Our Empirical Basis has no Empirical Basis

I read Graham Priest's book "Logic: A Very Short Introduction" ( http://www.oup.com/uk/catalogue/?ci=9780192893208 ). Very enjoyable it was too (and quite droll that a man called "Priest" uses refutations of arguments for the existence of God as his main examples!).


If the Argument from Design is hogwash.....

One argument got me thinking. This was Graham's take on the "Argument from Design". The "Argument from Design" takes the observation of the order ("O") in the universe as confirmation (to varying extents) of the existence of God ("G"). Using "P(A¦B)" as "The propbability of A given B" the claim that O confirms G is:

(1) P(G¦O) > P(¬G¦O) (the probability of God given the existence of Order is greater than the probability of not-God given the existence of order).

(2) This, Graham asserts, is equivalent to P(G) > P(¬G). The probability of God before any evidence is greater than the probablity of not-God before any evidence. If God is, prior to any evidence, more likely to exist than not exist then any evidence adds confirmation to his existence. If God is, prior to any evidence, more likely to not-exist then any evidence adss confirmation to his non-existence. Graham then disputes that, prior to any evidence, God is more likely to exist than not.

(a more detailed working is in M. Colyvan, J. Garfield and G. Priest, 'Problems with the Argument for Fine Tuning', Synthese 145 (2005), 325-38)

It got me thinking as to whether Graham's argument applied to other hypotheses, such as "I see a pint of beer" or "I see X" and, where X is "this is my empirical basis", whether the whole empirical basis disappears.


Lets Stick Some Numbers In

The probability of hypothesis (H) after some evidence (E), P(H¦E), is given by Bayes' Theorem:

(3) P(H¦E) = [P(E¦H) x P(H)]/P(E)

(The probability of a hypothesis given the evidence is

• the probability of the evidence given the hypothesis times the probability of the hypothesis ignoring the evidence (ie the probability that you are seeing the evidence because of the hypothesis)
• divided by the probabiliy that we would see the evidence anyway)

(The best explanation Bayes' Theorem on the web is Eliezer Yudkowsky's interactive explanation on http://yudkowsky.net/bayes/bayes.html)


Substituting

• G and ¬G for H (our hypothese are "God" and "Not-God"), and
• O for E (our evidence is the order in the universe

We get

• Probability of God given Order: [P(O¦G)xP(G)]/P(O)
• Probability of not-God given Order: [P(O¦¬G)xP(¬G)]/P(O)

Now can we add numbers to these? We can, at least have a go.

(4) P(G) and P(¬G) = 0.5. Before any evidence, before any consideration we should be completely indifferent between these two hypotheses. Thus the prior probabilites are both 0.5

(5) P(O¦G) = 1. Our conception of God is such that He would create order. So the probability of order, given the existence of God, is 1.

(6) P(O¦¬G) = 0.5 If there is no God, what are the chances of there being order? We have no ****ing idea again: so it is 0.5.

(7) P(O) = 0.75. The probability of order (given no evidence) is the probability of order, given God plus the probability of order given no God:

P(O) = [P(O¦G) x P(G)] + [P(O¦¬G) x P(¬G)] = (1 x 0.5) + (0.5 x 0.5) = 0.75

Substituting these numbers into our formulae:

• Probability of God given Order: (1x0.5)/0.75 = 2/3
• Probability of not-God given Order: (0.5x0.5)/0.75=1/3

It's considerably more likely that God exists given order than that God does not exist.

Ah, but....

What about a self-ordering non-divine universe (S)? On the same basis as steps (4) to (7) the probability of a self-ordering non-divine universe given that we have order is 2/3.

Or add something to the concept of God: "God is an Englishman". This can be translated as "there exists a God who is English" (say "(G&E)" and call this "ѱ"). ѱ and ¬ѱ will have the same relationship as G and ¬G; and S and ¬S. The probability of ѱ given there is order in the universe is 2/3

Or add something completely ridiculous to the concept of God: "God ordered the universe and koppites are not gobshites" ("ϕ").........

The posterior probabilities, where we have absolutely no idea before we gather evidence, are dependent upon the way we formulate the hypothesis. Careful formulation of any possible hypothesis will always make it more likely than not given the evidence. So what is determining our hypotheses appears to be something prior to evidence.

Now there may well be objections to the way that I have formulated the prior pobabilities of God, order, self-order or Koppites being Gobshites. Many may well be able to come up with different prior probabilites that are equally acceptable. They will, however, be equally acceptable. They will not overrule mine, nor yours. They will not be fixed and not, in any way, be fixed by evidence.

And what about the evidence itself?

What we commonly call 'evidence' is, itself, a theory dependent on the really basic evidence of our experience. I may be mistaken about the pint of beer in front of me (the "experiential statment, "ES"), despite the fact that I have all the experiences compatible with a pint of beer being in front of me (the experiences, "E"). It remains possible that E is true whilst ES is not. Thus the relation between E and ES is not "E if and only if ES" (E ↔ ES) but "E only if ES" (ES → E). Experience is inconclusive evidence for experiential statements. From above, prior to any evidence statements, the likelihood of our conclusion ES depends upon non-evidential statements about the likelihood of and formulation of ES itself.