White (1979) has proposed a principle that if only one putative explanation can be given of a phenomenon, that putative explanation is correct. As already mentioned, this is what Sherlock Holmes means by "when you have eliminated the impossible, whatever remains, however improbable, must be the truth" (Doyle 1890, p. 93; italics in the original). But only a necessary being's causal efficacy can explain global explananda like the BCCF. Hence, a necessarily existing First Cause exists.
The difficulty with this principle is that it is not clear how one counts putative explanations. Suppose a coroner sees a woman with a wound and can rule out all explanations except a stabbing with a knife. Does the POE apply? After all, one might say that there is still more than one explanation available. Maybe the woman was stabbed with a knife for profit or maybe she was stabbed out of a revenge. If the latter two count as alternative explanations and make the POE not applicable, then POE does not apply to the cosmological case, since more than one set of motives could be assigned to the First Cause.
Suppose instead that POE still applies in the murder case. Then POE must be understood as follows. If at some specific level of generality only one putative explanation can be given, then that one explanation must be correct. At one level of generality, we have a stabbing with a knife. All alternatives to that have been ruled out. Hence, we need to accept that a stabbing with a knife happened.
However, now POE becomes much more controversial, and it is not clear that it is a gain over versions of the CP. In fact, for a wide class of items, it implies a CP. For at a very high level of generality that "a cause caused E" seems to be an explanation, and for a large class of items E it seems plausible to suppose that no other explanation would be possible. This last claim requires ruling out conceptual explanations. To do that we would have to work with the ontologically most basic items, where further conceptual explanations are impossible so only causal ones are available, and then say that if the most basic items have causes, so do the less basic ones.
Perhaps more specificity is required than just "a cause caused E." Maybe "a necessary being caused E" has that kind of specificity, or maybe we can prove that only God's activity could explain the BCCF. But significant amounts of work would be needed here.
And, besides, it is not clear why someone who accepts the POE would deny the PSR. Supposing that the coroner rules out all explanations other than being poisoned or having a heart attack, the inference to the claim that the person was either poisoned or had a heart attack would be just as good as the inference to the claim that she was poisoned if only that option remained.
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