And God said, "I'm not going to be ordered about by a bunch of bloody logicians." and promptly ceased to exist.

(Plagiarised from Douglas Adams.)

If God does not exist, then it is false that if I pray, then my prayers will be answered. So I do not pray. Therefore God exists. — Banno

Nice stuff.

God exists if only I do not pray, but my prayers will not be answered then.

What about this one?

If God does not exist, then it is false that if I pray, then my prayers will be listened. So I don't pray to stop communicating with God. Therefore God exists.

A prayer doesn't necessarily need to be answered but listened, I guess.

G → ((P→A) ∨ G).

Rather than "Oramus, ergo non est deus", how about ...

If man is imago dei

and if the devil is imago hominis,

then god is imago diaboli;

ergo imagoes est. Amen.

:sparkle: :pray:

It’s remarkable how much time and effort silly things like this take up within “philosophy.” Still kind of fun as a game I guess.

“Everyone’s mad here. I’m mad; you’re mad.”
“How do you know I’m mad?”
“You must be — or you wouldn’t have come here.”

If God does not exist, then it is false that if I pray, then my prayers will be answered. So I do not pray. — Banno

How do you conclude "God exists" from this? Since the premise is "If God exists..", doesn't the conclusion of "God exists" involve an inversion fallacy?

If God does not exist, then it is false that if I pray, then my prayers will be answered. So I do not pray. Therefore God exists. — Banno

If God exists, then it is true that if I pray, then my prayers will not be answered. So I pray. Therefore God does not exist. :chin:

Sorry, sloppy mistake, or a strange sort of typo, in my last post. The question was meant to be, how do you proceed from the premise "if God does not exist..." to "therefore God exists" without an inversion fallacy?

Still makes no sense. What's an "inversion fallacy" in this context? Do you think that the argument is denying the antecedent? It isn't.

Presumably, the impetus of 'willing x makes it so' is either a tilt at a windmill or a collapse of all willing. The idea of a Supreme Being can play either side.

<"it is false that if I pray, then my prayers will be answered" translates to ~(P→A)>

We have scrutinized this sort of translation a great deal in the past months. This thread, for example:

However, what about ¬(A→B)? What can we say about this in English? — Lionino

The premise states a conditional concerning "if God does not exist". We cannot proceed logically, from that premise to make any conclusions about what would be the case "if God does exist". Such a conclusion would be an "inverse fallacy".

Here's Wikipedia:
"Confusion of the inverse, also called the conditional probability fallacy or the inverse fallacy, is a logical fallacy whereupon a conditional probability is equated with its inverse; that is, given two events A and B, the probability of A happening given that B has happened is assumed to be about the same as the probability of B given A, when there is actually no evidence for this assumption."

In other words, once you understand the relation between the antecedent and the consequent, in this type of conditional, as a relation of probability, you will see the argument in a completely different way. The relation between "if God does not exist", and "my prayers will not be answered" is a relation of probability.

- When the antecedent of a material conditional is false, the conditional itself is necessarily true. That's all that is happening here, and then the modus tollens draws 'G'.

We could add the implicit step:

~G→~(P→A)
~P
∴(P→A)
∴G

(As a proof this runs into some of the exact same difficulties that were discussed in this thread.)

I like your point of 'a tilt at a windmill', but I think it is not only for deities. Political theories can also be put into that rabbit hole. I can't even think of a better example. Politics are literally a tilt at a windmill.

The relation between "if God does not exist", and "my prayers will not be answered" is a relation of probability. — Metaphysician Undercover

Understood. But to what premise is the probability applied? God's existence or "my prayers will not be answered"? I mean, when the Wikipedia article states that "the probability of A happening given that B has happened," what are A and B here? Given that God needs to exist before a person's prayers, logically. But, in this philosophically tricky game, it is true that we could state that "when my prayers happened, there was a probability for God to exist."

Yep.

:roll:

Yep.

Understood. — javi2541997

It isn't anything to do with probability.

Consider:

@javi2541997 @Metaphysician Undercover
I'm not sure why the inversion fallacy is considered a separate fallacy from the fallacy of denying the antecedent. It only seems to differ in the assumption that if "If P, then Q" is true that therefore "if not P, then not Q" must also be true. But you get there if you analyse it as denying the antecedent as well.

Denying the Antecedent fallacy

If P, then Q
Not P
Therefore, not Q

If God does not exist, then it is false that if I pray, then my prayers will be answered. So I do not pray. Therefore God exists. — Banno

If P, then Q
Not P
Therefore, not Q

but really it says:

If not P, then not Q (if R, then S)
Q equals if R, then S
Not R
Therefore, not S
Therefore, Q (through double negation)
Therefore, P

But not "R" therefore not "S" is denying the antecedent in the secondary argument "if I pray, then my prayers will be answered". So this is still invalid if you ask me.

Cool. Very clear and nice explanation. It is somehow what I had in mind, but I don't know how to use the logic tree accordingly. Yes, it is a double negation, indeed. If God doesn't exist, then it is false that my prayers will be answered. The negation of the latter implies the negation of the first premise. Therefore, there is a double negation. OK. Everything it is starting to get clear, whether it is a valid statement or not, is another subject that I am not able to answer. Yet I think it is relevant to understand what comes first. My prayers or God's existence? Since they cause double effect, they are dependent on each other, but the order seems tricky to me.

I'm not sure why the inversion fallacy is considered a separate fallacy from the fallacy of denying the antecedent. It only seems to differ in the assumption that if "If P, then Q" is true that therefore "if not P, then not Q" must also be true. But you get there if you analyse it as denying the antecedent as well. — Benkei

I think it actually is the same, just different names for the same problem.

The quote I took from Wikipedia concerns what happens when the problem is carried into inductive premises which are naturally probabilistic. It throws a skeptic's curveball at the problem, by making the relation not "necessary" in either direction, because there is not the required relation between the two, in either direction. I think that's what says. In reality, there is no necessary relation between God's existence and prayers being answered, in either direction, because "fate" might answer the prayers, instead of God, and God could choose not to answer prayers. That's where freedom of choice throws the curveball at cause/effect relations.

If not P, then not Q (if R, then S)
Q equals if R, then S
Not R
Therefore, not S
Therefore, Q (through double negation)
Therefore, P

But not "R" therefore not "S" is denying the antecedent in the secondary argument "if I pray, then my prayers will be answered". So this is still invalid if you ask me. — Benkei

Thank you, that's a very nice, clear explanation as to why it is a case of "denying the antecedent", sometimes called an inversion fallacy. The issue is that the assumed necessary relation does not carry in both directions, and this is very significant in cases of cause/effect. We see that A causes B and we establish the necessary relation "if A then B", assuming A does not have freedom of choice in the matter. But we might be fooled if we do not allow that B could be caused by something else, therefore to prevent that possibility of being misled by the inversion, we cannot say if B then A.

So the issue here is that there is an assumed causal relation between God and prayers being answered, such that God causes prayers to be answered, The necessary relation is that God is the cause of prayers being answered, if G then PA. Where the fallacy lies, is in the assumption that this can be turned around, to say that if prayers are answered, then God must exist. We must maintain the possibility that the effect could have another cause. So the fallacy inheres within the claim "if God does not exist my prayers will not be answered". That primary premise, as an inversion of "God is the cause of prayers being answered", already has within it, the fallacy. According to the nature of cause/effect relations we must maintain the possibility that the effect can occur without the known cause.

I think that's what ↪javi2541997 says. In reality, there is no necessary relation between God's existence and prayers being answered, in either direction, because "fate" might answer the prayers, instead of God, and God could choose not to answer prayers. — Metaphysician Undercover

Could be but that doesn't invalidate an argument. Premisses do not have to be true or correct to reach a valid argument. It only means the argument is unsound.

I see what you mean. Your explanation shows the argument to be invalid though, because it puts a second instance of the same fallacy, in the second part. And that fallacy is required to carry out the procedure.

I think that's what ↪javi2541997 says. In reality, there is no necessary relation between God's existence and prayers being answered, in either direction, because "fate" might answer the prayers, instead of God, and God could choose not to answer prayers. That's where freedom of choice throws the curveball at cause/effect relations. — Metaphysician Undercover

Exactly. That's what I tried to say, but, as usual, I expressed myself very puzzled and particularly like a crackpot.

If not P, then not Q (if R, then S)
Q equals if R, then S
Not R
Therefore, not S
Therefore, Q (through double negation)
Therefore, P — Benkei

Nope. This is your mistake:

Not R
Therefore, not S
Therefore, Q

(Another mistake is that Q does not follow from ~S)

I disagree you can disregard the "not S" step, because the statement in its entirety must be false. If I say "if I pray then my prayers are answered", stating "I don't pray" says nothing about the consequent of that statement so we don't know what it means. Q is merely implied because if there are no prayers, they cannot be answered.

I can also interpret the statement as a regular modus tollens and I will be affirming the consequent as a result:

If God does not exist, then it is false that if I pray, my prayers will be answered. (If P, then Q)
I do not pray. (Implies Q)
Therefore, God exists. (Concludes not P)

So I agree this is valid:

~G→~(P→A)
~P
G

But the logical structure and the argument are not necessarily the same. There are different ways to interpret it.

I disagree you can disregard the "not S" step, because the statement in its entirety must be false. If I say "if I pray then my prayers are answered", stating "I don't pray" says nothing about the consequent of that statement so we don't know what it means. Q is merely implied because if there are no prayers, they cannot be answered. — Benkei

Did you read my post <here>?

Do you agree with this:

~P
∴(P→A)

Q is merely implied because if there are no prayers, they cannot be answered. — Benkei

I don't think that is quite right. Q is merely implied because of the way a material conditional works. The inference <~P; ∴(P→A)> is different from, "If there are no prayers, they cannot be answered." It says, "If there are no prayers, then it is true that (P→A)."

So I agree this is valid — Benkei

Okay, good.

But the logical structure and the argument are not necessarily the same. — Benkei

I agree.

I don't think that is quite right. Q is merely implied because of the way a material conditional works. The inference <~P; ∴(P→A)> is different from, "If there are no prayers, they cannot be answered." It says, "If there are no prayers, then it is true that (P→A)." — Leontiskos

Thank you for explaining that. That put me on the right track to understand what's going on. I found this via perplexity.ai:

Applications and Limitations

The material conditional is widely used in mathematics and formal logic. It serves as the basis for many programming language constructs. However, it's important to note that the material conditional doesn't always align perfectly with our intuitive understanding of "if-then" statements in natural language[1][2].

Paradoxes

The material conditional leads to some counterintuitive results when applied to natural language:

1. A conditional with a false antecedent is always true.
2. A conditional with a true consequent is always true.
3. There's no requirement for a logical connection between the antecedent and consequent[3].

These "paradoxes" arise from the truth-functional nature of the material conditional, which only considers the truth values of its components, not their meanings or relevance to each other[4].

Understanding these properties and limitations is crucial for correctly interpreting and applying the material conditional in logical reasoning and formal systems.

Citations:
[1] https://en.wikipedia.org/wiki/Material_conditional
[2] https://www.webpages.uidaho.edu/~morourke/202-phil/11-Fall/Handouts/Philosophical/Material-Conditional.htm
[3] https://open.conted.ox.ac.uk/sites/open.conted.ox.ac.uk/files/resources/Create%20Document/Note-ifthen.pdf
[4] https://rjh221.user.srcf.net/courses/1Aconditionals/Lecture1.pdf

So, I"m reading up right now. :smile:

- Yep, exactly right. :up:
That is literally the best AI-generated content I have ever seen. :smile:

(Edit: When I said, "We have scrutinized this sort of translation a great deal in the past months," that was a nice way of saying that the translation is problematic.)

I'm finding this hard to follow - is your claim that the argument is invalid? It isn't. Seems Leon addressed this.

The written form is, the formal notation isn't.

A correct written form could be—if I am not mistaken—and this is the closer I got using reason with words and not with logic symbols: If I pray and God exists, then my prayers will be answered. Otherwise, if I pray, but God doesn't exist, my prayers will not be answered. 

Because the 'consequent condition' is applied to my prayers and not to God's existence.