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.

OK.

~G→~(P→A)
~P
G — Banno

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.

I think this is the simplest explanation.

logic — Banno

Is that much different to or or to ? 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 $\mathbf{R}$? Does doing so better show the issue?

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

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

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.

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

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?

There must be a difference between implication and deduction — NotAristotle

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.

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

does not imply

G. — NotAristotle

In classical logic (but not intuitionistic logic),

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

is valid.

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. — NotAristotle

That's wrong.

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

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.

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

More simply:

~(P -> A)
therefore P

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

~G→~(P→A) — Banno

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.

Bye.

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

God prays to me. I just stopped listening ;)