I am starting a symbolic logic class in a few weeks so hopefully I will be able to understand this a little better then, but at this point (aside from understanding what the symbols are saying) I don't intuitively understand why they are equivalent.

I'm supposing that it is the case that the argument is valid (thought I am not familiar enough with a truth table yet to discern if that is the case or not) yet could never be sound, and it not being sound is what is intuitively raising flags? Or maybe it has to do with the definition of "equivalent" as it pertains to philosophy?

Edit: I think I get it, If God exists then God doesn't exist is a hypothetical in which it never follows that God exists. So it's equivalent to saying Either it is the case that God doesn't exist, or that God doesn't exist, because it never follows from that hypothetical that God exists.

"If A, then B" is always equivalent to "Either ~A or B."

"If...then..." is generally read as the weak conditional. They are logically equivalent because they necessarily have the same truth values:

If A is true and B is true, both statements are true
If A is true and B is false, both statements are false
If A is false and B is true, both statements are true
If A is false and B is false, both statements are true

One way to think about it is to ask "How would I demonstrate "If A then B" was false?" If you think about it, the way you would demonstrate that would be to show that A was true and yet B was false - if that state of affairs existed, we would be able to conclude that the implication itself didnt hold. (So if we wanted to demonstrate the falsity of "If a bird is a swan then it is white" we would do so by finding a bird which was a swan and which was not white).

There are other logical operators - the strong conditional (from rather ancient memory) requires that the antecedent, A, also be true. So that returns the following truth values:

If A then B...
If A is true and B is true, both statements are true
If A is true and B is false, both statements are false
If A is false and B is true, both statements are false
If A is false and B is false, both statements are false

Adopting this interpretation of If...Then... brings its own problems though. Note that if we use this account then we are forced to declare

"If I've thrown the brick through the window, the glass will be broken" to be false in those instances where we havent actually committed vandalism.

bunny -

I think the strict conditional is only available in modal logic. (Btw, in the second paragraph of your post, you wanted to distinguish between the truth values of A->B and ~AvB when A is false, right?)

Perhaps Original Position can clarify, but I don't think the strict conditional is a reasonable interpretation of any implications used in the arguments in this thread.

The strong conditional was an alternate semantic account (within first order logic) of if...then...

(though I may well be misremembering the terms). It isn't the strict conditional, but is rather an attempt within classical logic to overcome the difficulties people have accepting the weak conditional (or material conditional as your link termed it). It fails, but I was trying to point out to Acemanhattan that any interpretation within classical logic fails. It's a matter of which problem you're willing to accept. Having said that...

I'm pretty sure I'm misremembering it anyhow, since what I gave is just "A and B". perhaps the operator I recall was TFTF. Irrespective, the point was that TFTT is the best we can do, within first order logic.

It’s “always true” with a material conditional. In traditional logic it’s a violation of the law of non-contradiction to both affirm and deny the same predicate P to the same subject S.

bunny -

I think I'm misreading your post; I'll try again in the morning.

duffe -

So you're always making these sudden appeals to Aristolean logic. Can you please explain how your argument...

1. If God exists, then morality is objective in our world.
2. If morality is objective in our world, then gratuitous evil exists in our world.
3. If gratuitous evil exists in our world, then God does not exist.
4. Therefore, if God exists, then God does not exist.

...given traditional logic, would be consistent (all true premises) with the beliefs of a moral nihilist? I really don't see it how it can be. (I'll try to come to terms with Aristolean logic if it's really all you accept; but it appears you're just arbitrarily switching back and forth. Or using some strange hybrid. Or something else I don't understand.)

I haven't done any formal logic since 1998. It is entirely possible that what I said is just, plain wrong. Im certainly only working from hazy memory - i havent checked notes or anything. Nonetheless..

I understood Acemanhattan to be querying why we would interpret if...then... as the weak conditional. It isn't immediately obvious that "If A, then B" is always equivalent to "Either ~A or B." but only becomes so through looking at truth tables and asking what is the best semantic convention to adopt when trying to represent if...then... statements in simple, propositional logic.

TFTT (the weak conditional) was chosen from several possibilities - all of them are flawed in a sense (and modal logic was one way to improve on this state of affairs). Nonetheless, the weak conditional can perhaps be justified as a candidate based on asking "how would I demonstrate it's falsity?" the only way we can be sure if A then B is false is if A is true and B is false.

Adopting this interpretation, "If A, then B" is always equivalent to "Either ~A or B." is trivially true by inspection.

My "further illustration" perhaps muddied the waters (and was no doubt wrong). I'm also using quirky terminology, I suspect.

All in all, not a very helpful contribution, but I really like symbolic logic, so I had a hankering to participate.

Okay, I see that as presented my argument has been incomplete. As I said earlier, I think the proper form of this argument is as a reductio, but I'm leaving this part of it out of the form, so let's go ahead and put it in. So, reductio arguments work like this. You take the proposition you wish to prove, assert that it is false, and show that doing so leads to a contradiction. We don't generally include as premises in such arguments basic logical principles such as bivalence or the law of non-contradiction but just accept them as inference rules for the logic within which we are working. So, with that in mind, here is a more complete version.

1. God exists. (by assertion--this is the premise we are interested in)
2. If God exists, then morality is objective. (this is usually understood as being true by definition)
3. Therefore, morality is objective. (from (1) and (2) by MP)
4. If morality is objective, then gratuitous evils exist. (by assertion--this is supposed to be an empirical claim)
5. Therefore, gratuitous evils exist. (from (3) and (4) by MP)
6. If God exists, then gratuitous evils don't exist. (assertion--usually understood as an implication of God's nature).
7. Therefore, gratuitous evils don't exist. (from (1) and (6) by MP)
8. Therefore, gratuitous evils exist and gratuitous evils don't exist. (from (5) and (7) by conjunction)

Now, this argument is valid (it only uses MP and conjunction, both of which are valid inference rules), but leads to a contradiction. What this means is that one of the premises is false. We know this because, by definition, if an argument is valid and all of its premises are true, then the conclusion must be true as well. However, we know that this conclusion is false since it asserts a contradiction. So, which of these premises are false? Most theists will claim that (4) is false (although some will claim that (6) is false as well, but we'll ignore that for now). Most atheists will claim that (1) is false. This is why the actual debate about the problem of evil usually consists of atheists defending (4) and theists attacking (4). If you want to argue that (4) is false, fine go ahead, that is a real issue in my opinion.

However, you raised a different objection. You claimed that since the nihilist as a matter of fact believes that (5) is false, she cannot be asserting this argument in good faith. However, this is a misunderstanding of the nature of reductio arguments. The person presenting the argument is not claiming to actually accept the conclusions of the argument. Of course not. The point is to show the argument is unsound because it has a false premise--that god exists. You might as well argue that since the atheistic nihilist doesn't accept (1) she is arguing in bad faith.

For example, Anselm's version of the ontological argument is usually understood as a reductio. It begins by asserting something like, "God exists in the understanding, but not in reality" and then goes on to allegedly show that some contradiction ensues. As you know, I think this argument fails. However, I don't think it fails because Anselm is asserting something he himself doesn't accept (that God doesn't exist in reality). Rather, he is arguing that this cannot be true because it leads to falsehoods or contradictions.

Regarding your counterexample argument, as pointed out by Subfallen, the conclusion of that argument is accepted by the atheist. "If A, then ~A" is a true statement when A is false. Thus, if you accept that claim, then on pains of contradiction you must believe that A is false. This is very much like the reasoning in my argument.

I can't clarify this. In a prior thread I also suggested that perhaps duffe interprets certain kinds of conditional statements as strict conditionals, but that didn't end up being correct. To be honest, I am often confused by what duffe means when he uses logic.

As for strict conditionals in this thread, it is not a settled question (that I am aware of at least) how to interpret the logical structure of conditionals in natural language. Here is SEP for more.

I don't get it. Why does any theist who employs my argument have to assert that gratuitous evils don't exist?

In the original version of my argument I never included (3), but rather this claim "If God exists, then gratuitous evil doesn't exist in our world." In my revised version of the argument above, I do include (3), but only as a subconclusion based on a premise which the atheist thinks is false (that God exists).

define privileged.

appeals to mystery are great things





for science to poop on


i think when we 'recognise' a moral truth we either consciously or subconsciously recognise some utility of behavior, or have an evolved emotional/empathetic response to it, or both. i've never seen any evidence to make me suspect there is more than that is going on, or thought it was necessary that there is anything more to it.

it could be written in the sky and it wouldn't be a guide to behavior in any practical sense.

i think you in a sense effectively ARE arguing for objective beauty, or at least objective 'nobleness' or whatever.

1. If God doesn’t exist, then morality is subjective in our world.
2. If morality is subjective in our world, then gratuitous evil doesn't exist in our world.
3. If God doesn’t exist, then gratuitous evil doesn’t exist in our world.
4. Therefore, if God doesn’t exist, gratuitous evil both doesn’t and does exist in our world (which is absurd).

“... if one wishes to believe” God doesn’t exist “one must be prepared to believe absurdities.”

Now if upon reading my argument, the atheist doesn’t feel any compulsion whatsoever to “get religion,” why would you think the theist feels any compulsion whatsoever to abandon her belief based upon the original argument?

Because your argument is not valid?

Was (3) supposed to say, "If God doesn't exist, then gratuitous evil does exist in our world"?

The rule of the pure conditional in traditional logic is: if A is true then it necessarily follows that B is true. So, I’d accept 1 & 3 because the consequents seem (or they might per definition of terms) to follow the antecedents. However, I’d reject 2 as being a pure conditional since I don’t see a logical sequence between the antecedent and consequent.

So, do to that required logical sequence between the antecedent and consequent, if there’s a false or contradictory premise in the argument it will show up as a contradiction in the conclusion, or if there’s a contradiction in the conclusion the argument contains a contradictory premise.

My point, then, wasn’t to show that nihilism is absurd, but to show the consequences of allowing a premise into an argument that simply meets your standard of having an antecedent that one believes is false, i.e. the material conditional. Applying your standard, the theist, believing the antecedent false can simply assert:

If God doesn’t exist, then evil spirits are real.

From which,

Either God exists or evil spirits are real.

So faced with that disjunctive is the atheist then faced with a dilemma? Or, can it be said that it’s a consequence of denying the existence of God that one affirms the existence of evil spirits, or if one rejects evil spirits, then one accepts that God exists? I would say, no. And I’d argue that the reason why the atheist is not compelled to assent to any of those positions is because the train of logical inference was broken with your material conditional, hence the conclusion lacks any logically compelling force. In fact when looked at in another light, all one is essentially doing is begging the question:

If God doesn’t exist, then pigs can fly.
Pigs cannot fly.
Therefore, God exists.

In other words, the only reason the theist is assenting to the conditional is because she’s assumed the conclusion, which you noted, “...and of course a nihilist believes this!” as if some sort of logical triumph occurred.

Now, considering over the last couple of posts and using your standards, I’ve shown that:
Either God exists, or
(i) gratuitous evil both doesn’t and does exist in our world (which is absurd).
(ii) evil spirits are real.
(iii) pigs can fly.

And assuming you haven’t run off to church due to the force of my arguments, perhaps the workaround I mentioned earlier, the conditional proof (if A, then B; A, therefore B)* is a remedy for this particular issue.

from wiki:
*A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.

The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows. Note that the validity of a conditional proof does not require that the CPA is actually true, only that if it is true it leads to the consequent.

I’ll have to digest all that later tonight, but my real objection is introducing ‘objective morality’ into the argument to begin with. I’m not saying you shouldn’t, I just don’t see the reason for doing so. Theists will argue that by definition there’s more like ‘an absolute morality’ implicit in the definition of God’s existence which will make it unusable to the second premise in your original argument. So, my question is why aren’t you just saying something to the effect:

If God, then (x) gratuitous evil.
If God, then (~x) not gratuitous evil.
Therefore, if God, then x and ~x.

I guess what I’m getting at is what transformational power does objective morality have that when a theist asserts it exists then the conclusion is (~x), yet when the nihilist asserts it the conclusion is (x)?

Not at all. I was employing a rare and obscure method found in traditional logic called “the double reductio.”

but, yeah a typo thanks for noticing.

Sorry, typo in 3. Should read:
3) If God doesn’t exist, then gratuitous evil does exist in our world.

Qualitatively distinguishable from all other elements of the class. (in this case - complete and all true).
Or accept, for now.
I know.
Maybe. I'm not sure that a belief in the objectivity of something often believed to be subjective implies that I should deem everything objective. I'm not yet committed to the view that there are no inherently subjective concepts. (at this stage, I consider beauty to be closer to desirable than to truth).

Well, before we go any further, let me make sure that we're starting from the same place. You agree that, thus far, I have only been arguing against your view that a nihilist is unable to consistently present an argument from evil?

(And in particular, I've not spent a sentence arguing that a theist would find this argument convincing.)

4. If (a) morality is objective, then (b) gratuitous evils exist.

Logical consequences of affirming (4) is true:

A. Either (~a) morality is not objective or (b) gratuitous evils exist.

B. False: (~a) morality is not objective
C. True: (b) gratuitous evils exist.

D. Either (~a) morality is not objective or (a) morality is objective.
E. Either (b) gratuitous evils exist or (~b) gratuitous evils do not exist.

F. False: (~b) gratuitous evils do not exist.
G. True: (a) morality is objective.

I'm not really sure what you are trying to say here, but if by bolding (G) you mean to imply that it is a logical consequence of (A), then you are wrong. But mainly, I just don't understand what is going on here.

Aha, that makes sense. Googling "weak conditional" returns journal articles and books, so I'm not surprised I'm unfamiliar with it.

I’m saying that an atheist who asserts (4) is true must accept (A) as true.
(if a, then b = ~a or b)

So what member of (A) does the atheist affirm/deny? I assumed he’d opt for B & C, with the logical consequence of doing so being F & G, since electing otherwise nullifies the contradiction in the argument's conclusion.

What I mean is that upon entering the argument in good faith and meeting with…

6. If (a) God exists, then (b) gratuitous evils don't exist.
(Either (~a) God doesn’t exist or (b) gratuitous evils don’t exist.)

… the theist must affirm/deny one member or the other (~a or b) with (6), the proposition she is charged with supporting and defending as true. So in like manner what member of (A) does the atheist affirm/deny with (4), the proposition he is charged with supporting and defending as true?

What do you mean? At this point I'm not only confused by your logic, I'm also baffled by what work you expect an argument to do.

Say Claim X has a well-defined truth value. Do you believe there must exist an argument which will lead everyone reading in good faith to the same conclusion about X?