I do not pray. Therefore God exists.

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.)

Quoting Banno

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.

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.”

Quoting Banno

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

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?

Quoting Banno

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.

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:

Reply to Banno
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?

Reply to Metaphysician Undercover 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.

Reply to Banno

<"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:

Quoting Lionino

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

Reply to Banno
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.

Reply to Metaphysician Undercover - 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.)

Reply to Paine 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.

Quoting Metaphysician Undercover

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

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."

Reply to punos Yep.

Reply to Metaphysician Undercover :roll:

Reply to Leontiskos Yep.

Quoting javi2541997

Understood.

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

Quoting Banno

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.

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.

Reply to Benkei 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.

Quoting Benkei

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.

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 Reply to 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.

Quoting Benkei

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.

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.

Quoting Metaphysician Undercover

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.

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.

Reply to Benkei
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.

Quoting Metaphysician Undercover

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.

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

Quoting Benkei

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

Nope. This is your mistake:

Not R
[s]Therefore, not S[/s]
Therefore, Q

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

Reply to Leontiskos 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.

Quoting Benkei

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.

Did you read my post <here>?

Do you agree with this:

~P
?(P?A)

Quoting Benkei

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

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)."

Quoting Benkei

So I agree this is valid

Okay, good.

Quoting Benkei

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

I agree.

Quoting Leontiskos

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)."

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:

Reply to Benkei - 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.)

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

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

Reply to Benkei 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.

Why use this trick to find God when you can cash in?

If I am not a billionaire, then it is false that if I scream, my screams will be heard. So I do not scream. Therefore I am a billionaire.

This points out the real problem of the syllogism, which is that the premises in the God example are assumed by the reader to be contingent and not necessary and the truth value of the conclusion is then confused as actually saying something about the world as opposed to it just being a logical application of rules. The way it's structured is that you read the conclusion and forget it's just a tautology.

The "So" in "So I do not pray" is a clever twist, as it suggests the speaker has decided to do something to create God, leading the reader down the intended road that the syllogism means something beyond its logical structure.

Quoting Banno

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

I think it's more addressing that these mean different things:

1. ¬(P?A)
2. P?¬A

And so these mean different things:
3. (¬G?¬(P?A)?¬P)?G
4. (¬G?(P?¬A)?¬P)?G

(3) is valid but (4) isn't.

Translating (1) and (2) into ordinary language introduces a problem, because we would translate (1) as "it is not the case that if I pray then it will be answered" and (2) as "if I pray then it will not be answered" which seem to mean the same thing, but (1) and (2) don't mean the same thing.

Quoting Hanover

The "So" in "So I do not pray" is a clever twist, as it suggests the speaker has decided to do something to create God...

I want to say that this is off, and that the trick is the ambiguity of, "If God does not exist..." The valid argument looks like this:

1. Suppose God does not exist
2. Therefore, It is false that if I pray, then my prayers will be answered
3. Therefore, I do not pray

But the logical translation makes the "if" a logical condition, not a supposition (i.e. not a condition whose scope extends to (3)). "So I do not pray," is a hanger-on from the alternative English translation which the formal presentation opts out of. ...Of course the idiosyncrasy of the material conditional is also doing a lot of work here.

Quoting Michael

3. (¬G?¬(P?A)?¬P)?G

There is an ambiguity in the order of operations here which echoes my point to Reply to Hanover. Which has precedence? The '?' or the '?'? Depending on which, the nature of the falsum arguably changes.

Going back to this:

Quoting Leontiskos

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

Suppose the '?' has precedence: (¬G?(¬(P?A)?¬P))
Then we have (¬G?falsum)

But what happens if the '?' has precedence? : ((¬G?¬(P?A))?¬P)
Then the same paradox from the previous thread arises, where you have (¬G?¬(verum)), along with the quandary of whether ¬(verum) is the same as falsum (and also whether the consequent should be interpreted as ¬(verum), or as ¬(P?A) conjoined with the recognition that (P?A) happens to be true in this case).

(The difficulty is apparently that falsum is context-independent whereas propositional negation is not. Does the modus tollens require propositional negation, or will falsum also suffice? And then what about ¬(verum), which is a combination of the two?)

(CC: @Lionino, @TonesInDeepFreeze)

Reply to Leontiskos
Why isn't the conclusion just a non-sequitur?

Quoting schopenhauer1

Why isn't the conclusion just a non-sequitur?

Because [page 1]. :razz:

I tried to summarize why <here>.

Reply to Leontiskos
I don't see how the conclusion can be derived conditionally from the premises- it is tacked on.

Quoting Leontiskos

(CC: Lionino,

It seems that Lionino will not sign up ever again, sadly. :sad: I tried to interact with him through PM for the past months, and I hadn't any answer. I wish he could be back.

Quoting schopenhauer1

I don't see how the conclusion can be derived conditionally from the premises- it is tacked on.

Quoting Leontiskos

Do you agree with this:

~P
?(P?A)

Quoting javi2541997

It seems that Lionino will not sign up ever again, sadly. :sad:

Maybe. He left in frustration but will perhaps change his mind in time. I hope he returns.

Reply to schopenhauer1

The argument in Banno’s post is a link to a logic tree diagram that shows you why it’s valid.

Quoting Leontiskos

~P
?(P?A)

No.

Reply to schopenhauer1 - Then check out Reply to Benkei's response.

Quoting Michael

The argument in Banno’s post is a link to a logic tree diagram that shows you why it’s valid.

Can you have a non-sequitur critique of a structurally valid statement? Does content matter?

Reply to Leontiskos
Same question https://thephilosophyforum.com/discussion/comment/939929

Reply to schopenhauer1

Well, I suppose that’s what my first post above does. The (valid) formal logic is an improper translation of the English language sentence.

Quoting schopenhauer1

Same question

Same answer: Reply to Leontiskos, Reply to Leontiskos

Quoting Michael

Well, I suppose that’s what my first post above does. The (valid) formal logic is an improper translation of the English language sentence.

:up:

Reply to Michael Reply to Leontiskos
Does this whole exercise imply something about logic's usefulness with natural language? :chin:.

If there is a step before logical notation that is needed to translate, what is THIS?

Reply to schopenhauer1 Interpretation.

Quoting schopenhauer1

Can you have a non-sequitur critique of a structurally valid statement? Does content matter?

That was the point of my post. https://thephilosophyforum.com/discussion/comment/939830. You can say any ridiculous thing you want as long as you treat the statements as meaningless premises that are reducible to symbols. If you treat the premises as contingent statements that have a truth value of their own based upon empirical information or whatever you use to decide if a statement about the world is valid, then you end up with non-sequitur issues, but those non-sequiter issues are not deductive logic fallacies, but are inductive ones.

Deductively, the conclusion of the OP follows. Inductively not. That's the interesting part of the OP.

Reply to schopenhauer1 - Sure and Lionino's thread delved into this in some detail.

-

Reply to Hanover - No, I don't think so. The OP is nowhere near as "ridiculous" as your argument about billionaires. The English argument of the OP makes sense in a way that you haven't recognized. I don't see that any of this has to do with deduction vs. induction.

Quoting Leontiskos

No, I don't think so. The OP is nowhere near as "ridiculous" as your argument about billionaires. The English argument of the OP makes sense in a way that you haven't recognized. I don't see that any of this has to do with deduction vs. induction.

The two arguments (mine and the OP) are logically equivalent under deductive logic. They are represented symbolically the exact same. For one to be more ridiculous than the other means you are using some standard of measure other than deductive logic to measure them, which means you see one as a syllogism and the other as something else.'

Inductive logic references drawing a general conclusion from specific observations and it relates to gathering information about the world, not just simply maintaining the truth value of a sentence. To claim that statement of the OP is more logical than mine means that the conclusion of the OP bears some relationship to reality. If that is the case, it is entirely coincidental.

Deductive logic says nothing at all about the world.

(1) All dogs are cats, all cats are rats, therefore all dogs are rats. That is true, except for the fact that dogs aren't cats and cats aren't rats.

(2) All dogs are mammals and all mammals provide milk to their young; therefore, all dogs provide milk to their young. That is true, both deductively and inductively.

(1) and (2) are represented the exact same way deductively and are therefore both true deductively. (1) is inductively false and (2) is inductively true.

In a syllogism, the premise is a given. In an informal statement, it is a contigency.

That's what the OP plays upon.

Quoting Hanover

If you treat the premises as contingent statements that have a truth value of their own based upon empirical information or whatever you use to decide if a statement about the world is valid, then you end up with non-sequitur issues, but those non-sequiter issues are not deductive logic fallacies, but are inductive ones.

Reply to Leontiskos

Yep, makes sense. So I guess what's the bigger picture? We can do funny things with symbolic logic seems a bit arbitrary. We need more than symbolic logic to say anything meaningful seems a truism. So what then?

Quoting Banno

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

I, (or perhaps just this post,) am the answer to all you godless people's prayers.

Therefore the quoted premise is false.

Reply to unenlightened Yes, indeed.

If god does not exists, our prayers would be answered by Un, and hence it is false that if there is no god your prayers will not be answered.

If Un is not distracted by Hanover's screams, of course.

@Leontiskos @Hanover

I guess the silence speaks for itself :meh:

Quoting schopenhauer1

I guess the silence speaks for itself :meh:

Hanover's trying to tell us something?

Quoting Hanover

The two arguments (mine and the OP) are logically equivalent under deductive logic.

Except they're not, because your "So..." is entirely different than the OP's "So..." I explained this <here>.

Quoting Hanover

Deductive logic says nothing at all about the world.

Sure it does.

Quoting Hanover

(1) All dogs are cats, all cats are rats, therefore all dogs are rats. That is true, except for the fact that dogs aren't cats and cats aren't rats.

It is unsound, and that is why it fails to be informative. It is not uninformative because it is deductive.

Quoting Hanover

(1) and (2) are represented the exact same way deductively and are therefore both true deductively.

You're flubbing the difference between soundness and validity. A premise being true does not make it inductive.

The crux is that this claim of yours is entirely false:

Quoting Hanover

the speaker has decided to do something to create God

Quoting schopenhauer1

So I guess what's the bigger picture?

I'd say the main point of the OP was snark, hitting back at those ancient proofs for the existence of God that can't seem to go away. It points out that attempts to bootstrap something from from logic alone lead to whatever foolishness you desire.

Quoting Hanover

I'd say the main point of the OP was snark, hitting back at those ancient proofs for the existence of God that can't seem to go away. It points out that attempts to bootstrap something from from logic alone lead to

:up:

At what realm do you suppose symbolic logic makes sense besides mathematic proofs? Just philosophy journals as a way to gain street cred, that one knows the game?

Edit: I ask because clearly the reasoning and analysis matters more than turning the argument into symbolic logic. If anything, exercises like this show this.

Reply to Hanover Reply to schopenhauer1

I would give @Banno the credit of levity here, not snark. It is a philosophical joke, aptly placed in the lounge. The justifiable decision to not pray turns out to backfire and prove God's existence, given a logical translation that is initially plausible. Hanover is reading all sorts of strange things into the OP.

Quoting Leontiskos

I would give Banno the credit of levity here, not snark. It is a philosophical joke, aptly placed in the lounge. The justified decision to not pray turns out to prove God's existence, given a logical translation that is initially plausible. Hanover is reading all sorts of strange things into the OP.

I ask you the same:
At what realm do you suppose symbolic logic makes sense besides mathematic proofs? Just philosophy journals as a way to gain street cred, that one knows the game?

Edit: I ask because clearly the reasoning and analysis matters more than turning the argument into symbolic logic. If anything, exercises like this show this.

Reply to schopenhauer1 - This OP should not cause you to despair of logic, lol. As I've noted elsewhere, the material conditional is a disproportionately artificial logical construct. A false antecedent makes a material conditional true, and this is something like a bug or at least dross. Much like a bit of imperfect code, as long as you don't exploit the bug the logic is useful. The OP is a fun way of exploiting this bug, among other things. I don't think it is meant to be more than that.

Quoting Leontiskos

The OP is a fun way of exploiting this bug, among other things. I don't think it is meant to be more than that.

I get it, but was trying to see if there is a takeaway. My question still stands, what’s the use of symbolic logic if the analysis comes before the logic? I know the classic reason is clarity of presentation. But it would be misleading if it it’s seen as the actual catalyst behind the actual reasoning, like a computer language.

Quoting Hanover

I'd say the main point of the OP was snark,

"Just the place for a Snark!" the Bellman cried,
As he landed his crew with care;
Supporting each man on the top of the tide
By a finger entwined in his hair.

Quoting schopenhauer1

My question still stands, what’s the use of symbolic logic if the analysts comes before the logic?

Well, at the very least it is a useful aid for error-checking, even if it is not infallible. It represents a form of calcified analysis that is useful but limited. And it is useful for conceptualizing extended arguments that are difficult to capture succinctly. There are probably other uses as well. I have fought lots of battles against the folks in these parts who have a tendency to make formal logic an unimpeachable god, so I agree with the sort of objection you are considering.

(There is also a normative use in teaching reasoning skills, for we have some common sense intuitions which are fallacious, and which can be ironed out easily with formal logic. Reply to Moliere seems to overlook this latter point in his analysis of Aristotle.)

Quoting Banno

"Just the place for a Snark!" the Bellman cried,
As he landed his crew with care;
Supporting each man on the top of the tide
By a finger entwined in his hair.

And @Hanover, here we see Banno abandoning his Godless ways:

Snark = Jonah
Bellman = God
Crew = Jonah's shipmates

The Biblical allusion is too obvious to ignore. Banno made light of belief in his OP, and now a strange twist of fate has brought it about that his OP led to his belief, not unlike the subject of the OP. :grin:

Quoting Leontiskos

Well, at the very least it is a useful aid for error-checking, even if it is not infallible. It represents a form of calcified analysis that is useful but limited. And it is useful for conceptualizing extended arguments that are difficult to capture succinctly. There are probably other uses as well.

Sure, but as this exercise shows, the logic can stifle the analysis as well, if not used correctly, or even if used correctly.

Quoting Leontiskos

There are probably other uses as well. I have fought lots of battles against the folks in these parts who have a tendency to make formal logic an unimpeachable god, so I agree with the sort of objection you are considering.

:up:

I think we should be very careful when we throw around the word "logic", just like the word "rational". I try not to use "rational" too much, because it's often just a coded word for "I'm the one with the correct thinking and you are not, you're just not 'rational'". Similarly, logic can stand in for one's rationale, it can mean a formal logical system like Frege developed, a Hegelian-like totalizing feature of metaphysics, and a whole bunch of things.

Reply to schopenhauer1 - I agree. :up:

Reply to Leontiskos Interpretive monism to go with logical monism - the One True Understanding of Lewis Carrol. I prefer Martin Gardiner's analysis.

Besides, I haven't said it three times yet.

Reply to Banno - I was interpreting you, not Carroll.

Quoting Banno

yet

Yet. (!)

Quoting Hanover

The two arguments (mine and the OP) are logically equivalent under deductive logic. They are represented symbolically the exact same. For one to be more ridiculous than the other means you are using some standard of measure other than deductive logic to measure them, which means you see one as a syllogism and the other as something else.'

Logical equivalence is not determined solely by symbolic representation, especially in light of the interpretive choices made when translating from natural language to formal logical symbols. Even so, two arguments can be symbolically similar but not logically equivalent if their premises or conclusions differ in truth value or meaning. Logical equivalence requires that both arguments have the same truth value in all possible scenarios.

Quoting Hanover

Deductive logic says nothing at all about the world.

This statement is only partially correct. Deductive logic ensures that if the premises are true, the conclusion must also be true. Obviously when the premises are true, a valid deductive conclusion will say something about the world.

Quoting Hanover

Inductive logic references drawing a general conclusion from specific observations and it relates to gathering information about the world, not just simply maintaining the truth value of a sentence. To claim that statement of the OP is more logical than mine means that the conclusion of the OP bears some relationship to reality. If that is the case, it is entirely coincidental.

Inductive logic indeed involves drawing general conclusions from specific observations but they can never be proven true the way a deductive argument can. It merely deals in probabilities; the more observations you have the likelier your conclusion.

Your second argument is not inductively supported because the conclusion is supported by the definition of mammal. It's like saying, all bachelors are single, John is single and therefore a bachelor. There's no probability involved that a single man isn't a bachelor.

And yes, in formal logic, premises in syllogisms are assumed to be true for the sake of argumentation.

Reply to Benkei - :up:

-

Quoting Hanover

The two arguments (mine and the OP) are logically equivalent under deductive logic.

Quoting Leontiskos

Except they're not, because your "So..." is entirely different than the OP's "So..." I explained this .

Teasing this out a bit more, the OP contains an implicit move, "Supposing God does not exist..., I should not pray." The formal translation does not take this route, but the connotation is part of the parlor trick.

The parallel in your own example is, "Supposing I am not a billionaire..., I should not scream."

They are completely different. The implicit connotation in the OP makes perfect sense. Your parallel is perfect nonsense. Not all parlor tricks are created equal. The parlor trick of the OP is a great deal better than your attempt regarding billionaires. Your argument possesses no plausibility because it is so obviously unsound. You are trying to make yourself a billionaire with specious reasoning. The OP is not praying on the supposition that God does not exist.

Reply to Leontiskos

It's much simpler than that.

¬(P ? A) ? (P ? ¬A), so ¬G ? ¬(P ? A) means ¬G ? (P ? ¬A).

The argument is actually "if God does not exist then I pray [and it isn't answered], I don't pray, therefore God exists".

¬G ? (P ? ¬A) is a more appropriate premise and with it the conclusion no longer follows.

Quoting Benkei

Logical equivalence is not determined solely by symbolic representation, especially in light of the interpretive choices made when translating from natural language to formal logical symbols.

"If A then B" is logically equivalent to "if C then D." You're going to have offer a proof that is not the case without equivocating between deductive and inductive logic. I don't see how that can be done.

Quoting Benkei

Deductive logic ensures that if the premises are true, the conclusion must also be true. Obviously when the premises are true, a valid deductive conclusion will say something about the world.

This offers an equivocation of the term "true." The sylIogism "If A then B, A, therefore B" is true. The statement "I am at work today" is true. It's the analytic/synthetic distinction. It's for that reason why a statement can be deductively true and inductively false, which is what the OP showed. Analytic validity says nothing about synthetic validity.

Quoting Benkei

Your second argument is not inductively supported because the conclusion is supported by the definition of mammal. It's like saying, all bachelors are single, John is single and therefore a bachelor. There's no probability involved that a single man isn't a bachelor.

The definition of "mammal" was arrived at a posteriori as opposed to "bachelor" which, as you've used it, (i.e. there is no probability a bachelor can be married) is a purely analytic statement. That is, no amount of searching for the married bachelor will locate one. On the other hand, unless you've reduced all definitions to having a necessary element for them to be applicable (which would be an essentialist approach), the term "mammal" could be applied to a non-milk providing animal, assuming sufficient other attributes were satisfied. This might be the case should a new subspecies be found. For example, all mammals give birth to live young, except the platypus, which lays eggs. That exception is carved out because the users of the term "mammal" had other purposes for that word other than creation of a legalistic analytic term.

"All penguins are black" means something very different as an analytic statement versus a synthetic statement. The former holds it true as a matter of definition. The latter as a matter of fact. Necessary versus contingent.

Another hot button issue as an example, "Can a man give birth?"

Quoting Leontiskos

They are completely different. The implicit connotation in the OP makes perfect sense. Your parallel is perfect nonsense. Not all parlor tricks are created equal. The parlor trick of the OP is a great deal better than your attempt regarding billionaires. Your argument possesses no plausibility because it is so obviously unsound. You are trying to make yourself a billionaire with specious reasoning. The OP is not praying on the supposition that God does not exist.

I was trying to clear away the enticing parlor trick that made the OP appear plausible so that the error could be revealed. If it can be shown that the use of the logic within the OP will lead to absurd results in other instances, then that is a valid disproof of the logic within the OP. Such is a reductio ad absurdem.

Quoting Hanover

"If A then B" is logically equivalent to "if C then D." You're going to have offer a proof that is not the case without equivocating between deductive and inductive logic. I don't see how that can be done.

This is quite obviously not logically equivalent. The statements "if A then B" and "if C then D" involve different propositional variables (A, B, C, and D). Unless we have additional information about the relationship between these variables, we cannot assume they have any connection. The truth value of "if A then B" is determined solely by the truth values of A and B, while the truth value of "if C then D" depends only on C and D. These are independent of each other.

Without additional information, there's no reason to believe that the truth value of one statement would always match the other for all possible combinations of truth values. It's therefore entirely possible for "if A then B" to be true while "if C then D" is false, or vice versa, depending on the specific truth values of A, B, C, and D.

Quoting Hanover

This offers an equivocation of the term "true." The sylIogism "If A then B, A, therefore B" is true. The statement "I am at work today" is true. It's the analytic/synthetic distinction. It's for that reason why a statement can be deductively true and inductively false, which is what the OP showed. Analytic validity says nothing about synthetic validity.

Yes, you're right to point out some equivocation here but the point I was trying to make stands. If the premisses of a deductive argument are true (and I'm assuming a form of correspondence theory) then a valid argument will have a logically true conclusion and necessarily correspond with reality.

Quoting Hanover

The definition of "mammal" was arrived at a posteriori as opposed to "bachelor" which, as you've used it, (i.e. there is no probability a bachelor can be married) is a purely analytic statement. That is, no amount of searching for the married bachelor will locate one. On the other hand, unless you've reduced all definitions to having a necessary element for them to be applicable (which would be an essentialist approach), the term "mammal" could be applied to a non-milk providing animal, assuming sufficient other attributes were satisfied. This might be the case should a new subspecies be found. For example, all mammals give birth to live young, except the platypus, which lays eggs. That exception is carved out because the users of the term "mammal" had other purposes for that word other than creation of a legalistic analytic term.

While scientific terms do evolve, they do function as relatively fixed definitions within the scientific community. The fact that definitions can change doesn't necessarily mean they are probabilistic or inductive in nature during their period of use and "giving milk" is a rather necessary condition in that definition since the name is derived from breasts because of the mammary gland. So no, nice try but nobody has ever used the term for any animal that doesn't produce milk and they never will.

Quoting Benkei

So no, nice try but nobody has ever used the term for any animal that doesn't produce milk and they never will.

I don't produce milk?

Quoting Hanover

then that is a valid disproof of the logic within the OP

There isn't a problem with the logic. The problem is that the premise isn't saying what it superficially seems to be saying.

"it is not the case that if I pray then it will be answered" does not mean "if I pray then it will not be answered"; it means "I pray and it is not answered".

So the argument actually amounts to "if I do not pray then God exists, I do not pray, therefore God exists."

Formally:

¬G ? ¬(P ? A)
? ¬G ? P
? ¬P ? G
¬P
? G

Quoting Michael

There isn't a problem with the logic. The problem is that the premise isn't saying what it superficially seems to be saying.

I've agreed that the deductive logic within the OP is valid. I disagree that it's inductively valid. As in your reduction of the argument to:

"if I do not pray then God exists, I do not pray, therefore God exists."

that is deductively correct.

However, "if you do not pray then God exists" is a false statement if treated as a contingency. The reductio, for clarification purposes, was creating an absurdity, as in, "if I don't scream then I will be a billionaire, I do not scream, therefore I am a billionaire."

That is false because everyone knows that my defining characteristics are that I scream and that I am a billionaire.

If you don't produce milk, of what use are your nipples?

Reply to Hanover

The argument is valid but its first premise is false (or at least hasn't been proven to be true).

Reply to Michael :lol: Yes, yes, don't be too literal. You do have a mammary gland though.

Quoting Benkei

You do have a mammary gland though.

Well, I do, but those with congenital amazia don't. I assume they're still mammals.

Reply to Michael I don't know, are they?

Quoting Michael

The "parlor trick" is just that the antecedent contains the contradiction "¬(P ? A) ? ¬P".

My "parlor trick" includes the translation. The formalism is not very difficult to understand. What's fun is the way that the translation is intuitive. @Hanover's difficulty is this, "Why did we say, 'So I don't pray'?" The explanations I have been giving answer that question and give an account of why the translation is intuitive.

-

Quoting Hanover

I was trying to clear away the enticing parlor trick that made the OP appear plausible so that the error could be revealed. If it can be shown that the use of the logic within the OP will lead to absurd results in other instances, then that is a valid disproof of the logic within the OP. Such is a reductio ad absurdem.

You are failing to recognize the non-equivalence of the two. Whenever the "So" premise is justified the argument works. In the OP it is prima facie justified ("So I do not pray"). In your example it is not ("So I do not scream").

Reply to Michael also on bachelors:

How has the term bachelor evolved over time? Perplexity.ai:

The term "bachelor" has evolved significantly since its origins. Initially, in the 12th century, it referred to a "knight bachelor," a young squire training for knighthood. By the 14th century, it expanded to mean "unmarried man" and was also used for junior members of guilds and universities.

In the 13th century, it became associated with academic degrees, particularly the "bachelor's degree," indicating a low-level qualification. Over time, the term has taken on various connotations, including "eligible bachelor" in the Victorian era, referring to a financially and socially desirable unmarried man. Today, it primarily denotes an unmarried man without the historical implications of lower status.
---
So even bachelors are not as analytic as we like to pretend it is. But hey, everything frays at the edges of language. I'm not too worried about it.

And lo, God said "this should freak them scientists out for a century or Two." And verily, it was so.

https://www.britannica.com/story/why-is-the-platypus-a-mammal

Quoting Leontiskos

My "parlor trick" includes the translation. The formalism is not very difficult to understand. What's fun is the way that the translation is intuitive. Hanover's difficulty is this, "Why did we say, 'So I don't pray'?" The explanations I have been giving answer that question and give an account of why the translation is intuitive.

FYI I edited my post hours ago. Weird that you're seeing the old version.

I've corrected what I was trying to say.

See also this that might be even clearer.

Reply to Michael - I understand, but the same point applies to the edited post. You are prescinding from the translation and focusing entirely on the formalism. In fact you are back-engineering a new English sentence to better fit the formalism. Again, my "parlor trick" includes the translation itself. The levity of the OP derives in large part from the initial plausibility of the translation.

Again, Lionino's thread shows in some detail why there are no obvious English translations for ~(P?A).

Quoting Benkei

even bachelors are not as analytic as we like to pretend it is. But hey, everything frays at the edges of language. I'm not too worried about it.

That's the Quine argument. https://iep.utm.edu/quine-an/

While all mammals provide milk, not all providers of milk are mammals.

Gentlemen, I introduce you to pigeon milk: https://en.m.wikipedia.org/wiki/Crop_milk

Delicious in Froot Loops and a frothy cappuccino. Those birds will fight you though when you try to milk them.

Reply to Hanover Crop milk isn't from a maary gland.

Quoting Benkei

Crop milk isn't from a maary gland.

At least properly use you M key when you correct me.

Quoting Michael

The argument is valid but its first premise is false (or at least hasn't been proven to be true).

The first premise is the product of an inversion fallacy which I explained on the first page of this thread. There is an assumed cause/effect relation between God's existence and prayers being answered. We say that prayers being answered is the effect, and God's existence is the cause of this effect. God's existence causes prayers to be answered. However, it's an inverse fallacy to say that if prayers are answered then God exists. And saying "if God does not exist my prayers will not be answered" is another way of representing that same fallacious conclusion. So, the first premise, "If God does not exist, then it is false that if I pray, then my prayers will be answered" is a convoluted representation of that very same inversion fallacy.

The first premise is the product of a logical fallacy, and therefore can be considered to be false on that basis. I believe this is the fallacy which Hanover refers to as making the argument "inductively false".

Quoting Metaphysician Undercover

We say that prayers being answered is the effect, and God's existence is the cause of this effect. God's existence causes prayers to be answered. However, it's an inverse fallacy to say that if prayers are answered then God exists.

So you are saying that your prayers might still be answered even if God does not exist? So that an atheist could be justified in praying?

Quoting Leontiskos

So you are saying that your prayers might still be answered even if God does not exist? So that an atheist could be justified in praying?

There are all sorts of hypothetical entities that could answer prayers; devils, angels, fairies, wizards, extremely advanced aliens, the universe branching into a new timeline in accordance to one's will, etc. There's no reason to believe that it can only be the working of some sort of monotheistic creator deity (and certainly no reason to believe that it can only be the working of a specific religion's deity).

Reply to Metaphysician Undercover

None of that matters. Just assume that the premise is true. The conclusion is still (superficially) counterintuitive.

The issue concerns making sense of the argument's validity, not proving or disproving its soundness.

Quoting Leontiskos

So you are saying that your prayers might still be answered even if God does not exist? So that an atheist could be justified in praying?

The inverse fallacy is the perfect example of the need for skepticism. When we establish a cause/effect relationship between two types of events, A and B, this is based on either noticing that the first brings about the second, or in the case of the op, assuming that the first brings about the second. When the relationship is well known, and well documented, we get accustomed to it, and this produces a corresponding certitude surrounding those events.

The problem is that we never know for sure whether or not something other than A might bring about the occurrence of B. Because some degree of uncertainty lingers, even though we might say with a great degree of certainty that A always produces B, we cannot validly conclude that if we have B there must have been A.

There are many very good examples of this. For instance, the boiling point of water. We see that 100 degrees Celsius causes water to boil. But we cannot say that if water is boiling its temperature has reached that point, because pressure plays a role to decrease boiling temperature.

This is why ancient skeptics like Socrates and Plato were so persistent in warning us about how the senses mislead us. It is through this process whereby our inductively produced customs are held to high esteem. You can see that in those days it was assumed that the sun orbiting the earth caused the appearance of sunrise and sunset. If we do not allow the skeptic's premise, that possibly something other than the sun orbiting the earth could cause sunrise and sunset, we deny the possibility of advancements to scientific knowledge.

Quoting Michael

None of that matters. Just assume that the premise is true. The conclusion is still (superficially) counterintuitive.

The occurrence of a counterintuitive conclusion is the argument which Aristotle used against sophistry. This is why he placed Intuition as the highest form of knowledge. The sophists, such as Zeno, could use logic to produce absurd conclusions. When a conclusion produced from valid logic is strongly counterintuitive, this indicates the need to address the premises. It is very likely that there is hidden falsity, and that's what Socrates and Plato were demonstrating was the trick of sophistry, to veil falsity within the premises.

Quoting Michael

The issue concerns making sense of the argument's validity, not proving or disprove its soundness.

Nah, that's boring, Benkei went through that already on the first page, and as far as I'm concerned nothing more needs to be said. The real issue is the question of how this form of logic can produce seemingly absurd conclusions. And that was demonstrated by Hanover, it separates the form from the content.

This, I've argued in other places is the problem with "formalism" in general, it is an attempt to separate form from content, and this cannot actually be done without rendering the logic as totally meaningless and useless. So what happens is that little snippets of content get hidden within the logical form of the argument, or else there's be no argument. And, content always contains some degree of uncertainty. Then the form, being the logical process itself, has room for error inherent within it, rendering this a less than perfect form of logic. That is how formalism contaminates logic with uncertainty, in its attempt to do the impossible, remove all uncertainty (content).

Quoting Michael

There are all sorts of hypothetical entities that could answer prayers; devils, angels, fairies, wizards, extremely advanced aliens, the universe branching into a new timeline in accordance to one's will, etc. There's no reason to believe that it can only be the working of some sort of monotheistic creator deity (and certainly no reason to believe that it can only be the working of a specific religion's deity).

Eh. If I ask you to do something and someone else does it then you haven't fulfilled my request. Pretty basic. Has my petition been granted? No, I don't think so, unless the petition was somehow made to no one in particular.

Quoting Metaphysician Undercover

The problem is that we never know for sure whether or not something other than A might bring about the occurrence of B.

So you seem to think that atheists should go ahead and pray. It doesn't make sense. If someone believes that person X does not exist then they should not petition person X. A petition/prayer is not offered in generality, to no one in particular.

Reply to Leontiskos
You can pray to anything, it need not be God, it's called idolatry. So one might believe, that if you simply pray, in general, to no specific divinity, you'd have the highest probability of having your prayers responded to, because you are not limiting the possible respondents to one particular divinity.

Reply to Metaphysician Undercover - Prayer is just a special form of impetration or petition. I suppose one could send a petition to no one in particular—a kind of message in a bottle addressed to the universe at large—but that's really not what the word means. So if person X does not exist, you do not ask person X to do something for you.

There is idolatry, which is a case people praying to something other than God. If God can answer prayers why not something else as well?

Reply to Metaphysician Undercover - If you write a letter to Mike Tyson asking him to punch you in the face, and the next day a random guy on the street punches you in the face, has your petition been granted? Would you still await a response from Tyson?

The restricted sense of "pray" is just an accident of contemporary English. The concept traditionally has to do with petition:

Quoting Pray Etymology

early 13c., preien, "ask earnestly, beg (someone)," also (c. 1300) in a religious sense, "pray to a god or saint," from Old French preier "to pray" (c. 900, Modern French prier), from Vulgar Latin *precare (also source of Italian pregare), from Latin precari "ask earnestly, beg, entreat," from *prex (plural preces, genitive precis) "prayer, request, entreaty," from PIE root *prek- "to ask, request, entreat."

From early 14c. as "to invite." The deferential parenthetical expression I pray you, "please, if you will," attested from late 14c. (from c. 1300 as I pray thee), was contracted to pray in 16c. Related: Prayed; praying.

And as such, prayer is not restricted to God, worship (latria) is.

Quoting Leontiskos

If you write a letter to Mike Tyson asking him to punch you in the face, and the next day a random guy on the street punches you in the face, has your petition been granted? Would you still await a response from Tyson?

I don't understand your question. It does not seem to be comparable. If you ask God for something, or your favourite idol in the case of idolatry, and your wish comes true, how would you know whether this was caused by God, or the idol, some other cause, or just fate?

Quoting Leontiskos

And as such, prayer is not restricted to God, worship (latria) is.

Neither prayer nor worship is restricted to God. That's why the religious speak of false divinities, idols and heresy. And, that's part of the reason why the premise of the op is false.

But there is another, very serious issue I mentioned earlier, which has not been given attention in this thread. "God" is understood to have a will. And, because the will is understood to be free, there is no necessity between the intentional agent, and any described act, such that we could say that existence of the agent would necessitate that act. Therefore it is false to say that if God exists my prayers will be answered, or the inverted, if my prayers are not answered God does not exist.

Reply to Metaphysician Undercover - So apparently if you didn't get a good look at the guy who hit you, you would just assume it was Tyson. I still don't see how you would write him a letter if you don't believe he exists.

Quoting Banno

There's been a bunch of these around recently, so here's one that is actually valid...

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.

Attributed to Dorothy Eddington.

~G?~(P?A)
~P
G

That relies on conflating two different senses of "if then": an everyday sense and the material conditional. I'll use '-->' for the everyday sense and '->' for the material conditional:

(1) Everyday sense:

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

If ~G is true, then to have ~G --> ~(P --> A), even in the everyday sense, ~(P --> A) must be true. But why is ~(P --> A) true? Only because, unlike the material conditional, the everyday sense allows that a conditional may be false even when its antecedent is false.

(2) Material conditional:

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

That is a tautology. Because, unlike the everyday sense, a material conditional is false if and only if its antecednt is true and its consequent false.

/

With (1) we nod agreement with ~G --> ~(P --> A)) based on an everyday sense of the conditional by which a conditional (such as P --> A) may be false even when its antecedent is false.

With (2) we don't nod agreement with ~G -> ~(P -> A) since the material conditional (such as P -> A) is true when its antecedent is false.

Quoting Leontiskos

So apparently if you didn't get a good look at the guy who hit you, you would just assume it was Tyson. I still don't see how you would write him a letter if you don't believe he exists.

I don't pray myself, but I think that's how praying works. If your prayers are answered you assume it was God who did the answering. I don't understand the relevance of the last sentence though.

First, the argument is clearly valid.

So if the conclusion is false, one of the premises is false.

I do not pray, so the second premise is true.

Hence the first premise must be false. The first premise is "If God does not exist, then it is false that if I pray, then my prayers will be answered". ~G?~(P?A). Have a closer look at ~(P?A). Here's the truth table:

Notice that if "P" is false, ~(P?A) will also be false. ~P contradicts ~(P?A). But we know that ~P is true from the second premise. And if the consequent is false on a true implication, then the antecedent must also be false. That's how the logic works, and it's quite valid.

But that I don't pray can't imply that God exists. So something is amiss. Just not the logic.

If there is a god, then if you pray your prayers will be answered. This much seems true. So what can we conclude from this, if there is no god? We want to say that if there is no god, my prayers will not be answered. But this can be rendered in two ways.

Consider the difference between ""If God does not exist, then it is false that if I pray, then my prayers will be answered" and "If God does not exist, then if I pray, then it is false that my prayers will be answered". Between ~G?~(P?A) and ~G?(P?~A). These are not the same.

The simple answer is that, using material implication, it is not true that: if god does not exist then it is not true that if I pray then my prayers will be answered; but it is true that: if god does not exist then if I pray my prayers will not be answered.

But in ordinary English, we can say that it is not true that: if god does not exist then it is not true that if I pray then my prayers will be answered, Quoting TonesInDeepFreeze

Only because, unlike the material conditional, the everyday sense allows that a conditional may be false even when its antecedent is false.

Reply to Michael made much the same point.

The puzzle has nothing to do with the Inversion Fallacy, or the definition of God, or Denying the Antecedent fallacy, or the ambiguity of "If God does not exist..."; it's an ambiguity in the English use of "If...then" that, when done properly, formal logic sorts out.

Quoting Metaphysician Undercover

If your prayers are answered you assume it was God who did the answering.

And you think that one should still pray even if God doesn't exist?

Quoting Banno

@Michael made much the same point.

Then he's right. It takes only a moment to see that the salient feature of the argument is that it shifts from one sense of "if then" in one place to another sense of "if then" in another place.

Of course, "~G -> ~(P -> A) and ~P, therefore G" is classically valid. But what is interesting about the problem is that it has seemingly true premises and valid logic that lead to a conclusion that doesn't seem to follow from the truth of the premises.

"If there is no God then it is not the case that if I pray then my prayers are answered" seems true. It seems true based on an everyday sense of "if then" by which a conditional may be false when its antecedent is false.

But the inference "If there is no God then it is not the case that if I pray then my prayers are answered, and I do not pray, therefore there is a God" is valid based on a different sense of "if then" by which a conditional is false if and only if its antecedent is true and its consequent is false.

Noting that shift from one sense to another is a decisive and incisive explanation of how seemingly true premises and valid logic seem to lead to a conclusion that does not seem to follow from the truth of the premises.

Reply to TonesInDeepFreeze I think so. So if there is a god then my prays will be answered. If there is no god, they will not be answered. But how to pars this in propositional logic? It is better parsed as "If there is no god then if you pray then your prayers will not be answered" rather than "If there is no god then it is not the case that if you pray then your prayers will be answered".

Apparently Dorothy Eddington used this example in her logic classes to demonstrate the importance of taking care when interpreting natural languages.

Reply to Banno

I wouldn't assume that the everyday sense of "if then" in the problem has a truth table interpretation.

And, the premise is "If there is no God, then it is not the case that if I pray then my prayers are answered"; the premise is not stated as "If there is no God then if I pray then my prayers are not answered". But if it were stated that way, then, of course

(~G -> (P -> ~A)) & ~P, therefore G

is WRONG and there's not "puzzle" to it.

I took the problem to at least present a "puzzle".

Quoting TonesInDeepFreeze

I wouldn't assume that the everyday sense of "if then" in the problem has a truth table interpretation.

Fine by me. But if your logic teacher set parsing "If there is no god then your prayers will not be answered" into prop form, what would be the better choice? Which is why I thought it worth discussing. The creativity of the responses to this thread has been entertaining. :wink:

Reply to Banno

It depends on what the purpose of the translation is.

If the purpose is to directly emulate the sentence as literally said, then:

~G -> ~(P -> A)

If the purpose is to provide a reasonable guess as to what was meant when the sentence was said, then:

~G -> (P -> ~A)

Reply to TonesInDeepFreeze I edited my last post to make the meaning clearer. My apologies.

Quoting Leontiskos

And you think that one should still pray even if God doesn't exist?

As I said, I don't pray. And, I'll add that the existence or non-existence of God is irrelevant to that choice.

@fdrake shared some material with me on Relevant Logic, which is directly applicable here. There is a gap in relvance between the existence of god and my decision not to pray.

The variable sharing principle says that no formula of the form A?B can be proven in a relevance logic if A and B do not have at least one propositional variable (sometimes called a proposition letter) in common and that no inference can be shown valid if the premises and conclusion do not share at least one propositional variable.

Turns out to be not just a rabbit hole but a warren. Does anyone have a handle on this?

This does not belong in the lounge. This is a paradox that rest on a tricky difference between conditionals in language and conditionals in logic. The takeaway is, you have to be very careful translating language to logic, and very suspicious when others do so.

Quoting TonesInDeepFreeze

"If there is no God then it is not the case that if I pray then my prayers are answered" seems true. It seems true based on an everyday sense of "if then" by which a conditional may be false when its antecedent is false.

But the inference "If there is no God then it is not the case that if I pray then my prayers are answered, and I do not pray, therefore there is a God" is valid based on a different sense of "if then" by which a conditional is false if and only if its antecedent is true and its consequent is false.

I think this gets it right. There is simply no way to express the "everyday conditional" in propositional logic. I would call it the "real conditional"; it is what we actually mean by "if A then B". We certainly never mean A -> B, which is true whenever A is false. "If I were a billionaire I would grow 3 feet taller" is true in propositional logic, and clearly false in language.

The problem with the "everyday" or "real" conditional (given here by ?) is that it doesn't have a resolvable truth table (its truth is not determinable by the truth of its arguments alone):

A B A?B
F F ?
F T ?
T F F
T T ?

Only in one combination does A?B have a determinate truth value. Any logic that incorporated it would also have to incorporate indeterminate truth values. (Not a hard thing to do at all, it would probably be an interesting exercise for another post).

Quoting Michael

I think it's more addressing that these mean different things:

1. ¬(P?A)
2. P?¬A

The problem is that 2 does not express what the statement is saying either, which is that there is no relationship between praying and having the prayer answered. Note that "answering a prayer" here does not mean that God's fiery hand descends from the heavens, it means that whatever is prayed for comes to pass. If you pray for something, it might come to pass, or it might not. But if it does, the prayer would have had nothing to do with it. In terms of a truth table:

P A ¬(P?A)
F F ?
F T ?
T F T
T T ?

Quoting hypericin

This does not belong in the lounge. This is a paradox that rest on a tricky difference between conditionals in language and conditionals in logic.

I wouldn't restrict the lounge like that.

Off topic, but I think the various "wonderings" which are lounge-appropriate can lead to cool and interesting philosophical insights.

It's the creative space where as long as you're not a jerk go ahead -- random ideas, almost connected philosophical thoughts, conversational starting bits -- go for it!

So the philosophy bits do belong here -- I'd say especially because new philosophical thoughts often come from shooting the shit.

Reply to Moliere

I would move it. The thread seems more significant than the vast majority of mainline threads here; it reveals a huge landmine in propositional logic that I'm sure most aren't aware of (I sure wasn't), and is relevant to lots of other threads.

Reply to hypericin OK.

Reply to Banno Quoting Banno

~G?~(P?A)
~P
G

Mostly spitballing.

The offending equivalence (this is logically valid).

(¬G?¬(P?A))?((P?A)?G)

The latter: "If a prayer is answered by god, then that god exists"
The former: "If there is no god, then if something is a prayer then that prayer will be unanswered by that god."

Then you introduce ~P into the mix.

(((¬G?¬(P?A))?(¬P))?((P?A)?G)?(¬P))

Those are still equivalent, you just conjoin ~P to both sides. If you encountered ((P?A)?G)?(¬P)) out in the wild, you'd think "if something is a prayer, then it is an answered prayer, and that implication being true implied god existed" + "something isn't a prayer", you'd wonder why the hell anyone would be talking about something not being a prayer when it'd need to be an answered prayer to be relevant. It's a bit like trying to test a cat at the vet for a dog's illnesses.

Another thought regarding it is that the concept which makes the argument work is that if some prayers are answered by God, then God exists... Which looks a bit like (A?G). Rather than (P?A)?G. The equivalence between those two parsings isn't valid:

(((P?(A?G))?(¬P))?((P?A)?G)?(¬P))

since its countermodels are P false, A false, G false - IE no prayers, no answered prayers, no gods. The fact that A false G false is part of a countermodel to the equivalence and are also the facts which made the OP's argument seem paradoxical makes me believe that translating the natural language into (((¬G?¬(P?A)) makes us think we've translated (((P?(A?G))?(¬P)) into formal language, when we haven't. Which translation is of the two is not, in this instance, an innocuous choice.

The latter translation is also suspect - you can read it like "if I pray then all prayers answered are answered by god".

I prefer the latter analysis, an ambiguity between A->(B->C) and (A->B)->C that we don't notice much. But I get the impression that you could design other paradoxes to slip through this latter analysis.

Reply to fdrake

I think this is the simplest explanation.

Quoting Banno

logic

Reply to fdrake Is that much different to Reply to TonesInDeepFreeze or Reply to Banno or to Reply to Michael? Looks as if we have broad agreement. Always cause for concern.

We have that if you pray then your prayers will be answered, and that this will occur only if there is a god (leaving @unenlightened aside for a bit). We look to set out the consequence of there not being a god. Our natural language allows "If there is no god then your prayers will not be answered". This seems the same as "If there is no god then if you pray your prayers will not be answered". Then as "If there is no god then it is not the case that if you pray your prayers will be answered". But this last is subtly different, in a way brought out by formalising these last two sentences: ~G?(P?~A) against ~G?~(P?A). On this account the problem is that the English sentence "If there is no god then your prayers will not be answered" has an ambiguity that can lead to two different formalisations. That ability is the result of, as Tones puts it, "the everyday sense allows that a conditional may be false even when its antecedent is false".

Seems to me that if we are to go further with this we need a logic that will bring out the relation between prayer and god, such that @unenlightened is not the answer to our difficulties. Relevant Logic appears to offer such a possibility. Consider the example from that SEP article:

The moon is made of green cheese. Therefore, either it is raining in Ecuador now or it is not.

There are similarities to the present puzzle. Quite a valid conclusion, but it seems muddled. Similarly, whether I pray or not seems irrelevant to there being a god, although my prayers being answered is dependent on there being a god.

Can any of you parse the problem into [math]\mathbf{R}[/math]? Does doing so better show the issue?

And does this offer a way to formalise naive set theory?

Quoting Banno

?fdrake Is that much different to ?TonesInDeepFreeze or ?Banno or to ?Michael? Looks as if we have broad agreement. Always cause for concern.

It isn't much different no.

If x is a prayer answered by y, then x is a prayer, and y is a prayer answerer.

Axy -> (Px & Gy)

That's a real argument. Other versions are abusive.

The problem is that English doesn't adequately distinguish the counterfactual or hypothetical conditional from the logical conditional of the syllogism and so we confuse ourselves with the ambiguity.

1. Consider the sentence "If God exists, he will answer our prayers."

2. Consider this sentence "If God exist, he will answer our prayers."

Now represent these both formally.

Note the 2nd is not in the indicative, but the obsolete subjunctive and I'd submit incapable of being reduced formally. It does not say what will be. It hypothesizes. #1 has an antecedent. #2 has a hypothesis.

Or, to better clarify:

If I was President, I'd lower taxes.

I was president

I lowered taxes

P -> T.

P

T. Monus ponens.

But not:

If I were President, I'd lower taxes

I were President. (???)

I lowered taxes.

"I was President" can be represented as P.
"I were President" cannot.

The "were" becomes misplaced because it was a hypothetical as written and now it's being modified into an actual.

This is just to say our langauge poorly captures the distinction and the OP ridicules it

not-G -> ( not- (P -> A) )
not - P

does not imply

G.

in fact, the premises do not actually tell us anything. On the other hand,

not- G -> ( not- (P -> A) )
not- A

does seem to imply..

P.

But again, it still does not imply G.

On the other hand,

not- G -> ( not- (P -> A) )
A

does seem to imply

G.

Reply to NotAristotle https://www.umsu.de/trees/#(~3G~5~3(P~5A)~1~3P)~5G

Reply to Banno Okay, what about this argument -- https://www.umsu.de/trees/#((A~5~3A)~1A)~5~3A

A -> not-A
A
Therefore, not-A.

There must be a difference between implication and deduction, right?

Quoting NotAristotle

There must be a difference between implication and deduction

There is.

An argument is an ordered pair where the first coordinate is a set of formulas (the set of premises) and the second coordinate is a formula (the conclusion). (Or 'statement' instead of 'formula' if the context is less formal.)

A deduction is a certain kind of sequence of formulas (or a certain kind of sequence of formulas alongside numbered sets of previous entries), or tree, or sequent, or tableau, depending on the context).

An implication is a formula of the form 'P -> Q'. Or, an implication is an argument.

My view was characterized by posters recently.

I take the problem to be to explain the puzzle: How did we infer a seemingly false conclusion from seemingly true premises with seemingly correct logic?

My answer is that the argument uses two different senses of "if then".

And it is likely that ~(P -> Q) is interpreted by some people with the truth table for (P & ~Q) instead of the truth table for ~(P -> Q). But that is not the answer I provide to the puzzle, which is more general: Different senses of "if then" are used, whether a reinterpretation of the truth table or even an interpretation that is not truth-functional.

I have not necessarily signed on to the views or explanations of other posters.

Quoting NotAristotle

not-G -> ( not- (P -> A) )
not - P

does not imply

G.

In classical logic (but not intuitionistic logic),

~G -> ~(P -> A)
~P
therefore G

is valid.

Quoting NotAristotle

in fact, the premises do not actually tell us anything. On the other hand,

not- G -> ( not- (P -> A) )
not- A

does seem to imply..

P.

That's wrong.

Or, you're welcome to state your alternative logic.

Reply to TonesInDeepFreeze I can see why the premises imply G. I agree with Michael that there is a translation issue.

I think I meant to say:

1. not-G -> ( not (P->A) )
2. ( not (P->A) )
3. not-A
Therefore,
4. P

Following the events of The Brother's Karamazov Ivan Karamazov has a conversion experience and becomes a priest (he got better from the syphilis and insanity :grin: ). Years later, an atheist intellectual of much the sort that Ivan used to be moved to Ivan's village from St. Petersburg. One day, Ivan gets to talking apologetics with the man. The man says that he believes in science and logic, and that neither can show that God exists.

Ivan says, "well, if God does not exist everything is permitted, so I won't control myself and I'll sleep with your wife."*

"You can't do that!" the atheist replies.

He was inducted into the catechumenate the very next day baptized into the church the next Easter.

* We should note the implied premise that if God exists, everything is not permitted.

Quoting NotAristotle

1. not-G -> ( not (P->A) )
2. ( not (P->A) )
3. not-A
Therefore,
4. P

More simply:

~(P -> A)
therefore P

Quoting Banno

If God does not exist, then it is false that if I pray, then my prayers will be answered

Quoting Banno

~G?~(P?A)

These two are not the same thing.

What ¬G?¬(P?A) actually means is:
¬G?(P?¬A)
P and ¬A are necessary conditions of ¬G.
Since you say ¬P, one of the necessary conditions for ¬G are not there, so God exists by ¬¬G.
The argument is valid but unsound, P1 is false.

What you wanted to say by "It is false that if I pray, then my prayers will be answered", which is not two propositions P and A connected by material implication, but one single proposition containing the idea of causal implication, is ¬?(P?A) or ¬?(P?A). You can throw both of these into the logic checker and it will show that any conclusion about G is invalid. Besides, the premise would be false too.

https://slideplayer.com/slide/7419329/

Relevance logic is also irrelevant here. The premises are all thematically connected, and none of them are the LNC/LEM.

What you wanted to say...

I didn't want to say anything about possible worlds, nor "causal implication", whatever that might be.

This horse is dead.

That is exactly what you wanted to say by that phrase, unless you don't understand your own language, which is in fact the rule rather than the exception.

You have been given the answer to the "problem" and you don't like it.

Unsurprisingly, this website is still a waste of time.

Reply to Lionino Bye.

Folk who are interested can gather an idea of why Lionino was off-track from the SEP article on logical consequences.

Reply to Banno God prays to me. I just stopped listening ;)