Here is another objection to dwindling probabilities from Swinburne:

“A defender of the argument from dwindling probabilities may…emphasize that all the same the longer the route of the argument (or the more conjuncts involved in the conclusion), the less probable is the conclusion; and so suggest that it is not plausible to suppose that an argument of any length would yield a very probable conclusion. In rebuttal I make two points. The first is that the argument from dwindling probabilities applies, in so far as it does apply, not only to theological arguments, but to any argument of some length in history or science (or to any conjunction of propositions in these areas). Yet surely in these areas we can reach conclusions which are very probable.”

One cannot deny that historical and scientific arguments can in some cases show an historical or scientific conclusion to be very probable.  However, it is NOT clear that this is the case when there is just one argument for such a conclusion and that argument consists of deductive inferences and  multiple independent probable factual premises, where it would be correct to multiply the probability of the premises to determine the probability of the conclusion.

1. It will (probably) rain here this afternoon.

2. If it rains here this afternoon, then my lawn will (probably) be wet this evening at 8pm.

Therefore:

3. My lawn will (probably) be wet this evening at 8pm.

If the probability of (1) is .8  and the probability of (2) is .8, then we could multiply the probability of these premises to arrive at a probability of .64 for the conclusion (or about .6).

We need, however, to consider the possibility that it will not rain here this afternoon, and whethere there might be OTHER REASONS why my lawn might bet wet this evening.  One possibility is that it might not start raining until the early evening (between 6pm and 8pm).  Another possibility is that a sprinkler system might automatically water the lawn this afternoon.  There can be multiple reasons for thinking my lawn will be wet this evening.  Even if it does NOT rain this afternoon, my lawn might still be wet at 8pm this evening:

1. It will (probably) rain here this afternoon.

2. If it rains here this afternoon, then my lawn will (probably) be wet this evening at 8pm.

4. The sprinkler system will (probably) water the lawn for about an hour  this afternoon.

5. If the sprinkler system waters the lawn for about an hour this afternoon, then my lawn will (probably) be wet this evening at 8pm.

6. It will (possibly) rain here in the early evening between 6pm and 8pm.

7. If it rains here in the early evening between 6pm and 8pm, then my lawn will (probably) be wet this evening at 8pm.

Therefore:

3. My lawn will (probably) be wet this evening at 8pm.

In this case, the probability of the conclusion (3) is NOT limited to the probabilty either that it will rain in the afternoon, or to the probability that it will rain in the early evening, or to the probability that the sprinkler system will automatically water the lawn this afternoon.  Because there are multiple ways that my lawn can become wet, the probability of (3) can exceed the probability of any one particular way that my lawn might become wet.

In this argument which refers to multiple ways in which my lawn can become wet, it would be incorrect to simply multiply the probabilities of each of the premises together.  Multiplying the probabilities of these various premises would yeild a fairly small probability.  For example if the probability of each of these premises was .8, then multiplying their probabilities would result in a probability of .4096 or about .4 which is a significantly lower probability than we got with just the one possible way of the lawn getting wet (i.e. rain in the afternoon). But clearly, adding more ways for the lawn to get wet will INCREASE the probability of the concluson, not decrease it.  So, each additional way that the lawn can become wet will significantly INCREASE the probability of the conclusion (in the case of this argument).

Clearly, arguments involving probability can sometimes involve such multiple reasons or considerations each of which provides some additional support for the conclusion, and in the case of such arguments it would be incorrect to simply multiply the probabilities of the premises together to determine the probability of the conclusion.

Therefore, since there are some such arguments involving probability, it could be the case that MOST historical and scientific arguments that show their conclusions to be very probable are of this sort, and are not of the sort where it is appropriate to multiply the probabilities of the premises together to determine the probability of the conclusion.

Obviously, there could still be some historical or scientific arguments that show their conclusions to be very probable which are the sort of argument in which it is correct to simply multiply the probabilities of the premises to determine the probabiilty of the conclusion.  In such cases the probability of the premises will need to be very high, especially if there are four or five or more premises in the argument.  Consider, for example, an argument with four premises where each premise has a probabilty of .95.  If this argument was the sort where it would be correct to simply multiply the probabilities of the premises to determine the probability of the conclusion,then the probability of the conclusion would be: .8145 or about .8.  I would consider this conclusion to be “very probable”.  But if the probability of the premises were just a bit lower, say .9. then the probability of the conclusion would be only: .6561  or about .7.   Although this conclusion would clearly be probable, it is not clear to me that it would be “very probable”.

How does all this relate to my specific example of dwindling probabiilty?  There are at least two disanalogies that make it so that this point by Swinburne does not apply to my example.  First, in the case of ancient history it may be rather difficult to come up with factual premises that are highly probable, whereas it might well be easier to come up with highly probable factual premises in relation to arguments for conclusions in modern history or modern science.

Second, it might well be the case (Swinburne has not shown otherwise) that the probability arguments in history and science that show their conclusions to be very probable are of a differnt sort than the probability argument that I give as an example of dwindling probability.  It might be the case that most such historical and scientific arguments have premises that each provide some additional INCREASE to the probability of the conclusion.  These historical and scientific arguments might be such that it would be incorrect to simply multiply probabilities of the premises to determine the probability of the conclusion.

In previous posts, I discussed Swinburne’s idea that there can be multiple WAYS or ROUTES to a conclusion, and that each WAY or ROUTE can contribute to the overall probability of the conclusion.  Swinburne is correct on this point, but this implies that there are probability arguments in which the addition of premises INCREASES the probability of the conclusion rather than decreases the probability of the conclusion.  Thus, it is critical to distinguish between these different kinds of probability arguments.  The fact that there are SOME probability arguments where it is incorrect to simply multiply the probabilities of the premises to determine the probability of the conclusion does NOT show that there are NO probability arguments where it is correct to simply multiply the probabilities of the premises to determine the probability of the conclusion.

In fact, we KNOW that if the probability of the conclusion would be ZERO given that a premise is false, then the probability of the conclusion CANNOT exceed the probability of that premise.  In other words, if one claim is a necessary condition for another claim, then the probability of the latter claim cannot exceed the probability of the former claim.