**Notation**

*F*) < 0.5.[3]

- The Evidential Argument from Scale: compares naturalism to theism conjoined with various auxiliary hypotheses about God’s potential reasons for giving or not giving humans a privileged position in the universe.
- The Evidential Argument from the Flourishing and Languishing of Sentient Beings: compares naturalism conjoined with Darwnism to theism conjoined with Darwinism
- The Evidential Argument from the Self-Centeredness and Limited Altruism of Human Beings: compares naturalism conjoined with Darwnism to theism conjoined with Darwinism
- The (Evidential) Fine-Tuning Argument: compares theism to naturalism conjoined with the multiverse hypothesis

In the explanatory argument schema described above, it may be possible to defeat premise (3) by showing that an auxiliary hypothesis is evidentially significant in the relevant way, i.e., that A sufficiently raises Pr(F | H1) or lowers Pr(F | H2). In order to assess the evidential significance of such hypotheses, we would need to apply the theorem of the probability calculus known as the theorem of total probability.

*individually*is antecedently more probable on

*H1*than on

*H2,*i.e., Pr(F1| H1) > Pr(F1 | H2), Pr(F2 | H1) > Pr(F2 | H2), and Pr(F3 | H1) > Pr(F3| H2). But how could one show that the items of evidence

*collectively*are antecedently more probable on

*H1*than on

*H2?*In other words, how could one give a about cumulative case argument?

*H*&

*B*) = Pr(F1 & F2 & F3 |

*H*&

*B*)= Pr(F1|

*H*&

*B*) x Pr(F2| F1 &

*H*&

*B*) x Pr(F3 | F1& F2 & H &

*B*)

*H1*&

*B*) = Pr(F1 & F2 & F3 |

*H1*&

*B*)= Pr(F1|

*H1*&

*B*) x Pr(F2| F1 &

*H1*&

*B*) x Pr(F3 | F1& F2 & H1 &

*B*)

*H2*&

*B*) = Pr(F1 & F2 & F3 |

*H2*&

*B*)= Pr(F1|

*H2*&

*B*) x Pr(F2| F1 &

*H2*&

*B*) x Pr(F3 | F1& F2 & H2 &

*B*)Another way to compare these two values is to compare the

*ratio*of

*H1′s*final probability to

*H2′s*final probability. The ratio of these two formulas gives us Bayes’s Theorem in its compound odds form:

*H1′s*final probability is greater than

*H2′s*final probability; if the ratio is less than 1, then

*vice versa;*if the ratio equals one, then

*H1*and

*H2*have equal final probabilities.This insight gives us one way to show that one hypothesis has a higher final probability than the other hypothesis. For example, suppose want to show that

*H2*has a higher final probability than

*H1.*In other words, we want to show that:

One way to show that *H2 *has a higher final probability than *H1 *is to show that the ratio of each multiplicand on the right-hand side of Bayes’s Theorem in its compound odds form is *also *less than 1, i.e.,

and

and

and

.

This enables us to define the following schema for cumulative case arguments.

*B*). [from formula x]

*B*) is close to 1.

*Choice & Chance: An Introduction to Inductive Logic*(4th ed., Belmont: Wadsworth, 2000), 23.

[3] I owe this schema to Paul Draper.

[5] Draper 1989.

[6] Draper 1989.

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