The question addressed in this essay is: Does an analysis based on Bayes’ theorem nudge an agnostic toward belief? It is the second of two essays paired with essays by Bob Kurland. The first pair, published on August 25th, was a general consideration of probability. This second pair is on the probabilities of the anthropic principle of a fine-tuned universe and the existence of God. The main contrast in our essays is that of perspective.

#### Bayes’ Theorem

Bayes’ theorem refers to a data set, which, in its simplest formulation, is one divided into two subsets by the presence or absence of some property, Y, while being independently divided into two other subsets by the presence or absence of some other property, X. This results in four subsets. Each subset may be designated by the presence of X or its absence, NX, as well as the presence of Y or its absence, NY. The four subsets would then be (X,Y), (X,NY), (NX,Y), and (NX,NY). Each element of the population or data set is ID’d with any one of the four subset tags listed. The mathematics reduces the elements of the population, whatever they may be materially or intellectually, to their ID tags.

Bayes’ Theorem is often expressed as the probability of X given (i.e. restricted to the subset) Y. Symbolically, this is P(X/Y). It equals a likelihood factor times the probability of X for the set, which symbolically is P(X). The likelihood factor is term 2, as indicated by the brackets in Eq. 1.

P(X/Y) = [P(Y/X) / P(Y)] × P(X) Eq. 1

Term 1 = Term 2 × Term 3

This is the form used by Lataster and myself. Typically P(X) is said to be the prior probability, It is the generic or overall probability of X for the entire set. P(X/Y) is said to be the probability of X posterior to restriction to the specific subset, Y.

Bayes’ theorem may be expressed as:

P(X/Y) / P(NX/Y) = [P(X) / P(NX)] × [P(Y/X) / P(Y/NX)] Eq. 2

Term 1 = Term 2 × Term 3

This is the form used in an article by Bob Kurland on the anthropic principle of fine-tuning for life on earth. Both Eq. 1 and Eq. 2 are fully compatible with one another and represent a Bayesian population or data set divided into four subsets by two independent criteria X and Y as present or absent. Term 1 of Eq. 2 is the ratio of the probability of X, given Y, to the probability of NX, given Y. Term 2 is the ratio of the probability of X for the entire set to the probability of NX for the entire set. Term 3 is the ratio of the probability of Y given X to the probability of Y given NX. In this context, ‘given’ means ‘restricted to the subset’.

#### Belief in ‘God exists’ or No and Independently, Belief in ‘The Universe is Fine-tuned’ or No

In Eq. 2, let Y be replaced by F as the personal judgment, “the universe is fine-tuned” and X be replaced by G as the personal judgment, “God exists”. The result is:

P(G/F) / P(NG/F) = [P(G) / P(NG)] × [P(F/G) / P(F/NG)] Eq. 3

Term 1 = Term 2 × Term 3

The perfect agnostic would be one who was agnostic with respect to the probability of fine tuning and non-fine tuning, as well as with respect to the probability of God and non-God. For the perfect agnostic, each of the four subsets of the Bayesian population expressed as a fraction of the whole population, would be one-quarter of the total or 0.25. The three terms of Eq. 3 would each equal one.

Term 1 equals (.25/.5) / (.25/.5) = 1; Term 2 equals .5/.5 = 1; Term 3 = (.25/.5) / (.25/.5) = 1

#### The Bayesian Nudge Toward Belief

The referenced article indicates that an agnostic should be nudged toward belief in God because, for an agnostic, calculated term 1 should be greater than one. The article proposes that the calculated value of term 1 would be greater than one, because

“. . . certainly term 3 is a number much greater than 1, even if the exact value is indeterminate.”

and because it is implied that for an agnostic term 2 is numerically one. Consequently, term 1 will be greater than one.

“If you’re agnostic‒it’s a 50/50 proposition that God exists‒then certainly fine tuning should convince you that God exists.”

Implicitly, the rationale for considering term 3 to be greater than one is that an intelligible effect, namely fine tuning, is better explained by the existence of an intelligent agent, God, than by the naked probability of materialism/naturalism.

#### Possible Insufficiency of the Nudge

The increase in term 1, P(G/F) / P(NG/F), to a value greater than one is insufficient in itself to nudge an agnostic toward belief, if term 2, P(G) / P(NG), remains one. If it remains one, the increase in the probability of God to the probability of non-God, given fine-tuning could be offset by an increase in P(NG/NF) / P(G/NF), which is the probability of non-God to the probability of God, given non-fine tuning. A net retreat from agnosticism and a net movement toward belief would require that the increase in belief given fine-tuning be at the expense of non-belief. An increase in belief given fine-tuning is not sufficient in itself to nudge an agnostic toward belief.

The nudge toward belief in God initiated by an increase in term 3 of Eq. 3, would be mathematically insufficient within the context of Bayes’ theorem. However, its philosophical underpinning stated above could be sufficient. The effectiveness of the argument would then be an adaptation of the Socratic method, which elicits, by an indirect route, an explicit conclusion from a person (in this case, an agnostic) who was previously unaware of his implicit conviction of that conclusion (in this case, the existence of God as the intelligent agent of the intelligible effect of fine tuning).

#### Conclusion

The argument of a nudge toward belief, based on Eq. 3 alone, surely appears to be persuasive, but it is insufficient mathematically. The nudge toward belief may be offset by a nudge toward non-belief. The agnostic could be nudged toward non-belief as well, unless the increase in the probability of God, given fine-tuning, is accompanied by an increase in the overall probability of God.

‘Given that the universe is fine-tuned’ within the context of a Bayesian data set, does not mean ‘taking it for granted that the material universe in which we live is fine-tuned’. Thus, one could not maintain the nudge is only toward belief, given the material fact that the universe is fine-tuned. The Bayesian mathematics requires fine-tuning, F, and non-fine tuning, NF, to be hypothetical so that neither one is objectively factual to the exclusion of the other. The Bayesian population, by premise, is divisible into two subsets, F and NF. One of these may be a null set without denying the Bayesian nature of the population. This would render the data set a trivial, but not an invalid case of a Bayesian distributed population. However, to deny that either subset NF or subset F, cannot be hypothetical because the other is a fact, is to deny that the population is Bayesian. It would be denying the mathematical validity of Eq. 3 while basing an argument and its conclusion upon its mathematical validity.

The meaning of given in the context of a Bayesian data set is in accord with a principle proposed in the earlier essay on mathematical probability, “The IDs of elements, subsets, and sets, vis a vis probability, are purely nominal.”

Although mathematically insufficient, this argument from fine-tuning could be used as a Socratic vehicle to elicit an explicitly philosophical conclusion of the existence of God, based on the judgment that an intelligible effect (fine tuning) requires an intelligent agent.