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Pruss and Rasmussen on the Argument from Necessary Abstracta

Pruss and Rasmussen's seventh chapter puts forward an argument for the existence of a necessary concrete being from the existence of necessary abstracta. They connect this strategy with an argument of Leibniz's. The Leibnizian argument, usually known as the 'argument from necessary truths', is to some extent known in the contemporary literature, but it has not become part of the standard list of arguments for the existence of God. (For instance, it is not discussed in Jordan Howard Sobel's Logic and Theism or Graham Oppy's Arguing About Gods.) Leibniz himself always seems to run through this argument very fast, and there's a limited amount of recent philosophical literature on it, so the analysis provided here is a very welcome contribution.

As the name suggests, standard presentations of this argument normally focus on truths, but Pruss and Rasmussen's version focuses on the existence of abstracta. Issues of truth come in only in helping to motivate the claim that there are necessarily existing abstracta (see esp. sect. 7.3.1). It is rather difficult to say precisely how this relates to Leibniz's argument since Leibniz's various presentations of the argument differ from one another, and I am unaware of any place where Leibniz spells the argument out in detail. Pruss and Rasmussen do not address this question (their aims are, of course, not historical), so I won't say any more about it here.

Pruss and Rasmussen present the argument as follows:

1. Necessarily, there is an abstract object.


2. Necessarily, if there is an abstract object there is a concrete object.


3. Possibly, there are no contingent concrete objects.


4. Therefore, possibly there is an abstract object and no contingent concrete objects. ((1) and (3))


5. Therefore, possibly there is a concrete object and no contingent concrete objects. ((2) and (4)).


6. Therefore, possibly there is a necessary concrete object. ((5))


7. Therefore, there is a necessary concrete object. ((6) and S5) (p. 126)

Pruss and Rasmussen consider two main reasons in favor of (1): necessary abstracta are needed to make sense of necessary truths (sect. 7.3.1) and they are needed for a plausible ontology of mathematics (sect. 7.3.3). In defense of premise (2), Pruss and Rasmussen offer reasons for thinking that a reductive account of abstracta is preferable to either straight-up Platonism or straight-up (eliminative) nominalism (sect. 7.4). Premise (3) is needed because some might think that, while abstracta depend on concreta, a given abstract object need not depend on the same concrete objects in every possible world. If someone who held this view also thought that it was necessary that some concrete entity exist, but not necessary that any particular concrete entity exist, then that person could escape the argument.

Given (1), (2), and (3), the rest follows by S5.

In my view, this chapter is one of the best (perhaps the best) in the book, in terms of the success of the general project of showing that avoiding the existence of a necessary being involves some heavy-duty philosophical commitments. The authors conclude (p. 148) that nominalists and Platonists might both avoid the argument if they can solve the standard problems for nominalism and Platonism, respectively, but those who want an intermediate view will most likely have to accept the existence of a necessary being.

It seems to me, however, that the nominalist and Platonist positions here are not symmetric. The Platonist has only the standard objections to Platonism to address, whether she accepts a necessary being or not. (By 'Platonist' I here mean a proponent of necessary fundamental abstracta, not grounded in or otherwise dependent on anything concrete.) The nominalist, however, has extra problems, since most versions of nominalism make abstracta depend, in some way or other, on minds. The introduction of a necessary mind (e.g., God) therefore solves a bunch of problems for the nominalist, but not for the Platonist. So the conclusion is that anyone who wants to avoid commitment to a necessary being (remembering that in Pruss and Rasmussen's language 'necessary being' means 'necessarily existing causally capable thing') is going to be able to save a lot of standard claims about necessary truth (including the necessity of mathematical truths) only by endorsing full-strength Platonism.

That, at least, is how it looks to me. This chapter is especially dense with argument, so a full defense of my evaluation would require a lot more than a blog post!

Posted by Kenny at June 21, 2019 6:42 PM