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Something That Explains Everything… Explains Nothing

22 Sep
So let’s say that someone makes the argument “The existence of X is proof that god exists”. This is all right and good to do, but using god as an explanation in these cases usually follows this line of reasoning:
For example, Seth [of The Thinking Atheist] talks about a hypothetical American teenage boy who is the victim of a shooting. After being shot, the boy is rushed to the hospital, where one of several possible scenarios plays out.

  1. Imagine first that the boy makes a full recovery – the bullets missed all of his vital organs, thank God. He’ll need time to heal, but he won’t suffer any permanent damage. “It’s a miracle,” the religious will say. God is good.
  2. Second, imagine instead that one of the bullets had hit the boy’s spinal cord, leaving him paralyzed. He’ll have to live the rest of his life in a wheelchair… but he’s alive! “God must have had more work for him to do on this earth. Praise the Lord he’s still with us.” God is good.
  3. The worst-case scenario is that the boy dies. His wounds were too severe; the doctors couldn’t save him. “God must have been done with him here on this earth. He’s in a better place now, with no violence and no pain. He’s been called home.” God is good.

In each case down the line, the requirements are relaxed for what state of affairs would lead to the conclusion that God is good. By the time you get to the third scenario, you’re confronted with the fact that an innocent boy is dead, and God still gets credit for being good. At this point you have to admit that the statement has no requirements on its being true at all.

 
Tim’s critique of this type of reasoning is legitimate and cogent. If you admit that the survival of the boy is evidence for the goodness of the Christian god, then you have to admit that the death of the boy would be evidence against the goodness of the Christian god.
 
Though, to make this argument stronger we might be able to express why it fails in a mathematical fashion; why the post-hoc rationalization of many Christians is the equivalent of dividing by zero.
  
In probability theory there is something called Independence. If P(H|E) = P(H), or if P(H|~E) = P(H), then E and H are independent. So in the above scenario, we have three lines of evidence: The boy lives, the boy is permanently paralyzed, or the boy dies; So let’s say that E is the boy lives and ~E is that the boy is permanently paralyzed or dies. 
 
Remember that absence of evidence is evidence of absence? That same formula applies here as well, and is a logical inference from the formula to prove independence above. However, following the logic of the Christian above, we have P(H|E) > P(H) and P(H|~E) > P(H). That is, all three scenarios “prove” the goodness of the Christian god… even though all three scenarios are mutually independent. The boy cannot live free of injury, be paralyzed for life, and die all at the same time.
 
Now we have that equation for absence of evidence being evidence of absence, and we can arrange it in syllogistic format:
 
P1: P(H|E) > P(H)
P2: P(H|~E) < P(H)
C: P(H) = P(H) – [P(H|E) – P(H) + P(H|~E) – P(H)]
 
So in order to logically arrive at that conclusion, we have to have some negative numbers in there. If the difference between P(H|E) and P(H) is positive, then its opposite P(H|~E) and P(H) has to be negative. So if P(H) is 50%, and P(H|E) is 55%, then the difference is 5%; there has to be a comparable difference between P(H|~E) and P(H).
 
Thus we would have 50% [P(H)] = P(H) + [P(H|E) – P(H) + P(H|~E) – P(H)]
 
50% = 50% + [55% – 50% + ??? – 50%]
50% = 50% + 5% – 50% + ???
50% = 5% + ???
45% = ???
 
Basically the terms inside of the brackets [] should cancel each other out (i.e. add up to zero) so that we end up with P(H) = P(H). But what if we attempt to follow the logic of the original Christian, that both E and ~E are evidence for the goodness of the Christian god? We end up with the following scenario:
 
50% = 50% + [55% – 50% + 55% – 50%]
50% = 50% + 5% + 5%
50% = 60%
 
When was the other time you saw it argued that something like 1 = 2? That’s right: when you divide by zero.
 
In order to not end up trying to argue that 50% is equal to 60% you have to admit some evidence or outcome that argues against the goodness of the Christian god in this scenario. If you admit that E is evidence of H, then ~E has to be evidence against H. That, or you have to admit that goodness of the Christian god has no affect on whether the boy lives or dies; that P(H|E) – P(H) is equal to P(H|~E) – P(H). The only way that this can happen is if P(H|E) = P(H).
 
The Christian in the original scenario believes in the goodness of the Christian god no matter what happens to the boy. This means that the goodness of the Christian god has no relationship to whether the boy lives, is permanently injured, or dies. Just like Mars being the fourth planet from the sun has no relationship to whether the boy lives, is permanently injured, or dies. Thus, the goodness of the Christian god, just like Mars being the fourth planet from the sun, cannot be used as an explanation of why the boy lives, is permanently injured, or dies.
 
So if something is used to explain every outcome, if P(H|E) = P(H) or P(H|~E) = P(H), then it actually explains nothing.
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Posted by on September 22, 2011 in Bayes

 

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