Proof That God Exists

I've been linked by Vintage Piranha, so I'll formalize the valid argument for God's existence we've been discussing below.

Let:

G = God exists.
P = I pray to God.
A = God answers my prayers.

Here's the natural language version:

Premise 1:  If it is not the case that God exists, then it is not the case that if I pray to God, God answers my prayers.
Premise 2:  It it not the case that I pray to God.
Conclusion:  God exists.

And here's the proof in propositional logic:

Proof:

1. ~ P                                     (Premise)
2. ~ P V A                            (Disjunction Addition)
3.  P --> A                            (Material Implication)
4.  ~ G --> ~ ( P --> A)     (Premise)
5.  G                                       (3,4, Modus Tollens)

I'll post remarks later.  For now, please feel free to comment if you wish.

Comments

[David Ivy]

What a frustrating argument! I'll have to think about it more. My initial reaction is that it must be unsound because it can prove the existence of a vast multitude of prayer answering entities.

1. If it is not the case that the Flying Spaghetti Monster (FSM) exists, then it is not the case that if I pray to FSM, FSM answers my prayers.

2. It is not the case that I pray to FSM.

Therefore, FSM exists.

[Rad Geek]

This is obviously just material implication run amok.

If "-->" and "if-then" are read as material implication, there's no reason to believe that premise 4 is true, since what it says is logically equivalent to "If God does not exist, then both (1) I pray to God and (2) God does not answer my prayers," which hasn't got any plausibility independent of whether or not I do in fact pray to God.[*] You get the conclusion because whenever p-->q is false, where --> is material implicaiton, p has to be true. But the premise about the answers to your prayers is only plausible if you're using the nested "if-then" to express some sort of logical or causal entailment (to the effect that if God doesn't exist, my prayers aren't efficacious), which material implication doesn't express. If it is some kind of logical or causal entailment, then inferring 5 from 3 and 4 is just equivocation.

[*] "If both God does not exist and I pray to God, then God does not answer my prayers" is plausible independently of whether or not I pray to God, but that's logically equivalent to (~G --> (P --> ~A)), not (~G --> ~(P --> A)).

[anon]

I agree with Rad Geek. It seems to me that the problem with your argument is the same as the problem with this argument:

If God exists, then the sky is orange.
The sky is not orange.
Therefore, God exists.

Or with this argument:

If the sky is blue, then God exists.
The sky is blue.
Therefore, God exists.

There simply is no logical or causal connection between the antecedent and the consequent. So, though you have a formally valid argument, it is unsound because your conditional premise is false.

[Rad Geek]

Well, this argument:

"If God exists, then the sky is orange.
The sky is not orange.
Therefore, God exists."

... is invalid. Did you mean for the first premise to be "If God does not exist, then the sky is orange"?

In any case, I think there's an important difference here. Scott's argument seems plausible at first glance; the reason why is that premise 4 expresses something that seems like it ought to be true: "If God does not exist, then it's not the case that if I pray to God my prayers will be answered." Of course; after all, if there is no God then He can't be answering prayers. What I suggested to be the problem is that what is meant by the if-then nested in the consequent of the premise can't be captured by truth-functional material implication, because you can't deny a material implication except where you're willing to affirm that the hypothesis is true (~(p-->q) <-> (p & ~q)).

[Mike]

It is pretty obvious that (4) is the problem.

4. ~ G --> ~ ( P --> A)

(4) is of course equivalent to,

4' ~G --> (P & ~A)

According to (4') if God does not exist, then I pray to God. But that is false, and not just on empirical grounds. If God does not exist, then I certainly do not pray to him. Assuming I do pray, I rather *think* I pray to him or I *believe* I pray to him, but he is not there to be prayed to, or spoken to, or anything else to. So (4) is false.

[Scott Hagaman]

Wow. I didn't expect so many serious philosophers to respond. I expected to be getting a livejournal audience, since I was linked from LJ, so I thought it'd be good to have the argument up here as well. But it's nice to know that smart people sometimes read the blog! I'll comment more after class since this argument has quickly been killed before it had the opportunity to tempt any gullible LJ theists to deploy it.

[Rad Geek]

Incidentally, for an interesting compare and contrast, see Lewis Carroll's article A Logical Paradox, from the July 1894 issue of Mind, in which Carroll presents a puzzle with a different kind of nested conditional ("If C is true, then: if A is true, B is not true"). Russell and everybody thereafter thought that this puzzle was trivially solved if you based your theory of conditionals on material implication. Of course, Carroll couldn't be blamed for not seeing this, since there wasn't any widespread notion of material implication in English logic until a few years after he died; but (the idea goes) you could just chuck the puzzle out once you got rid of primitive vagueness about logical conditionals.

But, given that material implication causes its own problems with seemingly plausible nested conditionals, as seen here, this may just go to show that it's a bit harder to dismiss puzzles as relics of antiquated logical notation (made obsolete by the march of technical progress) than some philosophers in the last century were inclined to think. And that Carroll's questions about implication remain interesting (and open) after all these years. (For more, see my extended post on the puzzle.)

[Noumena]

Contra to Rad Geek, I think we only need to look at the truth-table for (4) and think a little to realise it's unacceptable as a premise. We don't need to worry about the adequacy of material representation as a representation of implication in English (though I do admit I think m.i. is inadequate in this respect; it's just that this example isn't an illustration of that).

G P A | (4)
===========
1) T T T | T
2) T T F | T
3) T F T | T
4) T F F | T
5) F T T | F
6) F T F | T
7) F F T | F
8) F F F | F

Now, the idea is that these rows are 'possible worlds', and we need to rule out all the ones where the last (rightmost) column is F -- we want our premise to be true 'in all possible worlds'. We have one other piece of data we can use for this -- A requires both G and P (I have to pray to have my prayers answered, and a god has to exist to answer them). Using this we can rule out rows 3), 5) and 7). That takes care of all the F rows except 8). But compare this with 4) -- the only difference is the value of G. Hence, we cannot rule out 8) and keep 4) except by presupposing or arguing independently that G must take the value T -- which would either be circular or make this argument superfluous. One could try to argue that both 8) and 4) ought to be jettisoned, but then (1) would be inadmissible.

For contrast, consider (4') =df ~G & P -> ~A

G P A | (4')
===========
1) T T T | T
2) T T F | T
3) T F T | T
4) T F F | T
5) F T T | F
6) F T F | T
7) F F T | T
8) F F F | T

And, as before, we rule out 3), 5), and 7), meaning (4') is acceptable as true for all 'possible worlds'.

There's an alternative way to cast this, using completeness for propositional logic, but I'm too lazy to work out the details of that right now. I find this approach far more intuitive.

NB The tables looked much better in fixed-width font. I'm also too lazy to look up the HTML tag to do that, so hopefully everyone can just cut-and-paste into Notepad if those tables are unclear.

[Rad Geek]

Noumena, I'm not claiming that pounding out a truth-table for (4) won't show why it's unacceptable as a premise (or, what amounts to the same thing, applying M.I. to transform (4) into ~G-->(P&~A), as I did above). I recognize that the argument, as formalized, is just unsound.

What I am suggesting is that the fact that (4) ever seemed plausible in the first place highlights one of the difficulties inherent in being trained to instinctively translate "if p, then q" into "p materially implies q" when you formalize your argument. The premise is intuitively true, but only as long as you're just reading it as "if-then," in one of the ordinary English senses, and not as "either not-P or A."

And I think there may be a moral to the story as to how far strictly technical advances in logic, such as the introduction and intensive use of material implication in foundational logic, deserves the kind of metaphilosophical fanfare that it got early in the last century.

[Noumena]

Sadly, too busy today to give this much more thought. But let me try this:

You reject (4) up above because it's equivalent to something that implies 'If a god does not exist, then I pray to god', which you find meaningless. Someone could object that one can pray without presupposing that there is a real entity that's the target of that prayer.

I would argue that (4) is unacceptable because it gets the truth values wrong -- in particular, it seems at least necessary that it take the value T on line 8). This might put us in agreement as far as the content of your last comment. But I think my approach can also explain why what I call (4') is a better formalisation of the English principle we find intuitively acceptable, at least in terms of getting the truth values right.

[Rad Geek]

Noumena, I think you've got me mixed up with Mike. I don't think that (4) is meaningless; I think it's false. Actually, Mike thinks it's false rather than meaningless, too; what he suggests that if God does not exist, then nothing you can do counts as praying to God. I think that's either untrue (if you're using "X prayed to Y" the same way that we often use "X wrote a letter to Santa Claus"), or uninteresting (since if you're using "X prayed to Y" in a way that presupposes that Y exists, we can always come up with a new proposition, e.g. "I say my prayers", that doesn't).

All that I've said about (4) is that, when read using material implication, it is only as plausible as the denial of premise (1) (as opposed to the original English sentence, which was intuitively plausible independently of whether I pray or not).

I agree with you that your (4') is plausibly true -- as I'd suggested already above. But it's not an accurate translation of what the original sentence means, either; it's actually logically a stronger claim than the original. (The original merely denied that there's a connection between saying prayers and those prayers being answered if God doesn't exist. 4' actually asserts that prayers won't be answered if you make them and God doesn't exist. (That's a claim you can justify apriori, but it's a different one from the original.)

[Scott]

So I think that was the most fun post I've ever thrown up, judging by the response in the comments, at least. Rad Geek has given the explanation I was going to to provide, so I'll refrain from providing that in detail now. Anon is close by when he points out that we can assess the truth value of conditionals with unrelated antecedents and consequents. Some are quite unhappy with the truth table for the material conditional, which guarantees that the conditional is true just in case the antecedent is false or the consequent is true. Conditionals such as "if God exists, then I'm eating dried fruit now" turn out to be true, since I am, right now, in fact eating dried fruit. Some take this to mean we should adopt relevance logic instead, while others (with whom I agree) are happy to point out that the price of truth-functionality is not really all that high.

But this isn't all there is the story, as Rad Geek pointed out. I didn't offer a conditional statement in which the antecedent was obviously unrelated to the consequence. Indeed, the translation, so long as one keeps it in material conditional form, is eminently plausible. That intuition doesn't reject the conditional translation shows that the conditional of English is not the material conditional. It also serves as a reminder to be very careful about how one translates English if..then constructions: make sure you're committed to all logical permutations!

Mike, I think Rad Geek's comment on your comment is interesting. Santa Claus does not exist. The Tooth Fairy does not exist. Mom or Dad are not Santa Clause and/or the Tooth Fairy. But a child does write a letter to Santa Claus, and since "writing a letter to" is not factive, this doesn't presuppose his existence. If someone's not convinced that "writing a letter to" is not factive, consider the (unfortunately all to commonplace) example of writing a letter to a dead soldier. Similarly, "praying to" shouldn't be factive, I'd think, and if it were, I could instead say "I pray (loosely) to God" where "praying loosely" is defined as the non-factive use of "pray". So the real problem with (4), or at least the problem I was getting it, is not that "praying to" is factive. Now my non-factive use of a relation here does presuppose that I can explain the "x is praying to y" relation without appeal to non-existent relata. No relations without existing relata, I say. (See John Bigelow's "Presentism and Properties" for more on this.)

Finally, nice work, Noumena. The truth tables have been crunched on this end as well. In fact, Tom Metcalf was the first person to notice that line eight poses a debiliating problem. (The argument has lots of problems. One shouldn't be able to infer a purportedly necessary conclusion from a contingent premise, for example, and one doesn't need the truth table to see that one of the argument's premises is contingently true. There's a more obvious candidate.) But I don't see how pointing that out shows that material implication is not an inadequate representation of the English conditional. I mean, it would very nice if English if..then's translated straightforwardly into logical if..then's. But since the logical if..then (in this case) is material, the translation fails. The argument is "initially persuasive" because the translation looks prima facie good.

[Mike]

Scott/Rad,

"... what [Mike] suggests that if God does not exist, then nothing you can do counts as praying to God. I think that's either untrue (if you're using "X prayed to Y" the same way that we often use "X wrote a letter to Santa Claus")..."

The premise in question (given obvious transformations) states that if I pray to God and if God answers the the prayer, then God exists. That is supposed to be (and seems to me) plausible on the face of it.
But Rad introduces standards of fictional discourse against my objection that 'praying to God' is factive. I think the analogy is not especially well-thought-out, but (to save us all some time) suppose I concede that the standards of fictional discourse are appropriate here.
Does anything change? No, it doesn't. The result is that Scott's premise (4) remains implausible. By fictional standards, of course, it might be true that I pray to God and God not exist (just as in the Santa case). But, by the same standards, it might be true that God answers my prayers, and God not exist. God's answering my prayers might be like Santa "answering my letter", as so often happens for children. Therefore, (4) remains implausible. More explcitly:
If I pray to God and if God answers my prayers, it might nonetheless be that God does not exist. And why is that? Well (invoking fictional standards) it can be that I write to Santa and Santa answers my letter and Santa does not exist. Analogously for God's answering my prayers. So the appeal to fictional standards leaves (4) implausible.

[Rad Geek]

"But Rad introduces standards of fictional discourse against my objection that 'praying to God' is factive."

I don't think the connection between "writing letters to Santa Claus" (to take the example) and standards of fictional discourse is quite so straightforward. The language-game surrounding Santa Claus (to take one example) is complicated by the fact that parents engage in fictional discourse that children are expected to treat as factual discourse about Santa Claus, for a few years of their life at least. A number of kids who might tell you that what they are doing is writing a letter to Santa Claus, wouldn't be saying it under an implicit fictional-that operator. You could, of course, just dig in and insist "Well, in that sense they aren't writing letters to Santa Claus. They just think they are, and you're only inclined to call it a letter to Santa Claus when you slip into the fictional context by playing along with the child's false beliefs." And I'd agree with you that there's a sense of "write to" (and "speak to," "pray to," and other forms of direct address) where the second person has to exist for you to count as having done it. What I'm more doubtful of is the idea that this is the only sense in which the terms can be used in direct discourse. I'd be interested to know why you think direct (non-fictional) uses of these phrases presupposes the existence of the indirect object.

I'd also note, in this connection, that it's a common use of English to say things like: "People pray to many different gods," without presupposing that all of the gods that people pray to exist.

[Weero]

Burden of proof. It's all I'm saying...