This debate is discussing the first premise of this moral argument:

- If God does not exist, then objective morality does not exist.
- Objective morality does exist.
- Therefore, God exists.

It’s notable how little Aleph Null addressed my

opening statement.

To support the truth of premise (1) I used the Argument from the Unbiased Atheist, and we’ve heard to no response to that yet. I also employed these two points:

- Given atheism, we would not have good reasons for believing in objective morality.
- Given atheism, we would have (at least)
*prima facia* grounds for believing objective morality doesn’t exist.

So far we’ve heard no rebuttal to point (1). We’ve seen no evidence for evolved intuitions of moral oughtness being any less delusory than intuitions of gods existing. Recall I’ve also given reasons for why we don’t have rational grounds for believing objective moral oughtness on atheism (zero empirical evidence for moral oughtness, our best explanation for why we believe in morality has no need for morality’s existence, especially since belief in moral oughtness has evolutionary value whether it exists or not, etc.).

Nor have we heard any rebuttal to point (2), though I provided reasons for it (argument from queerness with the invisible unicorn illustration, objective morality’s queerness is akin to nonphysical gods, the precedent of evolution giving humans delusory intuitions of gods existing etc.).

**General Approach**Aleph did attack my General Approach though. To illustrate how one can be justified in accepting if-then statements, I used the following scenario: I know Sue parked her car out in the open air ten minutes ago, she has not left my side since then or given her car keys to anyone, and she’s the only one with the car keys. I claimed I’d be justified in believing “If it rained recently, then Sue’s car is wet.” Incredibly, Aleph Null says not only is this if-then statement unjustified, it’s actually

*false!*Some logic lingo: an “If

*P*, then

*Q*” statement is called a

*conditional*, where the

*P* part is called the

*antecedent* and the

*Q* part is called the

*consequent*. Aleph Null suggested that if it’s possible for there to be a true antecedent with a false consequent, then the conditional is itself false, e.g. it’s possible (though unlikely) some weird person is randomly throwing tarps on people’s cars, including Sue’s car, and thus it’s possible (but unlikely) for “If it rained recently, then Sue’s car is wet” to have a true antecedent with a false consequent.

To illustrate the folly of this approach, consider the following. A mayor concerned with the safety of bridge X wants to know whether we’re justified in thinking, “If an ordinary car travels over bridge X, the bridge will support the weight.” She hires a large team of the best, most reliable, most scientifically minded bridge inspectors on earth to inspect the bridge, and the inspectors unanimously ensure the bridge can handle the weight and much more. Ordinarily we’d take this as superb grounds for accepting, “If an ordinary car travels over bridge X, then the bridge will support the weight,” but Aleph Null’s reasoning leads us to believe that if-then statement is not only unjustified but also

*false!*Why? Because it is

*possible* (though unlikely) that the bridge inspectors made a mistake. Even if our bridge inspectors are superhumanly reliable, it is possible (though unlikely) that the scientific grounds we use for making judgments like this are false (scientific theories are often rational to believe but they are not strictly proven) in a way that the bridge would collapse under the car’s weight. It is possible, albeit unlikely, that the antecedent is true and the consequent is false for “If an ordinary car travels over bridge X, then the bridge will support the weight,” and thus this if-then statement is unjustified and false, even with superhumanly reliable bridge inspectors saying otherwise. But clearly this is absurd; Aleph has raised the standard of proof for accepting if-then statements to an unreasonably high level. “If an ordinary car travels over bridge X, then the bridge will support the weight” is a justified belief in this scenario, even if it’s not known with absolute certainty.

**The Actual Claim**I am not claiming that God is necessary for morality in this debate; in my opening statement’s “What the claim is not” section I pointed out the claim is not “that objective morality

*can’t* exist if God doesn’t exist, the claim is that objective morality

*doesn’t* exist if God doesn’t exist.”

For sake of brevity, let’s use these symbols:

**A** = atheism is true**¬M** = objective morality does not exist.**Pr(¬M|A)** = the probability of ¬M given A.

My actual claim (from my opening statement):

…given atheism, objective morality probably doesn’t exist—to the point where given atheism, one would be justified in believing objective morality does not exist.

I believe that given A, ¬M is probably true, i.e. I believe Pr(¬M|A) is high, and I believe Pr(¬M|A) is high enough to be justified in believing that “If A, then ¬M.”

**Mathematics and Propositional Logic**I believe that if Pr(¬M|A) is high, then the probability of “If A, then ¬M” is high. More generically, I believe Pr(Q|P) being high entails a high probability of “If P, then Q,” at least in propositional logic, and I’ll provide a mathematical proof for this.

In propositional logic, the “If P, then Q” material conditional says it is not the case that antecedent is true and consequent is false. This is often good enough for philosophical arguments (such as the moral argument), since in a true material conditional, if the antecedent

*is* true, the consequent is true as well—because a true material conditional prohibits the consequent from being

*false* when the antecedent is true. (Note: I’ve proved the following theorem before

at my blog, and one can read that to see a quick refresher of basic set theory and probability).

In math and logic, a

*lemma* is a claim that is proved to demonstrate something else later in a proof. For this proof I’ll be using several lemmas.

Recall that the “If P, then Q” material conditional means it is not the case that P is true and Q is false. Thus the probability that the material conditional is true can be mathematically depicted as this:

Pr(¬(P ∩ ¬Q))

Which means “The probability of it not being the case that P and ¬Q are both true.”

By “If P, then probably Q” I mean “Given P, Q is probably true,” which in turn means that Pr(Q|P) is high. So to show that high Pr(Q|P) entails a high Pr(¬(P ∩ ¬Q)), I want to prove the following:

Pr(¬(P ∩ ¬Q)) ≥ Pr(Q|P)

Lemma (1): (P ∩ Q) and (P ∩ ¬Q) are disjoint (mutually exclusive). We can show that no element in the universe can be a member of both (P ∩ Q) and (P ∩ ¬Q). Let

*x* be an arbitrary element and let’s suppose

*x* is a member of both (P ∩ Q) and (P ∩ ¬Q). With a bit of math logic, we show that there can’t be any

*x* such that

*x* ∈ (P ∩ Q) and

*x* ∈ (P ∩ ¬Q) by assuming there is such an

*x* and deriving an impossibility, like so:

(1)

*x* ∈ (P ∩ Q) and

*x* ∈ (P ∩ ¬Q)

(2) (x ∈ P and

*x* ∈ Q) and (x ∈ P and

*x* ∈ ¬Q), from (1) and definition of ∩

(3)

*x* ∈ P and

*x* ∈ Q and

*x* ∈ P and

*x* ∈ ¬Q, from (2)

(4)

*x* ∈ Q and

*x* ∈ ¬Q, from (3)

Of course, it’s impossible for there to be an element that is a member of a set and its complement, since (Q ∩ ¬Q) = ∅. Thus (P ∩ Q) and (P ∩ ¬Q) are disjoint, i.e. (P ∩ Q) ∩ (P ∩ ¬Q) = ∅.

With this in mind, let ξ be the universal set.

P ∩ ξ = P

⇔ P ∩ (Q ∪ ¬Q) = P

⇔ (P ∩ Q) ∪ (P ∩ ¬Q) = P

Lemma (2): Since (P ∩ Q) and (P ∩ ¬Q) and are mutually exclusive, by the rules of probability:

Pr(P ∩ Q) + Pr(P ∩ ¬Q) = Pr(P)

With those two lemmas in mind, consider this statement:

Pr(Q|P) = Pr(Q ∩ P)/Pr(P)

Now we swap Pr(P) for Pr(P ∩ Q) + Pr(P ∩ ¬Q), and this is a legitimate move thanks to the equality proved in lemma (2):

Pr(Q|P) = Pr(Q ∩ P)/[Pr(P ∩ Q) + Pr(P ∩ ¬Q)]

⇔ Pr(Q|P) = Pr(P ∩ Q)/[Pr(P ∩ Q) + Pr(P ∩ ¬Q)]

Just to make this easier to read, let’s have

*x* represent Pr(P ∩ Q) like so:

Pr(Q|P) =

*x*/[

*x* + Pr(P ∩ ¬Q)]

Given some value for Pr(P ∩ ¬Q), what is the highest Pr(Q|P) possible? One hint is this: given some Pr(P ∩ ¬Q), when

*x* goes to zero, so does Pr(Q|P); a smaller

*x* means a smaller Pr(Q|P) (

some calculus rigorously proves this). So to get the highest Pr(Q|P) value given some Pr(P ∩ ¬Q), we want

*x* to be as big as possible. Now since this is true:

Pr(P ∩ Q) + Pr(P ∩ ¬Q) = Pr(P) ≤ 1

⇔ Pr(P ∩ Q) + Pr(P ∩ ¬Q) ≤ 1

⇔ Pr(P ∩ Q) ≤ 1 − Pr(P ∩ ¬Q)

The highest Pr(P ∩ Q) (and thus

*x*) can be is 1 − Pr(P ∩ ¬Q). So substituting the maximum value for

*x* to obtain an upper limit for Pr(Q|P) gives us this:

Pr(Q|P) ≤ [1 − Pr(P ∩ ¬Q)]/[1 − Pr(P ∩ ¬Q) + Pr(P ∩ ¬Q)]

⇔ Pr(Q|P) ≤ [1 − Pr(P ∩ ¬Q)]/{1 + [−Pr(P ∩ ¬Q)] + Pr(P ∩ ¬Q)}

⇔ Pr(Q|P) ≤ [1 − Pr(P ∩ ¬Q)]/[1 + 0]

⇔ Pr(Q|P) ≤ [1 − Pr(P ∩ ¬Q)]/1

⇔ Pr(Q|P) ≤ 1 − Pr(P ∩ ¬Q)

⇔ Pr(Q|P) ≤ Pr(¬(P ∩ ¬Q))

⇔ Pr(¬(P ∩ ¬Q)) ≥ Pr(Q|P)

This means that Pr(¬(P ∩ ¬Q)) must be at least as great as Pr(Q|P), which means Pr(Q|P) being high entails Pr(¬(P ∩ ¬Q)) being high. This in turn means “Given P, then probably Q” entails “Probably, if P then Q.” Thus, “Given A, probably ¬M” entails “Probably, If A then ¬M.”

In response one could say that premise (1) as a material conditional doesn’t match the ordinary language if-then statement. This is true; in addition to being a true material conditional, the natural language if-then statement must have the consequent follow from the antecedent in some relevant way. I think this holds though; notice that God (as commonly conceived) is morally good independently of whether we think he is (e.g. God was good prior to humans existing) which would entail objective moral values (confer how I defined moral objectivism in my

opening statement) and thus entails objective morality. Replace God with atheism however, and we get a worldview that suggests objective morality doesn’t exist for reasons I explained in my opening statement.

Even if premise (1) of the moral argument weren’t a true if-then statement of ordinary language, the fact that premise (1) is a justifiably true material conditional is enough to intellectually trouble the atheist moral objectivist, since (1) and (2) together entail “God exists” even when (1) is “merely” a justifiably true material conditional.

**Aleph’s proof**What about Aleph’s proof? If I were to make the claim that “If P, then probably Q” entails “Probably, if P then Q” I’d be using “If P, then probably Q” to mean “Given P, probably Q” which means “P(Q|P) is high.” Aleph attempts a counterexample in which “If P, then probably Q” is true when “Probably, if P then Q” is not. Paradoxically, in Aleph’s counterexample “If P, then probably Q” is true (on his interpretation) when “Given P, probably Q” is

*false* since P(Q|P) = 0 due to Q being a contradiction. The paradox thus results from Aleph interpreting “If P, then probably Q” differently from how I’d be using the phrase.

**Tu quoque?**Aleph suggests my arguments against moral objectivism in my

opening statement apply for theism, not just atheism. Does it? Not for a God like the Christian God. As noted earlier, God as commonly conceived (e.g. in Christianity) entails objective moral values. The Christian God gives commandments (e.g. thou shalt not steal) we ought to obey even if we disagree with them, and this implies objective moral duties. Thus the argument from queerness wouldn’t apply if we knew this God exists (objective morality isn’t at all queer given the existence of this God). On atheism however, objective morality is suspiciously queer for reasons I described in my opening statement.

What about the attack on moral knowledge? The Christian God makes moral knowledge likely (confer Romans 2:14-15). Notably, these theological beliefs are hardly unique to Christianity; even if I were a deist I’d still believe God is objectively morally good and I’d believe that God designed us (via evolution) in such a way that when our cognitive faculties are functioning properly we intuitively apprehend elementary moral truths. On atheism however, it is (obviously) unlikely that there was a supernatural omniscient intelligence superintending evolution. Given atheism, it is far more likely that we don’t have rational grounds for believing in morality, for reasons I explained in my opening statement.

**Conclusion**By and large, Aleph Null did little to attack my opening statement (all my reasons for thinking “Given A, then probably ¬M” went unaddressed; note also my questions at the end of my

opening statement went unanswered), and his attack on my “General Approach” is unsuccessful.