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406 GEROGIORGAKIS
number which rather exceeds the value 4. And due to the principle of indifference Swinburne
gives the value ½ for P (e | h & k) (≈ given that there is a God, it is as probable that we would
possess our available evidence for His existence, as that we would not possess it). But then the
value for the probability of God’s existence given the evidence is absurdly high: P (h | e & k) =
P (e | h & k) • P (h | k) / P (e | k) ≥ 2, which leads Bayesian Theism as a whole ad absurdum.
To fix the calculation, i.e. to avoid the absurdity of the likeliness of God’s existence given the
total evidence being assigned a value higher than one (i.e. P (h | e & k) ≥ 1), Gwiazda
considers the option of abandoning the value ½ for P (e | h & k). So, if, by Gwiazda’s
calculations, P (h | k) / P (e | k) ≥ 4 then, in order for the likeliness of God’s existence given
the total evidence to approximate the value 1, the probability for our evidence for God’s
existence to be available given God’s existence should be less than 25 per cent (i.e. P (e | h &
k) < 0,25). This means that God would rather avoid to let us possess our available evidence
for His existence. For one thing, the justification for such an assumption would be very
debatable and suspect of being ad hoc. Why would God do something like this? For another,
the aforementioned justification would not be based on any generally accepted principle of
inductive logic. But what is worse is that an ad hoc assumption of an at-most-25-per-centprobability
by which God would let us have some evidence for His existence, would substitute
Swinburne’s initial one-half-probability for the same state of affairs, a probability which was
at least based on the principle of indifference. Since we feel justified to assume ad hoc that it
would be rather unlikely (less than 25 per cent) for God to let us possess some evidence for
His existence, there is not any particular reason why the atheist should not expect this
likeliness or rather unlikeliness to be extremely low, say P (e | h & k) ≤ 0,00000000025, to
make P (h | e & k) very low as well. If you do not care if your assumptions are ad hoc, you
cannot very reliably argue that your opponent should care!
Swinburne (2011) has argued against this criticism that the principle of simplicity is to be
applied in comparing between explanatory hypotheses, not however between explanatory
hypotheses and events. This means that P (h | k), despite its pertaining to a much simpler
assumption, does not have to be much higher than P (e | k). Swinburne insists in the value ½
for P (e | h & k), and gives the value 0,01/0,005 (= 2) for the ratio: P (h | k) / P (e | k), which
results in the value 1 for P (h | e & k).
Swinburne’s counter-argument against Gwiazda leaves much room for criticism. It
presupposes a selective application of the principle of simplicity which can be easily
considered to be ad hoc. Moreover, Swinburne’s insistence in very high values for P (h | e & k)
fails to account for those facets of religious faith, which maintain that it would be rational to
affirm God’s existence independently of how probable or improbable His existence is. Pascal’s
Wager exemplifies this kind of religious faith. The history of religion has many examples of
religious traditions to offer, which support Pascal’s line of argument. I propose to affirm a
Bayesian Theism which supports Pascal’s line of argument. Among other things, Pascal’s line
of argument can be analysed in terms of Bayesian probabilities and shown to be immune
against Gwiazda’s criticism (cf. section 4 and 5).
2. Gambling Odds in Bayes’s Theorem
Gambling odds express the payouts from a bet on the occurrence of an event. The higher the
likeliness of the event, the lower are the gambling odds for its occurrence. The lower the
likeliness of the event, the higher are the gambling odds. For example, ideally, when an event
is as likely as it is unlikely to happen, i.e. P (e) = 0,5, then the gambling odds for this event
amount to 1/0,5 = 2. I.e. if one bets that the event occurs and this happens, then one doubles
her stake.