## Friday, June 24, 2022

### Conjunction and Conditional Probability

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### Intro

In this article I will prove that the probability of “Given A, probably C” is greater than or equal to the probability of “A and C are both true.” This has implications for the moral argument for the existence of God, for reasons I’ll explain next.

### Background

Behold the following moral argument:
1. If God does not exist, then objective morality does not exist.
2. Objective morality does exist.
3. Therefore, God exists.
In a previous article I argued that “Given A, probably C” (e.g., “Given God’s nonexistence, objective morality does not exist”) entails “Probably, if A then C” (e.g., “If God does not exist, then objective morality does not exist”). A lot of atheists who disagree with the argument concede that “Given God’s nonexistence, objective morality does not exist” are thus rationally committed to believing the first premise of the moral argument.

However, some atheists who believe objective morality does not exist say they don’t agree with the first premise despite rationally committed to doing so. In this article I argue that atheists who believe that objective morality does not exist are rationally committed to believing Given God’s nonexistence, objective morality probably doesn’t exist which in turn implies If God does not exist, then objective morality does not exist is probably true. More precisely, I will use the power of mathematics to show that anyone who believes God does not exist and objective morality does not exist is rationally committed to believing Given God’s nonexistence, it is unlikely that objective morality’s existence.

### The Proof

For brevity’s sake, I’ll use a bit of symbolic logic, where ∧ represents “and” and ¬ represents “not” as part of my abbreviations:

 G = God exists. ¬G = God does not exist. M = Objective morality exists. ¬M = Objective morality does not exist. ¬G ∧ ¬M = God does not exist and objective morality does not exist. P(¬G ∧ ¬M) = The probability that ¬G and ¬M are both true, i.e., the probability that God does not exist and objective morality does not exist is true. P(¬M|¬G) = The probability of ¬M given ¬G, i.e. the probability of objective morality not existing given God’s nonexistence.

One assumption I will make is the person who believes that ¬G ∧ ¬M is true also believes that ¬G ∧ ¬M is at least probably true. Generally speaking, to be rationally consistent, if you are to believe that some proposition P is true, you should also believe that P is at least probably true.

To generalize, I’ll use two propositions A and C. The goal is to prove this is true:
P(C|A) ≥ P(A ∧ C )
To start with, note the following equation familiar to many high school graduates and intelligent middle school students:
P(A ∧ C) = P(C|A) × P(A)
What is the highest P(A ∧ C) can be if we hold P(C|A) constant? Well, we would want P(A) to be as high as it can be, which is 1. Thus, in finding an upper limit for P(A ∧ C), this inequality is true:
P(A ∧ C) ≤ P(C|A) × 1
⇔ P(A ∧ C) ≤ P(C|A)
After finding this upper limit for P(A ∧ C), there’s just one more step:
P(A ∧ C) ≤ P(C|A)
⇔ P(C|A) ≥ P(A ∧ C)
Since A and C were of course arbitrary placeholders, we can use all sorts of propositions, including ¬G and ¬M. So for example this is true:
P(¬M|¬G) ≥ P(¬G ∧ ¬M)

### Conclusion

Thus anyone who believes God does not exist and objective morality does not exist is rationally committed to believing Given God’s nonexistence, it is unlikely that objective morality’s existence, since P(¬M|¬G) is going to be greater than or equal to the probability of ¬G ∧ ¬M.

Since God does not exist and objective morality does not exist being probably true entails Given God’s nonexistence, it is unlikely that objective morality’s existence, which in turn entails that If God does not exist, then objective morality does not exist, one who believes God does not exist and objective morality does not exist should also believe that the first premise of the moral argument (If God does not exist, then objective morality does not exist) is at least probably true.