VAN INWAGEN'S ARGUMENT

Peter van Inwagen (1983, pp. 202–204) has formulated an influential and elegant reductio ad absurdum of the PSR. Let p be the conjunction of all contingent truths. If p has an explanation, say, q, then q will itself be a contingent truth, and hence a conjunct of p. But then q will end up explaining itself, and that would be absurd. We can formulate this precisely as follows:

1. (33) If the PSR holds, then every true contingent proposition has an explanation. (Premise)

2. (34) No necessary proposition explains a contingent proposition. (Premise)

3. (35) No contingent proposition explains itself. (Premise)

4. (36) If a proposition explains a conjunction, it explains every conjunct. (Premise)

5. (37) A proposition q only explains a proposition p if q is true. (Premise)

6. (38) There is a Big Conjunctive Contingent Fact (BCCF) that is the conjunction of all true contingent propositions, perhaps with logical redundancies removed, and the BCCF is contingent. (Premise)

7. (39) Suppose the PSR holds. (For reductio)

8. (40) Then, the BCCF has an explanation, q. (By (33), (38) and (39))

9. (41) The proposition q is not necessary. (By (34), (38), and (40) and as the conjunction of true contingent propositions is contingent)

10. (42) Therefore, q is a contingent true proposition. (By (37), (40), and (41))

11. (43) Thus, q is a conjunct in the BCCF. (By (38) and (42))

12. (44) Thus, q is self-explanatory. (By (36), (40), and (43))

13. (45) But q is not self-explanatory. (By (35) and (42))

14. (46) Thus, q is and is not self-explanatory, and that is absurd. Hence, the PSR is false.