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.

Crop milk isn't from a maary gland.

Crop milk isn't from a maary gland. — Benkei

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

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

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".

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. — Metaphysician Undercover

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?

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? — Leontiskos

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).

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.

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? — Leontiskos

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.

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

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.

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

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).

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). — Michael

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.

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

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.

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.

- 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?

- 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:

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. — Pray Etymology

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

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? — Leontiskos

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?

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

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.

- 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.

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 — Banno

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.

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. — Leontiskos

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,

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

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.

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

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

@Michael made much the same point. — Banno

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.

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.

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".

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

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:

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)

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

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

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.

"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. — TonesInDeepFreeze

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).

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

1. ¬(P→A)
2. P→¬A — Michael

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 ?

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. — hypericin

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.