## More Logical Trickery

I recently came across this logical "proof" of the existence of God at Robert Oerter's blog, Somewhat Abnormal.

Consider the following sentence:When I read it, it struck me as just the kind of thing that Christians would buy (note that Robert Oerter does not buy it). It is a logical sleight of hand. It tricks the reader into accepting its conclusion. And that's the basis of many theistic proofs. Let's examine exactly why this argument fails.

(S) If this sentence is true, then God exists.

Suppose sentence S is true. Then the first clause is satisfied, so the second clause is true. Thus, God exists.

What the preceding paragraph proves is that if sentence S is true, then God exists. But that is exactly what sentence S asserts. So that means we have proved that sentence S is true! And therefore God really does exist.

We are asked to make an assumption - namely that (S) is true. And then, based on that assumption, we conclude that God exists. But is this valid? Let us consider what it means to assume that (S) is true. The proposition (S) is called a material conditional. A material conditional is generally expressed in the form "If A, then C." The clause "If A" is called the antecedent, where A is some assertion. The clause "then C" is called the consequent, where C is the condition that is entailed by the truth of A. It may be said that a proposition of this form is true if it expresses a valid logical relationship between the antecedent and the consequent. To assume that (S) is true is not merely to assume that the consequent it true, but to assume that the consequent is entailed by the antecedent.

To help determine the validity of (S), let's take a different material conditional as an example.

(X) If the sky is blue, then my dog has four legs.We can see immediately that this proposition does not express a valid logical relationship between the antecedent and the consequent. The consequent is the condition of my dog having four legs. But my dog may or may not have four legs (he could be an amputee), and the truth of that is not entailed in any way by the antecedent. The sky could be gray from clouds or red from smoke particles, and it has no effect on the number of legs my dog has. So even though the sky might in fact be blue, and my dog might in fact have four legs, (X) does not express a valid logical relationship between antecedent and consequent. Therefore, it is not reasonable to assume that (X) is true. If you do assume that it is true, you may well be led to a false conclusion. In other words, you haven't proven anything at all. It makes no difference what the consequent is. You can't prove it by simply assuming the truth of an invalid logical relationship.

Now, to get bask to the original (S), it may not be immediately apparent that this doesn't express a valid logical relationship between antecedent and consequent. Starting with the basic form of the material conditional, we have

(S1) If A, then C. - where A is "this sentence is true" and C is "God exists".But the proposition A the assertion of the truth of (S), so we can say

(S2) If (S), then C. - where C is "God exists".Now if you expand (S), you can see that the antecedent becomes an infinite regress assertions that the assertion is true. This alone should reveal that the proposition (S) is not formed in a valid way. But more importantly, it becomes a little easier to examine the relationship between antecedent and consequent. To say "this sentence is true" is to assert that the existence of God is logically entailed by (S) being true. But that's not a valid assertion, in the same way that (X) is not valid. The consequent is not implied by the antecedent, and it is not reasonable to assume that (S) is true, any more than it is reasonable to assume that (X) is true. In other words, (S) asserts a valid logical relationship between antecedent and consequent where there is none, and therefore (S) is not true. So assuming the truth of (S) is like assuming the truth of "If P, then not P".

This argument is reminiscent of other logical proofs of the existence of God that I have encountered. In particular, Anselm's ontological argument comes to mind, which also uses a logical sleight of hand to trick the reader into making a false conclusion. As a self-described skeptic, I try to be on guard against such trickery in logical arguments. Theists are free to disagree.

Your analysis is factually and demonstrably wrong on several points. But I am not going to waste my time in what would be a completely fruitless and pointless discussion, and simply say that it really does not matter, because it is very easy to spot where the proof goes wrong. Denote by P the proposition "if this proposition is True then God exists" and Q the proposition "God exists". Then P is the proposition

ReplyDeleteP => Q

with => the symbol for implication. The proof of any implication usually proceeds by assuming the truth of the antecedent and then proving the truth of the consequent (*). In this particular case, the assumption of the truth of the antecedent is the assumption of the truth of P which is precisely what we want to to prove, so the proof is circular and thus completely useless.

As a suggestion (which I am completely sure you will disregard, which is why I give it so freely) you really should not comment on that which you do not understand -- it just makes you look like an ignorant fool.

(*) This can be justified in various ways; e.g. in first-order propositional calculus by looking at truth tables.

grodrigues,

DeleteI generally try to avoid excessive use of technical terminology and symbology because I feel that it might come off as pretentious. However, you should not mistake that for ignorance on a particular topic. I, too, have studied advanced mathematics, and done my share of mathematical proofs. Your lack of charity toward my analysis prevents you from seeing the truth of what I said.

"The proof of any implication usually proceeds by assuming the truth of the antecedent and then proving the truth of the consequent"

- That is demonstrably wrong, and it shows that perhaps you don't know what you're talking about. There are other methods of proceeding with the proof. For example, you could prove the contrapositive, which means to show that the falsity of the consequent entails the falsity of the antecedent. So I don't know where you get this "proof of any implication usually proceeds by ...". It doesn't usually proceed by one method in particular. It sounds like you are just pulling this out of your ass.

What you didn't state explicitly in your method of proof is to show that the consequent

necessarily followsfrom the antecedent (which is what we mean by 'implies'). That's precisely what I was trying to demonstrate with my example proposition (X). You can assume that the sky is blue, and you can even show that the dog has four legs, but this does not prove the proposition (X). But if I demonstrate that my dog could have three legs despite the fact that the sky is blue, I have succeeded in disproving proposition (X). The whole idea of the consequent following from the antecedent was what I focused on in my discussion. But somehow, that seems to have escaped you.My suggestion to you is that you should not assume that my lack of religious belief makes me wrong on every topic, and secondly, you should not assume that your own academic credentials give you the right to unquestioned authority.

"That is demonstrably wrong, and it shows that perhaps you don't know what you're talking about. There are other methods of proceeding with the proof. For example, you could prove the contrapositive, which means to show that the falsity of the consequent entails the falsity of the antecedent."

DeleteFirst, I said "usually" not "always" (the adverb is there because it really depends on the details of the deductive calculus); you might want to read more carefully (not that you will). And the reason why I pointed this out has to do with the logical structure of the proof, to show that some responses to it do not work. Unfortunately, but not unexpectedly, you missed the point. Second, "falsity of the consequent", for a consequent Q, is logically equivalent to truth of not-Q. In other words you are just making the point for me.

"But somehow, that seems to have escaped you."

No, it did not escape me, I just did not comment on it. But if you want to me to say something I will say this; not only there are mistakes in what you say, the response does not work. But as I said, and will repeat it, I really have no wish to waste my time or yours in a pointless discussion that will go nowhere, except perhaps in the gradient of aggravation. You are ignorant and ineducable; your last paragraph is nonsense pulled out from God knows where (where did I make any assumptions or mention of your atheism? And while you did not fail to mention that "I, too, have studied advanced mathematics" where did I mentioned my academic credentials?); but I do agree with you that the tendency will be for me to be unduly harsh and uncharitable. And after all this, the blog that is, is your home. It would be pointless, and also supremely rude, to disabuse you (whether you deserve it or not).

If grodrigues has academic credentials in mathematics, then God exists. :o)

DeleteIf grodrigues has academic credentials in mathematics, then God exists. :o)

Delete"the reason why I pointed this out has to do with the logical structure of the proof, to show that some responses to it do not work."

Delete- It all depends on how you look at it. I approached the issue from a different perspective. The existence of God does not depend on the truth of a proposition. But You're too intent on your own agenda to see what I was saying.

"(where did I make any assumptions or mention of your atheism? And while you did not fail to mention that "I, too, have studied advanced mathematics" where did I mentioned my academic credentials?)"

- Now you're pretending that we've never exchanged words before? As I said before, you wield your degree like a cudgel.

"It would be pointless, and also supremely rude, to disabuse you (whether you deserve it or not)."

- And yet, here you are. But please, don't spare my sensibilities. All I ask is that you be civil.

@im-skeptical:

Delete"It all depends on how you look at it. I approached the issue from a different perspective. The existence of God does not depend on the truth of a proposition."

I do not know exactly what you mean by "The existence of God does not depend on the truth of a proposition". If what you mean is that there must be something "fishy" with deriving a substantial metaphysical conclusion (e.g. God exists) from what on all appearances looks like mere word play, then sure. But once again this is irrelevant as a response to the "argument". Neither does it have anything to do with whether God exists or not, or even with Anselm's argument, but with its logical structure, for if it did work then anything (and therefore nothing) could be proved true. There is a world of difference between "smelling fishy" and actually spotting the fallacy. Which you did not.

To compare, there are various objections lodged against Anselm's argument(s) and its variants. Most of them are spurious and wrong, when not plainly ignorant and idiotic. The one I think it works (that is, I am inclined to think the argument does not work -- inclined because the issue is subtle and the argument even if wrong is telling us something non-trivial) is a variation of the question-begging fallacy.

"Now you're pretending that we've never exchanged words before?"

Indeed I was only appealing to what passed on this thread, but fair enough. Although I would like you to show me where did I assumed, implicitly or explicitly, that (1) you being an atheist implied you were "wrong on every topic" and (2) my "academic credentials" gives me "the right to unquestioned authority". I am pretty confident that you will find it nowhere, but I have made mistakes before.

"As I said before, you wield your degree like a cudgel."

I find it a natural reaction to know-nothings, but I can see how it can be annoying. On the other hand, when the question is one of authority it does make sense to bring up one's "academic credentials" -- e.g. when you said in response to my comment, mimicking my turn of phrase, "That is demonstrably wrong, and it shows that perhaps you don't know what you're talking about" -- it would be perfectly legitimate to point out my "academic credentials". Unlike you, I do know what I am talking about. But the obsession with "academic credentials" is all yours (maybe an inferiority complex?) as it is you who keeps bringing it up (maybe as a pre-emptive rhetorical move?), and instead I *showed* that your response was wrong -- not only you misread me, but you made the point for me.

"And yet, here you are."

And yet here I am. Boredom got the better of me. Now I am not.

grodrigues,

Delete"I do not know exactly what you mean by "The existence of God does not depend on the truth of a proposition"."

- Nor do you care to try to discern the meaning of it. But I will spell it out for you. By most any theistic understanding of God (as an eternal being who is the creator of all things, including mankind), the fact of God's existence (or non-existence) is independent of mankind or any idea that mankind may propose. If God exists, there is no proposition that we can make that would render God's existence dependent on the truth of that proposition. Likewise, if God doesn't exist, there is no proposition we can make that could possibly change that fact.

"But once again this is irrelevant as a response to the "argument"."

- It is absolutely relevant to my response, since it shows that the proposition is false.

"There is a world of difference between "smelling fishy" and actually spotting the fallacy. Which you did not."

- That's your ill-considered opinion. I didn't mention anything about "smelling fishy". I argued that the proposition was false. The fact that you think there is only one way to attack it is evidence of your narrow-mindedness.

"To compare, there are various objections lodged against Anselm's argument(s) and its variants. Most of them are spurious and wrong, when not plainly ignorant and idiotic."

- Yes, it would appear that any line of thinking that doesn't explicitly match your own is ignorant and idiotic.

"I am pretty confident that you will find it nowhere, but I have made mistakes before."

- It shows every time you have spoken to me. You have never once viewed any argument I make with any measure of charity. You are so stuck on your own superiority that you can only see your own perspective.

"I find it a natural reaction to know-nothings"

- And I find it egotistical and intolerant. There are certain academics who, like you, display this air of superiority, and it has less to do with their actual knowledge than the fact that they can't stand their ideas or beliefs being challenged. Many others don't share your attitude of superiority. They are willing to discuss issues amicably, and if they feel that their opponent is wrong, help them to understand why without trying to belittle them, or simply disagree without hurling insults. Fortunately the latter are the majority.

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