2. Some preliminaries on formulation
2.1 The Likelihood Principle
This version of the argument will use the Likelihood Principle (LP) – this roughly states that, if some observation O is to be expected under a hypothesis h1, and if it not to be expected (or expected to a lesser extent) under h2, then O counts as evidence in favour of h1 over h2. This is not to say that h1 is made more overall probable than not, nor even that h1 is overall more probable than h2. It is only to say that O confirms h1 over h2.
More formally, then, the likelihood principle says that if the conditional probability P(O|h1 & k’) > P(O|h2 & k’) – where k’ represents appropriately chosen background information – then h1 is confirmed over h2. It is not necessarily the case, however, that P(h1|O & k’) > 0.5 or that P(h1|O & k’) > P(h2|O & k’). Similarly, ¬h1 can be substituted for h2 so that, if a particular observation is more likely given h1 than given ¬h1, then h1 is confirmed over ¬h1, and thus h1 is made more probable than otherwise. This demonstrates quite neatly how one might use the fine tuning argument in a Bayesian framework, where P(h1|O & k’) = [P(h1|k’)·P(O|h1 & k’)]/P(O|k’). Here, P(h1|k’) represents the prior probability of h1 – the probability of h1 being true based on the background information alone. It follows from Bayes’ Theorem that, if P(O|h1 & k’) > P(O|¬h1 & k), and therefore P(O|h1 & k’) > P(O|k’), then P(h1|O & k’) > P(h1|k’). That is, the probability of h1, given O, is greater than it was before O was considered. Thus, O constitutes some evidence for h1. Note also that the ratio P(O|h1 & k’)/P(O|k’) determines the degree to which h1 is confirmed, and thus the strength of the evidence. The main point of contention with Bayesian arguments is their use of prior probability – it is a controversial issue as to how meaningful prior probability is, and how it can be reliably judged. If one is happy with prior probability and Bayesian arguments, then the argument can be interpreted as Bayesian. If not, then one can avoid the problematic issue of prior probability by simply using the Likelihood Principle.
The Likelihood Principle is relatively uncontroversial, and seems to me a foundational aspect of our reasoning about the world. When we talk about evidence for a hypothesis, what we generally mean is the observation of things which we would expect to observe if the hypothesis were true, and which we would expect less if the hypothesis were not true. Thus, fingerprints of X at a crime scene ‘confirm’ X’s culpability – if X were guilty, we would expect there to be fingerprints, and this would be less likely if X were not guilty. This does not prove culpability conclusively, but it does constitute some evidence. The LP is used in prospective, experimental science (through prediction) and in retrospective, historical science; the latter will be of relevance for the ‘old evidence’ problem, as well as the ‘anthropic principle’ objection. The principle is exemplified most strongly in prospective science: we consider a hypothesis h, and make predictions based on what we would expect if the hypothesis were true. We then perform experiments to see if the predictions are correct. If they are, then we have an observation which satisfies the LP criterion. If they are not, then the hypothesis is disconfirmed. But, importantly, the principle is also fundamental to retrospective science. Much evidence for scientific study of the past is not found through experiment, but through discovery. Thus, fossils often constitute evidence for Darwinism despite not being the result of human experiment. This is because, if Darwinism were true, we would expect to find certain types of fossils, especially compared to the likelihood of finding them if Darwinism were not true. It does not matter that this evidence was not found through prospective prediction and experiment – it still constitutes evidence.
Though the LP is uncontroversial, there are two objections that might be made, and which invite clarification. The first is that this seems to suggest that absurd, ad hoc hypotheses are confirmed by certain observations. For example, the fact that the lottery numbers this past week were 4, 10, 37, 41 and 49 is more likely given that there is an omnipotent, invisible hippopotamus who has a proclivity towards making lotteries come up with those results, than it is otherwise. But does it not seem counter-intuitive to suggest that this confirms the hypothesis? I do not think so. For any hypothesis with such arbitrary contingencies has an extraordinarily low prior probability, such that even the very strong evidence here would be insufficient to make it all probable. But, even if theism has a reasonably low prior probability, it does not seem at all clear that it has a comparably low prior probability, since it does not incorporate similarly arbitrary contingencies.
Though I think it ought, this account of prior probability may not satisfy a non-Bayesian, and so we might tackle the problem by creating a restricted version of the likelihood principle (cf. Collins, 2009). This excludes hypotheses which are particularly ad hoc, which have no independent motivation for their advocacy and which generate obscure contrivances in retrospect, in order to explain the observation under scrutiny. If there had been a group of hippopotamus-followers who declared, prior to the lottery, that they had good reason to believe their hero would predict these exact numbers, we would be far more inclined to take them seriously. We would not necessarily accept their claims, but we would typically give them far more credence than if it had been advanced after the fact. So I think there is good reason to accept the LP despite this purported difficulty.
Secondly, the LP (along with Bayes’ Theorem) has been criticised for its lack of quantitative precision – but this seems trivial. We can’t give exact numerical values to the probability of finding particular fossils (PF) given Darwinism (D) and given ¬D, but that doesn’t mean that we can’t roughly compare the probabilities: we can still say that P(PF|D & k’) > P(PF|¬D & k’) and therefore that PF constitutes evidence for D. To undermine the LP, then, would undermine much of the scientific enterprise, as well as an enormous deal of our ordinary, elementary reasoning.
1. Which background information this includes will be discussed in a later section. ^