September 28, 2010

Explanatory Principles and Infinite Propositions

In the course of his discussion of cosmological arguments, Sobel argues against the Principle of Sufficient Reason and similar strong explanatory principles. In particular, he argues that even a weak principle like "there is a deductive explanation that has only true premises for every contingent truth" will result in modal collapse (p. 218). In Sobel's terminology, an argument 'deductively explains' its conclusion iff (1) the argument is sound, and (2) the conclusion does not entail the premises (p. 219; condition (2) applies to contingent conclusions only). Sobel now introduces the following two premises:

(3) If there is any true contingent proposition, then there is a true contingent proposition that entails every true contingent proposition.
(4) For any true propositions, there is a true proposition that entails exactly these propositions, itself, and every necessary proposition. (p. 219)

Sobel uses these premises to derive the conclusion that there is a contingent truth C that entails every contingent truth. C must then have a deductive explanation whose premise, Pr, is not entailed by C. Since it is not entailed by C, it is not contingent, but if it were necessary, C would be necessary, contrary to hypothesis.

It is not clear to me that we should accept (3) and (4). Sobel provides no defense of them. Obviously for finite sets of truths this will work: we can just conjoin them. But standard logic doesn't allow infinite conjunctions. Should we believe in infinitely conjunctive propositions? I don't know.

It seems unlikely to me that C would be a finite proposition. As such, we can escape Sobel's argument either by rejecting infinite propositions altogether, and so rejecting (3) and restricting (4), or by restricting the principle of deductive explanation to finite propositions. Either of these courses will cause problems for cosmological arguments that deal in contingent facts, but it will not effect those that deal in contingent events or beings.

If anything, the effect this section of Sobel's book has had on me has been to make me think that perhaps Leibniz's definitions of necessity and contingency are less crazy than I previously thought. Perhaps there is something odd - in a metaphysically significant way - about infinitary logic.

Posted by Kenny at September 28, 2010 10:13 PM
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