Several commentators have attempted to reduce Hume's maxim about miracles to a formula in the language of probability theory. This paper examines two such attempts, one of which is based on the probability of the alleged miracle conditioned by the testimony for it, and the other on its unconditional probability. The conditional probability leads to a formula that is valid—though only when qualified—but not useful, while the unconditional probability results in an invalid formula. The utility of expressing Hume's maxim in terms of probability theory is shown to be questionable, and an alternative approach is presented.

A paper written over twenty years proves a mathematical theorem purporting to show that a supernaturalistic explanation for the universe is not supported by the anthropic principle, the notion that the observed properties of the universe must be compatible with its observers since otherwise the observers couldn't exist. Although this theorem is undoubtedly correct, it is not a very useful argument against the fine-tuning argument, whose standard premise is that fine-tuning is extremely improbable if naturalism is true. In the present paper Stephen Nygaard explains this mathematical theorem, presents some criticisms of it, and examines some counterarguments to the fine-tuning argument in which this theorem does not play a significant role. Nygaard shows that other aspects of probability theory, particularly the odds form of Bayes' theorem, are much more useful in uncovering the shortcomings of the fine-tuning argument. In particular, the fine-tuning argument ultimately fails because theism is not an explanation of apparent fine-tuning at all in any practical sense, so even if naturalism were unable to explain apparent fine-tuning, theism would not be a better alternative.

William Lane Craig's kalam cosmological argument maintains that the universe had a beginning. One of his arguments for this premise aims to show that a beginningless universe is metaphysically impossible, either because an actual infinite cannot exist because it would result in counterintuitive absurdities, or because time consists of a temporal series of events formed by successive addition, and that it's not possible for any such series to be an actual infinite. In the first of two previous papers, Arnold T. Guminski presents his solution to the problem of counterintuitive absurdities, which he believes results from applying Cantorian theory to the real world. However, his alternative version of the application of Cantorian theory to the real world attempts to achieve by a priori methods what can only be accomplished a posteriori, raises the question of whether a set theory can be fully developed that is consistent with it, and addresses "counterintuitive absurdities" that are not absurdities at all. In his second paper, Guminski correctly argues that it's possible for time to have no beginning by showing that the totality of all time need not be formed by successive addition, but this argument succeeds independently of his alternative version of the application of Cantorian theory to the real world, rendering it unnecessary.