How to Construct a Cumulative Case Argument

by Max Andrews

The cumulative case uses the prime principle of confirmation: Whenever we are considering two competing hypotheses, an observation counts as evidence in favor of the hypothesis under which the observation has the highest probability. This principle is sound under all interpretations of probability.  Each argument must be taken on its own grounds and one cannot arrive at “God” at the end of each argument.  The conjunction of arguments is what is needed to make a cumulative case for the existence of God.

The Likelihood Principle of Confirmation theory states as follows.  Let h1 and h2 be two be competing hypothesis (in this case the existence of X and ~X, with X being a first cause, fine-tuner, etc.).  According to the Likelihood Principle, an observation e counts as evidence in favor of hypothesis h1 over h2 if the observation is more probable under h1 than h2.  Thus, e counts in favor of h1 over h2 if P(e|h1) > P(e|h2), where P(e|h1) and P(e|h2) depict a conditional probability of e on h1 and h2, respectively.  The degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which e is more probable under h1 than h2: particularly, it is proportional to P(e|h1)/P(e|h2) .  The Likelihood Principle seems to be sound under all interpretations of probability.  This form is concerned with epistemic probability.

The Likelihood Principle can be derived from the so-called odds form of Bayes’ Theorem, which also allows one to give a precise statement to the degree to which evidence counts in favor of one hypothesis over another.  The odds form of Bayes’ Theorem is P(h1|e)/P(h2|e) = [P(h1)/P(h2)] x [P(e|h1)/P(e|h2)].  The Likelihood Principle, however, does not require the applicability or truth of Bayes’ Theorem and can be given independent justification by appeal to our normal epistemic practices.[1]

So, what about the problem of dwindling probabilities in a cumulative case?  When combining probabilities the end product is smaller (.5 multiplied by .5 = .25).  This issue concerns the restricted conjunction rule for probability: P (A and B) = P (A) x P (B) (when A and B are independent).  It also appears in the general conjunction rule of probability: P (A and B) = P (A) x P (B given A).  When using a probability calculus the only time you would add probabilities is in disjunctive calculus (.25+ .25 = .5).  This occurs in the restricted disjunction rule for probability and the general disjunction rule for probability, respectively: P (A or B) = P (A) + P (B) (when A and B are mutually exclusive), and P (A or B) = P (A)+ P (B) – P (A and B).  Now, in order to avoid the problem of dwindling probabilities the conjunction of arguments must be used as one probability calculus.  Even if these arguments weren’t used in the cumulative case form the converse probabilities would make the probability for the non-existence of God congruently smaller.

[1] Robin Collins, “The Teleological Argument,” in The Blackwell Companion to Natural Theology Eds. William Lane Craig and J.P. Moreland (Oxford, UK: Blackwell, 2009), 205.

2 Responses to “How to Construct a Cumulative Case Argument”

  1. right, I’ll leave this deeper apologetics stuff to you guys…….

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