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Leibnizian CA: Pruss

Another formulation of the Leibnizian cosmological argument comes from philosopher Alexander Pruss.

Formulation​

  1. Every contingent fact has an explanation.
  2. There is a contingent fact that includes all other contingent facts.
  3. Therefore, there is an explanation of this fact. (1, 2)
  4. This explanation cannot itself be contingent.
  5. Therefore, the explanation of all contingent facts is a necessary being. (3, 4)

The Second Premise and the Big Conjunctive Contingent Fact (BCCF)​

The second premise is not as obvious as it may seem. Pruss treats a fact as a proposition that is true, meaning the first premise is then "every contingent true proposition has an explanation."

The second premise then posits the existence of a Big Conjunctive Contingent Fact (BCCF) that includes all contingent facts. There is a problem here.

Let be the BCCF. By the Principle of Sufficient Reason (PSR), must have an explanation, say . is either contingent or necessary. If is contingent, then is part of the BCCF, which means is self-explanatory. This cannot be the case, so must be necessary.

However, not all conjuctions like the BCCF make sense. Consider the conjuction of all non-self-conjunctive propositions, where a self-conjunctive proposition is one that is conjunctive of itself. is either self-conjunctive or not. If is self-conjunctive, since contains non-self-conjunctive propositions, it is not self-conjunctive. If is not self-conjunctive, since contains all non-self-conjunctive propositions, then contains itself, and is hence self-conjunctive. This presents a major contradiction, and so the BCCF is not necessarily a coherent concept.

The BCCF is highly criticized. For instance, this paper by Christopher M. P. Tomaszewski, titled "The Principle of Sufficient Reason Defended: There Is No Conjunction of All Contingently True Propositions", completely rejects the idea of a BCCF. The paper refutes Peter van Inwagen's argument that the PSR entails "modal collapse", where absolutely everything is necessary. Tomazewski states that a critical flaw in van Inwagen's argument is the assumption that there is a conjuction of all contingently true propositions.

He argues that there are too many propositions to conjunct through the use of Cantor's Theorem (directly quoted from the paper):

  1. There is a conjunction of all contingently true propositions . (Assumed)
  2. For each non-empty collection of propositions which are conjuncts of , there is a unique contingently true proposition to which it corresponds.
  3. Every such contingently true proposition is a conjunct of . (1)
  4. There are strictly more non-empty collections of propositions which are conjuncts of than there are propositions which are conjuncts of . (From Cantor's Theorem)
  5. There are strictly more propositions which are conjuncts of than there are propositions which are conjuncts of . (2, 3, 4)
  6. Therefore, there is no conjunction of all contingently true propositions.

The paper also includes a defense of the second premise in the refutation of van Inwagen's argument. Start with a collection of conjuncts of , say , and let be the proposition that is the conjunction of all the conjuncts in . Let be an arbitrary necessary proposition. Then, consider the proposition :

Each is a proposition that belongs to this collection. The inclusion of means that is not logically equivalent to any conjunction of the propositions in , and must hence be included in .

What Tomaszewski is saying, essentially, is that for any conjuncts, you can find a unique proposition that is not in the conjuncts, hence defending the second premise.

An elaboration of this argument is as follows:

  1. Assume there is a conjunction of all contingently true propositions. This is , and it can be written as . A conjunct is a proposition that is a part of the conjunction - , , etc.
  2. Now consider taking any collection of conjuncts, say , and forming a conjunction of them, . Since is a proposition, it must be a conjunct of .
  3. This means that all are conjuncts of .
  4. This is the key step. Cantor's Theorem states that for any set, the set of all of its subsets is strictly larger than the set itself. (Intuitively, think of it analogous to how there are 52 cards in a deck, but far more than 52 ways to choose an arbitrary number of cards from the deck.) This means that there are more collections of conjuncts than there are conjuncts.
  5. By Cantor's Theorem, there are more collections of conjuncts, like , than there are conjuncts, like , , etc. But by (3), all are part of . So this is actually saying that there are more than there are . This is a contradiction.
  6. This argument shows that by assuming that the BCCF exists, we run into a contradiction, and hence the BCCF does not exist.

The Conclusion (Again)​

Just like the previous cosmological arguments, the conclusions do not really lead to "God".