I don't see the problem caused by the following entailment

Question

From Forall X (p.74):

(3) ‘It’s not the case that, if God exists, She answers malevolent prayers.’

Symbolizing this in TFL, we would offer something like ‘¬(G → M )’. Now, ‘¬(G → M )’ entails ‘G’ (again, check this with a truth table). So if we symbolize sentence 3 in TFL, it seems to entail that God exists. But that’s strange: surely even an atheist can accept sentence 3, without contradicting herself!”

The textbook says that the statement tautologically entails “God exists” & that this creates a problem but I don’t see where the problem lies. The above statement is only true when “god exists” is true & “She answers malevolent prayers” is false. So when the above statement is true, it logically follows that “god exists” must be true.

Answer 1

An atheist might be convinced that there is no God but accept that hypothetically if there were a God, God would be benevolent and not malevolent. Such a possibility should not entail the existence of God. The problem is that conditionals are not typically true just because their antecedent is false. Hence, ordinary conditionals are not typically material implications.

Let's modify your example and turn it into an example given by Dorothy Edgington. If I utter a prayer (to God) will my prayer be answered (by God)? Clearly not if there is no God. So we have:

1. ¬G → ¬(P → A)

But I don't pray, so we have:

2. ¬P

From this it follows:

3. G

So now we have true premises and a valid argument with the conclusion that God exists. The problem is that our conditionals here are not material implications.

Answer 2

¬(G → M) ⊨ G

is correct: we can check it with truth table.

If we assume that G is False, then we have that also ¬(G → M) is: thus, we cannot have that the conclusion is False and the premise True, satisfying the definition of entailment.

For an atheist G is False and thus (G → M) is true.

This means that for an atheist the premise is False: but this fact does not contradict the definition of valid argument:

In logic an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false

For an atheist the premise is false and thus the validity of the argument does not implies the truth of the conclusion.

---

Maybe the issue is that we can read the sentence as a counterfactual: atheists do no believe in the existence of God, but they will agree that the "concept" of God implies that he must not answer malevolent prayers.

Answer 3

At a pure formal logic level, this is a triviality because in classical logic ¬(G → M) is indeed equivalent with G ∧ ¬M, so that entails G; e.g. "proof" using deduction detachment to internalize the entailment to implication as (G ∧ ¬M) → G. The latter is a theorem (i.e. true) for any truth values of M and G; fairly obvious because ¬(G ∧ ¬M) ∨ G = ¬G ∨ ¬M ∨ G. So that (latter) statement is implied by the law of excluded middle (¬G ∨ G) in classical logic, plus the introduction of something "irrelevant" to the truth of the whole formula by disjunction introduction (of ¬M) next to a tautology.

The subtlety here is that the statement (3) alone is not the same as ‘(3) entails G’. I.e. G ∧ ¬M (or equivalently ¬(G → M)) can be false (if G is false for example), but (G ∧ ¬M) → G is still true even in that case (of G being false), simply because "the false implies anything" (ex falso quodlibet) in classical logic.

So where/what is the problem that the book is trying to illustrate? It's perhaps this: the natural language formulation of (3) is such that "everyone would agree it's true"...

---

For what's that worth, here's a slightly less confusing exposition of what the problem is supposed to be, found in another paper:

The problem for psychological theories based on logic (e.g., Braine & O'Brien, 1998; Grice, 1989; Rips, 1994) is that the truth‐functional meaning for conditionals fails for indicative conditionals in daily life. For example, it has the unfortunate consequence that false conditionals have if‐clauses that are true. The following conditional is false:

If God exists then atheism is correct.

Its negation is therefore true in logic:

It is not the case that if God exists then atheism is correct.

Given the truth‐functional meaning of conditionals, this assertion validly implies:

God exists.

But it is absurd to prove the existence of God merely from the negation of a false conditional. Some proponents of logical rules of inference (e.g., Rips, 1994), therefore, exclude the rule:

Not (If A then B). Therefore, A.

As a result, no clear meaning for conditionals exists within the theory.

Unfortunately the paper then goes on to discuss other topics (like counterfactuals). But the above formulation cuts to the chase, so to speak, by removing the more confusing part about benevolence, prayers etc.

Basically, the problem in this latter formulation seems to be that we have this (implicit/received) mental model that A <-> not G; i.e. atheism and (the assertion that) "God exists" are logical opposites. So the paper asserts that therefore when we hear

If God exists then atheism is correct.

it immediately seems false to us. Whereas with the usual "material conditional" implication from classical logic, a false premise implies everything, so the above implication is actually true in classical logic if God does not exist, because the false implies everything.

At least in this formulation, it's more clear where the problem springs from... I don't have the references on hand here, but (IIRC) there are some psychology papers that discuss that many people when they hear a conditional only think of the possibility of the premise being true, i.e. they only form a mental model of "half of the truth table" of classical implication ("material conditional"). On quick search I found another paper that asserts that:

Given a basic conditional of the form, If A then C, individuals usually list three cases as possible: A and C, not‐A and not‐C, not‐A and C. This result corroborates the theory of mental models. By contrast, individuals often judge that the conditional is true only in the case of A and C, and that cases of not‐A are irrelevant to its truth or falsity.

I didn't have time to read it thoroughly, so I'm not sure if that's a mere assertion or there is substantial experimental data to back it up. Most of that paper itself is actually about a different/related issue.

Anyway, I think this is the problem that the book 'Forall X' is trying to illustrate there, but they could have chosen a simpler example for this purpose, as in the above...

In the 'Forall X' example, there's an additional psychological trick here that once you assume God exists, you also agree with "usual assumptions" (from Christian theology) that God is benevolent (thus not malevolent), so you reflexively agree that both prongs (of the conjunction G ∧ ¬M = ¬(G → M)) are true. If Satanism (or in any case some monotheistic religion that assumes God is malevolent) were [more] predominant in society, you'd not intuitively agree with (3) anymore. So really there's a good level of "system 1" tricks being played there. What really happens (IMHO) is that you think/process that as if you've read "assuming Christian theology true, God exists [and is benevolent] and [thus] does not answer malevolent prayers". So that's how you "intuitively agree" with it.

Answer 4

It’s not the case that, if God exists, She answers malevolent prayers.

As others mentioned, when symbolized as ¬(G → M), an atheist would say this is a false statement; the antecedent G is false, therefore G → M must be true, therefore ¬(G → M) must be false.

But that's only half the puzzle. The other half is that despite the if-then wording, ¬(G → M) is simply not the correct way to capture this sentiment. A better symbolization is ¬(G ∧ M), i.e. "It's not the case that there is a God that answers malevolent prayers." How do we get from the if-then to a conjunction?

First the sentence ought to be reworded. This changes the meaning, but as described earlier, the original sentence taken literally does not capture the intended sentiment. It should be reworded like this:

If there is a God, it is not the case that there is a God that answers malevolent prayers.

This can be written, "G → ¬M." This is equivalent to ¬G ∨ ¬M (rule of material implication) which is equivalent to ¬(G ∧ M), by DeMorgan.