Help Wanted – Part 3

A key premise in Swinburne’s (deductive) argument in defense of his inductive version of the Cosmological argument (TCA) goes like this:


(TCA9) The probability that there will be a complex physical universe given that God does not exist is low. (EOG, p.151)

Based on Swinburne’s explanation of his reasoning in support of this premise (in email dated 10/24/11), I understand his argument for (TCA9) to be (roughly) as follows:

1. P(e&~h&~c&k;) is approximately equal to P(eI~h&~c&k;)

2. P(eI~h&~c&k;) is approximately equal to P(eI~h&k;)
Therefore:

3. P(e&~h&~c&k;) is approximately equal to P(eI~h&k;)
4. P(e&~h&~c&k;) is a very low probability.

Therefore:

5. P(eI~h&k;) is a low probability (at most).

I take it that (5) is equivalent to (TCA9). So, a good argument for (5) would be a good argument for (TCA9).

Swinburne is very knowledgeable about conditional probability and Bayes’s Theorem, so his reasoning is a bit condensed. It took me a few days to work out the detailed steps of logic and math required to get to premise (1) from his assumptions. Having figured this out, I can walk through Swinburne’s reasoning, showing the step-by-step logical and mathematical details that validate this reasoning. Fortunately, the level of math and logic involved is pretty basic and easy, at least once it has been layed out.

[Swinburne:] Let c be ‘there is a personal creator other than God’.

This first sentence from Swinburne’s email needs no explanation. I will just point out that c in conjunction with h ( meaning ‘God exists’) are supposed to exhaust the logical possibilities concerning a personal explanation of the existence of a complex physical universe.

[Swinburne:] Then (given the sentence on p.149, ‘e could not, as we have seen..’), with k as a mere tautology, P(e&~h&~c&k;) will be the probability that a complex physical universe exists without an explanation.
I explained this in the previous post (Part 2), but will repeat the explanation here. Swinburne has previously concluded that there can be no scientific explanation for the existence of a complex physical universe, so assuming that the only other kind of explanation that can be given is a personal explanation (in terms of a creator, or group of creators, who has some purpose or purposes for making a complex physical universe) the denial of the existence of God (~h) combined with the denial of any other personal creator (~c) eliminates the possibility of a personal explanation, and thus there would be no explanation for the existence of a complex physical universe.

[Swinburne:] By the calculus this equals P(eI~h&~c&k;) P(~h&~c&k;).

The pronoun ‘this’ here refers to the probability statement that, in Swinburne’s view, relates to the possibility that a complex physical universe exists without an explanation: P(e&~h&~c&k;). So the first bit of reasoning (‘by the calculus’) gets us to this equation:

P(e&~h&~c&k;) = P(eI~h&~c&k;) P(~h&~c&k;)

1. P(AIB) = P(A&B;)/P(B)……………………………..Conditional Probability Formula*
2. P(AIB) x P(B) = [P(A&B;)/P(B)] x P(B)………….1, multiply both sides by P(B)
3. c/d x d = c……………………………………………………………..See proof below
4. [P(A&B;)/P(B)] x P(B) = P(A&B;)……………………….3, instance of the formula
5. P(AIB) x P(B) = P(A&B;)…………………………………….2,4 transitivity of equality
6. P(A&B;) = P(AIB) x P(B)………………………………………5, symmetry of equality
7. P(e&~h&~c&k;) = P(eI~h&~c&k;) x P(~h&~c&k;)…..6, instance of the formula


* requires that P(B) not be equal to zero (to avoid division by zero, which is undefined).

Prove: c/d x d = c

1. c/d = c x 1/d…………………………….multiplication by reciprocal
2. c/d x d = (c x 1/d) x d…………………1, multiply both sides by d
3. (c x 1/d) x d = c x (1/d x d)……..associative prop. of multiplication
4. c/d x d = c x (1/d x d)………………2,3 transitivity of equality
5. 1/d x d = d x 1/d………………..commutative prop. of multiplication
6. d x 1/d = 1…………………………………..axiom of multiplicative inverse
7. 1/d x d = 1……………………………….5,6 transitivity of equality
8. c/d x d = c x 1……………………………7,4 substitution of equals
9. c x 1 = c………………………………….identity element for multiplication
10. c/d x d = c…………………………..8,9 transitivity of equality

symmetry of equality: If a = b, then b = a.
transitivity of equality: If a = b and b = c, then a = c
multiplication property of equality: If a = b, then a x c = b x c
associative property of multiplication: (a x b) x c = a x (b x c)
commutative property of multiplication: a x b = b x a
identity element for multiplication: a x 1 = a
axiom of multiplicative inverse: a x 1/a = 1

To be continued…

About Bradley Bowen
  • http://www.blogger.com/profile/14479224236264150172 Ben

    His background knowledge of all metaphysical reality we know nothing of is zero. You'd have to have insider information apart from this universe on all that exists and humans are in no position to assess that. His equation will be crap as a result, so you don't really have to sort out all the details there. And of course, he doesn't apply the same issue to his complex and arbitrary notion of a god with all sorts of obnoxious anthropomorphic and impossible characteristics, and magical powers.

  • http://www.blogger.com/profile/05211466026535549638 Bradley Bowen

    Ben said…

    His background knowledge of all metaphysical reality we know nothing of is zero. You'd have to have insider information apart from this universe on all that exists and humans are in no position to assess that.
    ===========
    Response:

    I don't understand your objection. Can you say a bit more to clarify? I would like to understand your objection.

    When you say "His background knowlege…is zero." are you implying that we have no relevant a priori background knowledge? If so, then your claim seems false to me.

    We know (or have a good idea about) the meanings of the words involved: God, universe, etc. And to the extent that there are ambiguities or unclarity in these terms, Swinburne has provided analysis and clarification of the key words, so unless you want to object to some aspect of his analysis of the key terms, those are bits of a priori 'background knowledge' that are relevant here.

  • http://www.blogger.com/profile/06221062723275477038 harun

    Thanks for rules and tricks of multiplication,multiplication is a part of maths and I want to discuss associative property of multiplication as-he Associative property of multiplication say that it doesn't matter how you group the numbers,maening will we same.example-(a × b) × c = a × (b × c)
    associative property of multiplication worksheets


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