.
Tyrell got it right: The self-reference allows the circular argument here! That’s where you can see, that the self-reference is the problem.
Now how does step 2) work in the OP?
Simple:
From 1) we have S, which equals S->B
Now from S and S->B modus ponens gives us B, as you said, hansel. Nothing wrong here.
This completes step 2.
Therefore we proved the conditional S->B. Which is S.
(There is no “pending” assumption here anymore, as hansel wants us to believe.)
Now we can do the same trick again.
We have S. That is S->B. Therefore B.
The conditional is true if, by assuming the precedent term, it can be shown that the consequent term is true. This proves the conditional, which discharges the hypothesis, which supplies a proof of the precedent term that we lacked at the beginning, which allows us to assert the consequent.
Yep, it’s the self-reference, which allows the circular argument to occur: assuming the truth of the conditional proves the truth of the conditional.
So what’s the error in logic here? Or have we stumbled into Godel’s Incompleteness Theorem?
This is bringing back nightmares about the class in the philosophy of truth I took. Tarski would handle this by claiming that predicating truth of a statement automatically bumps that predicate to a meta-language: any logic allowing a truth predicate on its own terms allows paradoxes like this one to occur. That would disqualify the conditional in the argument. But then you get into an infinite regression of meta-languages, which is an uncomfortable property for truth to entail.
[QUOTE]
*Originally posted by hansel * … So what’s the error in logic here?
[QUOTE]
The problem is that we implicitly assume, that our informal rules of logic (I’m a mathematician, I can call them informal.)
apply to every sentence of natural language. But some of these sentences are just pathological.
The only way out of this, is restricting our language, when we want a consistent logic.
Not at all. Godel’s Theorem belong to a strictly formalized logic with a formal language that does not allow these kind of sentences.
And Godel’s Theorem definitely doesn’t produce paradoxa, inconsisties, proofs of god, etc.
waít a second… What do you mean by infinite regression?
What we get is infinitely many levels of meta languages.
And every sentence belongs to exactly one of them.
This may be uncomfortable, but surely not as bad as logical inconsistencies.
is, that “A” is replaced with “S”.