Craig (2016) asserts:

“........why can’t a person believe something based on 51% probability? The claim that he can’t seems to me just a matter of personal psychology.”

This is undoubtedly true, but is it a prudent prescription for belief across the board? Is there any correct threshold of probability that justifies and recommends theistic belief? I suggest not, whether we employ relativistic or quantitative approaches to probability.

Sincerely held beliefs are possible based on any level of subjective vs. objective probability, as plenty of psychological studies of the cognitive biases exhibited during gambling demonstrate (Griffiths, 1990). In terms of relative probabilities, a believer might argue that theism (defined here as the claim that a morally perfect, omnipotent, omniscient being who has created the universe necessarily exists; though, in effect, a blanket term for a large number of contradictory concepts and claims) holds inherently more probability of being true than naturalism (the hypothesis that the natural world is a closed system unaffected by the kind of being claimed by theism). But things are not so simple because theism is a highly specified version of supernaturalism (although theism entails supernaturalism, supernaturalism does not entail theism; Draper, 2009. Draper’s argument might appear to reflect an a-priori ideological bias toward naturalism so it’s only fair to note that he views supernaturalism and naturalism as holding equal prior probability) and so the theist further needs to consider why theism is more probably true than either some generic view of supernaturalism or a vague belief such as ietsism. Monotheism, of course, is a specific version of theism so even if they are able to successfully dispatch naturalism and generic supernaturalism, the theist has the task of providing evidence as to why other versions of theism, such as pantheism, panentheism or polytheism etc., are less probable than monotheism. If the believer is a mainstream Christian they would further have to debate why specific Christian theology (such as the triune claim) is more probable than other, non-triune monotheistic theology.

To identify which particular religious beliefs are considered as rational by adherents but not so by others, theist or atheist, often one has to look no further than the particular social-epistemic environment in which those beliefs are prevalent. Believers may initially form specific religious beliefs in rational ways. Although nth hand evidence emanating from ancient sources is notoriously unreliable (Schum, 1992) cognitively-immature children placing credence in what trusted adults tell them about the world is not in itself an irrational stance given the circumstances. However, those same children will often proceed to sustain those beliefs in later life in clearly irrational ways such as not accepting that epistemic probability (the subjective degree of belief in an event given at least some evidence) is often at variance with the actual probability of that same event given the entirety of available evidence (Skyrms, 2000).

Here are two mundane quantitative examples of what I’m getting at. In the first example, someone may sincerely yet erroneously believe that the die used in a game of chance has been purposely weighted to land on a ‘6’ at greater than chance levels, in which case their epistemic probability that a ‘6’ will result from the next throw would be somewhat greater than an otherwise rational expectation of approximately .17. Similarly, it is commonplace to find arguments for the existence of the classical monotheistic God that contain fatal flaws (Frances, 2015; 2016, Swinburne, 1991), yet they are widely perceived as being ‘good enough’ to sustain belief regardless of whether believers understand or are even aware of these flaws, i.e., they are ‘weighted’ and believers endow them with a higher degree of epistemic probability than they objectively deserve. Indeed, large numbers of theistic believers hold strongly to the opinion that the evidence surrounding some the issues contained in the premises (concerning e.g., biological evolution) has been and is being systematically tampered with.

In the second example, imagine a sack containing 1000 balls. 510 balls are black and 490 balls are white. If someone were to randomly, blindly, choose a single ball from the sack we can easily estimate the probability of the ball being black (p = .51). Would we then be justified in believing that on any single random pick the ball chosen would be black? Surely most people would think not, even though it would be entirely plausible and certainly meet Craig’s level of justification for belief. What then of events in which we justifiably have much higher levels of confidence in their probability? Throw a die once. The probability that it will land on ‘1’, ‘2’, ‘3’, or ‘4’ is .67. Would it be justified to stake, say, a month’s earnings on the warranted belief that the next throw of the die will plausibly result in your favour, i.e., it is 67% likely to be true? And how many people would not consider it justified but be prepared to change their belief should they receive further information that the previous two throws had been a ‘5’ and/or’ 6’? Undoubtedly, many would do so but their belief in the outcome of the next throw would definitely not be rational, as any casino owner would aver.

Preferring P-inductive arguments requiring quantitative measures, Swinburne (2011) has calculated the intrinsic, inherent (prior) probability of a generic monotheistic God’s existence, obtained from the hypothetical situation in which there is zero additional information to be factored in, to be as low as .01 (and polytheism even lower at .0005). There are good arguments made, however, that prior probabilities should not be used as criteria for acceptability. It is easy to imagine events with an extremely minuscule prior probability that are nevertheless manifestly true and events with a seemingly large prior probability that are manifestly false. Take the first case, for example; what are the chances of every single one of your thousands and thousands of ancestors having met, procreated and their billions and billions of sperm and many thousands of egg cells being subject to those specific random mutations that have eventually resulted in your personal genome?

Despite such a low prior probability, Swinburne devoutly believes that monotheism obtains because he subsequently factors in carefully selected empirically-derived facts that he considers relevant to his a-priori view that God exists. The systematic addition of each fact to the probability equation then has a cumulative effect, increasing the probability that God exists. This process continues until he achieves a probability of >50% at which point, and beyond, he is able to claim that he has calculated a more concrete epistemic probability, i.e., given what we know about the universe, the probability of God's existence becomes more plausible given that knowledge (Swinburne, 1991). (Achinstein, 2001) describes the process:

“An observation A is evidence for a hypothesis H if the probability that H is true given A (along with background information) is larger than the probability that H is true given background information alone.”

Here, Swinburne’s hypothesis (H) is God’s existence and he bolsters the probability of that hypothesis being true by each observation (A) that some particular state obtains. This technique can be inherently flawed, which can be demonstrated in the following way:

P1: H is the hypothesis that I will win the lottery next Saturday
P2: A is the observation that I hold a valid ticket in next Saturday’s lottery
C1: Therefore: the probability that I will win the lottery next Saturday (H) is increased by my holding a ticket (A);

Pr[HA] > Pr[H]

This seems a perfectly reasonable conclusion, except for this fact: holding a lottery ticket in next Saturday’s lottery entails the far stronger probability that I will not win the lottery than I will win the lottery. If the probability of any single ticket winning the lottery is w then the probability of that ticket losing must be:

Pr = [1-w].

Thus the probability that n tickets will lose is:

Pr = [1-w]^n

This demonstrates something that Christian apologists often wilfully ignore when arguing that some miraculous or otherwise allegedly historical event is more likely to have occurred than not: negative statements tend to hold higher prior probabilities than affirmative statements (Achinstein, 200; Nolt, 1984). If we plug the same reasoning back into Swinburne’s methodology, we can see that, for example, he calculates the prior probability that the universe exists to be .5 (Swinburne, 1991; on the grounds that it either exists or it doesn’t) and further considers his observation that the universe does exist to increase the probability of his hypothesis that God exists being true (on the grounds that it is more likely that a God created the universe than it has occurred via natural means). But a binary sample space with equal probability can be considered a specious assumption for two reasons; (i) it presupposes that the existence of the universe is a physical fact; and (ii) it further presupposes that, even if the existence of the universe is a physical fact, then the fact of that existence along with carefully selected current states of the universe are best accounted for by underlying affirmative statements, i.e., there is a responsible agent and that agent is the classical monotheistic God. A number of other formulations are, of course, possible and therein lies a further problem with P-inductive approaches. Normally we would contrast a proposition P with its negation, not-P (i.e., a .5/.5 probability) giving us necessary and sufficient reason to either accept or reject P.

In the specific case of starting an analysis with the claim that God either exists or does not exist (P or not-P) theologians again assume a binary sample space with equal probability. But here P is not a singular claim. It is a compound claim. For example, P (God exists) is necessarily included in the veracity of a prior proposition that monotheism and not polytheism obtains which, in turn, is necessarily included in the veracity of a prior proposition that supernaturalism obtains. Thus when considering many theological and scientific claims side by side a .5/.5 null hypothesis is not always possible. Further, those who use the P-inductive method to rationalise their faith are being forced to accept P, not because it is obviously more plausible than its negation, but on the basis that it also appears to be more plausible than each possible alternative P, as well as any lack of belief in either P or non-P (Schellenberg, 2016). In the case of God exists/does not exist, the theologian is weighting the argument in their favour by presenting: P (God exists) versus not-P (all possible propositions that can counter P; all but one arbitrarily excluded) and then claiming that the probability of P being true is effectively equal to the probability of any single proposition included in not-P being true.

Nevertheless, no matter which P is accepted there are important caveats: First, and most obviously, the degree of belief must obey the laws of probability. Second, it should comport in some demonstrable way with data we currently hold. Third, it must act to confirm the nature of beliefs already held. In other words, it must be internally consistent with other held beliefs. When data is conflicting, Swinburne's approach is not to remain agnostic or further analyse the data but to accept the views of those who have similarly held beliefs or whose views are closest to those beliefs. The first and second caveats are empirically justified, but the third is not because there is no particular requirement that the belief be true. It opens the door to those kinds of soteriological belief which do not depend on either universally agreed plausibility or knowledge as Swinburne (2005) seems to acknowledge:

"To pursue a way in order to achieve the goals of religion, someone needs to believe that it is at least as probable that pursuit of that way will attain those goals as that pursuit of any other way will.........It would not merely be foolish or irrational but logically impossible to pursue a religious way in order to obtain a certain goal, if you believed that pursuing that way would make you less likely to obtain the goal than you would otherwise"

In the case of (i) the simulation hypothesis (Bostrom, 2003; 2005; who calculates the probability of our existing within a simulation to be approximately .33; see also my essay 'Is God a Finite Being or a Simulation?) throws a spanner into the works of P-inductive arguments that assume that God directly created the universe as the prior probability that the universe physically exists must now be viewed as something less than 1. As far as (ii) goes, this claim runs contrary to the views of the majority of cosmologists and their several models, which demonstrate that the universe is better explained by naturalism rather than theism (e.g., Carroll, 2005; 2012). Swinburne's reasoning on this point boils down to P or not-P:

P1: If God does not exist, then the universe does not exist
P2: The universe exists.
C: Therefore, God exists.

This surely begs the question. P1 is simply assumed without proof (note: this differs to claims that P1 is false until proven true or even true until proven false) and it is doubtful that anyone would lend credence to P1 without having already accepted C. Yet it would only take one of the current scientific cosmological models to be made more plausible (due to additional empirical information) to cast doubt on P1. This, in turn, allows a concomitant argument that the epistemic probability of C, that God exists, may actually be decreased by P2, the fact of the existence of the universe. Coming at it from another angle, Schwitzgebel (2015; see also Schwitzgebel & Moore, 2015) calculates the same probability as Swinburne does for the prior probability of God’s existence for “the disjunction of all radically skeptical scenarios” (including the simulation hypothesis) and so also accepts that a physical universe exists. Yet he denies that the existence of the universe supposes God.

The obvious problem here (as Schwitzgebel readily acknowledges) is that P-inductive arguments can only really work with empirical data resulting from objective observation and experimentation and not with subjective degrees of belief. Or any other observer-dependent assumptions; Swinburne concedes that all sub-arguments he factors into his P-inductive argument are dependent on assumptions about God's motivation and purpose for acting as claimed. There are no naturally existing prior degrees of belief. Just because beliefs can sometimes be expressed in quantitative terms (i.e., accepted on the basis of some probability threshold that makes the belief plausible) does not confer accuracy or reliability. This is not a blanket condemnation of the P-inductive theorem. It appears less at fault than the motives and beliefs behind it's application. Nevertheless, even in those cases where we are able to input verifiable quantitative data in the form of a premise, a logically valid P-inductive argument may still furnish a conclusion which would be false if considered true and acted upon. Consider this example (actually based on someone I know, in the county in which I live, though anonymised):

P1: 99% of the residents of Happy County self-identify as being of white European ancestry
P2: Sam Jones has lived in Happy County for 55 years and eight months
C1: Therefore: Sam Jones’ ancestry is white European (Pr = .99)

Both premises are empirical facts. The P-inductive argument is logically valid; it is indeed far more probable than not that Sam Jones shares ancestry with the bulk of the population of Happy County. However, if we accepted the conclusion as being probably true and plausible and then employed it as a basis for further belief it would nevertheless be a mistaken belief because the conclusion is factually wrong. And therefore by definition not knowledge; no matter how justified one feels a belief to be, it does not constitute knowledge unless it is true. Indeed, believing in something that is false carries you further from knowledge than does a lack of belief. We could, of course, add further premises until a sound argument is obtained, but to reach a true conclusion that is not mere accident we would need to add empirically verified premises only, and not merely ‘plausible’ claims. Yet despite the availability of such trivial examples some people are seduced into believing that when a theologian plugs subjectively acquired quantitative data (usually masquerading as objective) into a P-inductive argument and then concludes that God exists with a probability somewhere between .51 - .99, then they have provided sufficiently plausible evidence that God exists that they can then claim to be a warranted belief. But they have done no such thing. The original question (does God exist or not?) is really no closer to being answered in anything like an objective fashion. Likewise for other theological claims. Plausibility arguments suffer from this inherent weakness: they can be inverted simply by proffering an equally plausible counter-argument. Ultimately something exists or it doesn’t. In which case P or not-P.