Compared to Part II, I am going to make the premises even more favorable to the Theist and show his desired conclusion still does not follow. To do this I will prove two interesting probability results - called PR1 and PR2. See later.

Premise - The probability or chance of a Life Permitting Universe given supernatural, is greater than the probability of a Life Permitting Universe given solely nature (no supernatural, no god(s)).

Conclusion: Given our life permitting universe, the probability of “only nature” is less than the probability of supernatural. We should invoke the existence of a god fine tuner to explain why our Universe is such that it permits life.

The Premise seems very difficult to support and I wholly reject it. For example determining the probability or chance of a Life Permitting Universe, given supernatural seems impossible. But but for the sake of argument I will accept the Premise. Nevertheless, we will see the argument fails.

First A Short Digression On Probability, Plus PR1 and PR2

Assume no divisions by zero.

For event A, we denote it’s probability as P(A)

For two events A and B, the Probability of A given B is denoted by P(A/B) and equals P(A and B)/P(B).

Also, P(B/A)= P(A and B)/P(A)

Notice that in general P(B/A) does not equal P(A/B).

Rather P(B/A)= P(A/B)*P(B)/P(A).

Thus if P(A/B) is very low, it does not follow P(B/A) is very low. If the ratio P(B)/P(A) is very high, then P(B/A) may not be low and in fact can be quite high.

Also, P(~B/A)= 1-P(B/A). The symbol ‘~’ means negation or not.

__Proving PR1__

PR1 is stated as follows

If P(A/~B) > P(A/B) then P(~B/A) > P(B/A)*P(~B)/P(B)

Proof: Multiply the left and right side of the inequality by P(~B) then use definitions of conditional probabilities as given above.

P(~B)* P(A/~B) > P(~B)* P(A/B)

P(~B and A) > P(~B)*P(A/B)*P(B)/P(B) , I multiplied by unity.

So P(~B/A) * P(A) > P(~B)*P(B/A)*P(A)/P(B) I used definition of conditional probability to restate P(A/B)*P(B) as P(B/A)*P(A)

Divide by P(A) to complete the proof.

__Deriving PR2__

PR2 Relates P(~B/A)/ P(B/A) to the value chosen for P(~B)/P(B).

From PR1, notice P(~B/A)/ P(B/A) > P(~B)/P(B). Let X = P(~B)/P(B). If X is greater than or equal to unity we know P(~B/A) > P(B/A) . If X is less than unity then we can not draw conclusions. For example if X is say .2, then P(~B/A) is greater than .17 and P(B/A) is less than 0.83. Or if say X = .8, then P(~B/A) > .44 and P(B/A) is less than .56. These results follow from P(~B/A) + P(B/A) = 1 and solving the inequality. If X is less than unity then P(B/A) < 1/(1+X) and P(~B/A) > X/(1+X)

Here is a table

X P(~B/A) P(B/A)

---- ------------ -----------

.01 >.01 <.99

.20 >.17 <.83

.50 >.33 <.67

.80 >.44 <.56

.99 >.4975 <.5025

If X is less than unity, we can not determine if P(~B/A)> P(B/A) or if P(~B/A) < P(B/A). We only have lower bounds and upper bounds for P(~B/A), (and also for P(B/A)), and their ranges overlap. We are stuck.

Clarification of the Fine Tuning Argument in terms of Probability

With the basic probability theory understood, we can state the Premise as follows

Let A be the event Life Permitting Universe

Let B be the event Only Nature

Then the Premise is P(Life Permitting Universe/Supernatural) > P(Life Permitting Universe/Only Nature). The Premise seems very difficult to support and I wholly reject it. For example determining the probability or chance of a Life Permitting Universe, given supernatural seems impossible. But but for the sake of argument I will accept the Premise. Nevertheless, we will see the argument fails.

The conclusion can be stated as P(Supernatural /Life Permitting Universe) > P(Nature Only /Life Permitting Universe). That is the conclusion a religious person would like to reach.

Hopefully, I got the math right and now we can begin to apply it.

Let A be the event Life Permitting Universe

Let B be the event Only Nature

Using the premise and PR1 we have

P(Supernatural/Life Permitting Universe)> P( Only Nature/Life Permitting Universe)*P(Supernatural)/P(Only nature)

Now use PR2. If we assume P(Supernatural)/P(Only nature) > 1 i.e X >1, then the Fine Tuning Argument would work. But if we made that assumption there would be no need to bother with the Fine Tune argument at all, since the assumption would state P(Supernatural) > P(Only nature). {ETA - What I mean is Theologians would have no need for the Fine Tune argument if it was assumed P(Supernatural) > P(Only nature). However, Theologians could attempt to use a Fine Tuning argument to increase the odds of Supernatural vs (Only nature).}

Furthermore, I would reject P(Supernatural) > P(Only nature) for lack of support. If we assume X=1, that would give equal probability for 'Supernatural' and 'Only Nature' and then the Fine Tuning argument would work. However I do not think it is reasonable to give equal probability for supernatural and 'Only Nature' since we have plenty of observations of 'Only Nature' compared to Supernatural. It seems more reasonable to choose X<1, (arguably much less than unity), and if X is less than unity the fine tuning argument is inconclusive as shown by PR2.