Reply to Gale and Pruss

Philo, 7, 2004, 114-1121

by John F. Post

Comments welcome! || john.f.post@vanderbilt.edu || Home Place

I am indebted to Richard Gale and Alexander Pruss for the excellent questions they raise in their separate responses to my comments on Gale's book, On the Nature and Existence of God."⁽¹⁾ They focus on aspects of my discussion that need at least to be clarified, if not retracted, in ways I hope to explain in what follows. But first let me call attention to a couple of arguments they do not mention.

One is a positive argument that expands on, and is meant to be in solidarity with, a largely unappreciated theme of Gale's formidable book. I argue that when applied to theism, "revisionary theory-construction" (as I call it) is actually an advantageous extension of his own revisionary remarks. The other argument, likewise revisionary, advances a positive account of omniscience (in Section II). This positive account, I argue, not only avoids the Cantorian troubles that afflict accounts in terms of "all truths" and the like, it allows God's knowledge to be suitably supreme in kind and extent. The account thus bypasses the swamp of technical complexities involved in the debate over whether talk literally of "all truths" is logically incoherent; theism need not be made hostage to the outcome of such arcane disputes. Not all my arguments are negative, and I would hate for readers to get the contrary impression.

That said, Gale is right that the brevity of my Section IV makes it difficult to understand just what "revisionary theory-construction" is supposed to be. I had hoped that the methodological remarks I made along the way in earlier sections, taken together with references to half a dozen other writings, would make things sufficiently clear. Evidently they did not. So let me add some clarificatory remarks here.

According to Gale, "revisionary theory-construction," which I advance as an alternative to analytic theistic argumentation, "seems similar to inference to the best explanation" (Gale 2003, 143). It may indeed seem similar, but what I have in mind is not inference to the best explanation, indeed not even an inference from a theory's explanatory success to its probable truth, or even to its being more likely true than its competitors. Rather, I have in mind the comparatively greater adequacy of a theory that has survived better than its competitors in the ongoing "trial by prerequisites for truth" -- namely, the ongoing comparison of the theory and its competitors as regards their satisfaction, as best we can tell, of conditions that are logically necessary conditions of truth, not sufficient conditions or criteria (which typically are epistemically problematic at best). The survivor may not be true, or even more likely true than its competitors, but at least it is the best we have so far.⁽²⁾

To be sure, Gale is right that as thus characterized, my approach need not dispense with conceptual analysis. But earlier, in Section II, I presented independent reasons for limiting the role of conceptual analysis, if not dispensing with it altogether. Insofar as the aim of conceptual analysis is to capture existing concepts -- say those employed in extant creeds or other established language games⁽³⁾ -- it is at odds with our creating new or revised concepts that may have a better chance of conforming to real-world affairs than do the received concepts.⁽⁴⁾

Gale also mentions "The currently fashionable global or agglomerative defenses of theism," and, as he suggests, they may seem compatible with what I have in mind. But as I understand them, these defenses are not compatible with the spirit of my approach, nor with the letter, though it would take too long to explain here the ways in which they are incompatible or at least diverge.⁽⁵⁾

Finally, as regards alternatives to analytic theistic argumentation, I am grateful to Gale for noting that

After rightly stating that we must avoid making theism a competitor to non-reductive [naturalism], [Post] say[s] that the way to do this is "is to demetaphysicalize our concept of God, say by constructing a theory in which theistic claims hold not in virtue of extra entities or processes but in virtue of an objectively, solidly true normative vision of ourselves and the world in theistic terms" (Gale 2003, 143).

However, I do not think, nor did I say, that this is the way to avoid making competitors of theism and non-reductive naturalism, but one way; I am confident that eventually there will be other ways, quite possibly preferable.

Immediately after this passage, Gale goes on to say that he does "not see why our attempt to make theism compatible with non-reductive naturalism should have this non-realist consequence." But as I explain in the chapter to which I referred in footnote 29, I do not regard this as a non-realist consequence but as robustly realist: the normative vision involved is an "objectively, solidly true normative vision . . . in theistic terms." Granted, this normative realism is teamed with non-realism about extra entities -- "extra" from the point of view of non-reductive naturalism; in this sense my approach is metaphysically non-realist. But how else could it be if it is to avoid competing with non-reductive naturalism, which discredits traffic in theological and other extra entities? ⁽⁶⁾ Furthermore, a normative truth that does not posit extra entities can be just as vivid and compelling as one that does; we can and do have inalienable rights and implacable obligations, and can be steadfast realists about having them, without there being such entities as "rights" or "obligations" (perhaps in Plato's heaven along with the form of the good).

In any case I think the crucial issue lies elsewhere, and I suspect Gale thinks so as well. He raises the issue in his next remark: "such [metaphysical] nonrealism fails to give the working theist a religiously available God" (Gale 2003, 143). It can certainly look that way, but I don't think the metaphysical non-realism need fail in this respect. After all, non-realism about entified rights and obligations is compatible not only with normative realism about rights and obligations, but with morally available action in their name, indeed urgent, categorical action. But of course much more remains to be done to explain just why the metaphysical non-realism, combined with realism about a normative vision of ourselves and the world in theistic terms, does after all give the working theist a religiously available God. Elsewhere I have a go at an explanation (Post 1987, Ch. 8), but more remains to be done.

In an agreed division of labor with Gale, Pruss concentrates on technical questions about the Cantorian argument that the notion of "all true propositions" is logically incoherent. He claims that Grim's Cantorian argument depends on there being set-like collections of propositions, contrary to Grim and Post. In order to establish his claim, Pruss argues that the Cantorian argument presupposes the Schröder-Bernstein theorem about sets, hence is committed to there being sets or at least set-like collections of propositions or other truth-bearers, or of anything else an omniscient being might plausibly be supposed to know (states of affairs, facts, conditions, whatever).⁽⁷⁾

Pruss begins his argument by asking what happens when we apply the Cantorian argument not to propositions but to sentences. In the propositional case, the Cantorian argument involves what amount to a couple of lemmas. In Pruss's numbering, these are

(3) There are more propositional properties than there are true propositions, and

(4) There are at least as many true propositions as there are propositional properties.

Pruss then instructs us to systematically replace certain terms in (3) and (4), which replacement yields

(3a) There are more sentential [predicates] than true sentences, ⁽⁸⁾ and

(4a) There are at least as many true sentences as there are sentential predicates.

Unfortunately, (3a) and (4a) are not lemmas in an argument Grim or I would recognize as an application of the Cantorian argument to sentences. The reason is that in this context, we both adhere consistently to talk not of predicates but of properties of the truth bearers or of anything else an omniscient being might plausibly be supposed to know.

The correct way to apply the Cantorian argument to sentences is presented in my comments on Gale's book (Post 2003, 36). There the relevant lemmas are

(3s) There are more sentential properties than true sentences, and

(4s) There are at least as many true sentences as there are sentential properties.

The flaw in this argument, as I pointed out in my comments, is that while (3s) is true (being easily proved by a diagonal argument), (4s) is pretty obviously false. While there are uncountably many sentential properties, there are only countably many sentences of a standard language. (On how to explicate such cardinality comparisons, see below.) Thus what does properly count as an application of the Cantorian argument to sentences pretty obviously fails, hence pretty obviously should not tempt us to conclude, absurdly, that it makes no sense to talk about all true sentences in English.

Nonetheless, the argument Pruss counts as an application of the Cantorian argument to sentences remains an interesting argument, in view of the important issues he raises in connection with it. Two are crucial. One is whether we may assume, as he does, that "a 1-1 pairing of A's with some B's is, surely, nothing but a collection of ordered pairs" (Pruss 2003, 54). The other is whether, as Pruss claims, lemmas (3) and (4) can be shown to entail a contradiction only if one assumes the Schröder-Bernstein theorem, application of which presupposes there are sets or at least set-like collections of truths and properties (Pruss 2003, 56).

These two issues are hardly new, having been discussed at length by Grim and Plantinga (1993), and earlier still by Grim (1991), as I pointed out in my comments on Gale's book (Post 2003, 37 and 37n9). Plantinga, like Pruss, argues at one point that Grim's Cantorian argument "demands that there be a set of all truths" (Grim and Plantinga 1993, 267). Grim's response, which I had very much in mind throughout my comments on Gale, is that

the Cantorian argument against 'all truths' can be constructed using only quantification and some basic intuitions regarding truths -- without, in particular, any explicit appeal to sets, classes, or collections of any kind.... [T]he argument can for example be phrased entirely in terms of properties, relations, and quantification (Grim and Plantinga 1993, 270, 274).

Grim goes on to explain how the needed notions of mapping and comparative cardinality can be defined entirely in terms of quantification and properties (where relations are just properties that apply to things taken in order; thus the relation of being taller than is the property of being taller than, which applies to x and y in that order if and only if x is taller than y). Plantinga's reply is

Right: we can define mappings and cardinalities as you suggest, in terms of properties rather than sets (Grim and Plantinga 1993, 274).

That a philosopher so astute as Plantinga agrees with Grim on this point is itself good reason for thinking Grim is right: we can indeed define a 1-1 mapping (what I more colloquially call a 1-1 pairing) in terms of properties rather than sets or other collections.

Provided Grim's definition works, a 1-1 pairing need not be nothing but a collection of ordered pairs, contrary to Pruss, nor do cardinality comparisons like "more than" and "at least as many" -- which are crucially involved in lemmas (3) and (4) -- require reference to sets or set-like collections, say by way of the Schröder-Bernstein theorem, in order to make sense of such comparisons and to show that there is a contradiction between (3) and (4). To see why, we need to look carefully at the details of Grim's definition.

In his published exchange with Plantinga in 1993, Grim repeated his 1991 definition, essentially verbatim. Here is the 1993 version. Where again a relation is just a property that applies to things taken in order,

I. A relation R gives us a one-to-one mapping from those things that have a property P1 into those things that have a property P2 just in case:

(x)(y)[P1x & P1y & Ez(P2z & Rxz & Ryz) --> x = y] & (x)[P1x --> Ey(z)(P2z & Rxz <--> z = y)].

II. A relation R gives us a mapping from those things that are P1 that is one-to-one and onto those things that are P2 just in case (here we merely add a conjunct):

(x)(y)[P1x & P1y & Ez(P2z & Rxz & Ryz) --> x = y] & (x)[P1x > Ey(z)(P2z & Rxz <--> z = y)] & (y)[P2y --> Ex(P1x & Rxy)].

We can outline cardinality, finally, simply in terms of whether there is or is not a relation that satisfies the first condition but doesn't satisfy the second (Grim in Grim and Plantinga (1993), 273).

This definition of a 1-1 mapping or pairing is the one presupposed in my comments on Gale. The definition makes clear how a 1-1 pairing, whether into or onto, need not be "nothing but a collection of ordered pairs"; the quantifiers in I and II range not over pairs but over things that have the properties P1, P2 and R. It follows that talk of pairings need not refer to ordered pairs or to collections of them.

Furthermore, Grim's definition enables us to unpack the assertion that there are more P2's than P1's as the assertion that bothe (a) there is a 1-1 R pairing from the P1's into the P2's, and (b) for no property G is there a 1-1 pairing R from the P1's onto the P2's. This implies, as it should, that there are more P2's than P1's if there are not enough P1's to "cover" all the P2's, where to say that there are enough is to say that there is a property G such that for each of the P2's there is a distinct P1 that has G.

The assertion that there are at least as many P1's as P2's is equivalent to the assertion that there are not more P2's than P1's. One way to show that there are at least as many P1's as P2's, hence that there are not more P2's than P1's, is to show that there is a property G such that for each of the P2's there is a distinct P1 that has G. The idea is that if there are enough P1's that have G to cover the P2's, then there are enough P1's to cover the P2's. Finally, that there are countably many P1's can be unpacked as the assertion that there is a 1-1 pairing from the P1's into the natural numbers; that there are countably infinitely many P1's can be unpacked as the assertion that there is a 1-1 pairing from the P1's onto the natural numbers.

It follows that cardinality comparisons like "more than" and "at least as many," as they function in lemmas (3) and (4), do not require appeal to sets or set-like collections, nor do they require appeal to the Schröder-Bernstein theorem. Thanks to Grim's definition we can unpack Lemma (3) -- "There are more propositional properties than there are true propositions" -- as:

(3u) There is a 1-1 pairing R from the true propositions into the propositional properties, but for no property G is there a 1-1 pairing R from the true propositions that have G onto the propositional properties.

Lemma (4) -- "There are at least as many true propositions as there are propositional properties" -- is equivalent to the denial of (3), hence of (3u); there are not more propositional properties than there are true propositions. ⁽⁹⁾

As lately noted, one way to show that there are at least as many P1's as P2's, hence that there are not more P2's than P1's, is to show that there is a property G such that for each of the P2's there is a distinct P1 that has G. Hence one way to show that there are not more propositional properties than there are true propositions, contrary to (3), is to show that there is a G such that for each propositional property y there is a distinct true proposition x that has G. The proof of Lemma (4) in my comments on Gale's book, like Grim's proof, proceeds in effect by noting that for each propositional property y, there is a distinct true proposition x that has the property G of being the proposition that y is a property. Furthermore, for each true proposition x that y is a property there is the distinct propositional property y. So there is a 1-1 pairing from the true propositions x that y is a property onto the propositional properties. It follows that there are at least as many true propositions as propositional properties, hence that there are not more propositional properties than there are true propositions, contrary to Lemma (3).

Lemmas (3) and (4) can therefore be shown to contradict each other without assuming the Schröder-Bernstein theorem or requiring in some other way that there be sets or set-like collections of truths, properties or pairs. Grim and Plantinga evidently knew what they were talking about (not that it's any surprise). If there is a flaw in the Cantorian argument, it lies elsewhere. If there is no flaw, and talk literally of all of whatever an omniscient being might plausibly be supposed to know is logically incoherent, as I believe, then omniscience traditionally construed is likewise incoherent. Fortunately there is at least one account of omniscience -- the revisionary account advanced in my comments on Gale -- that incurs none of these Cantorian troubles yet allows God's knowledge to be suitably supreme in kind and extent. We can thereby avoid the swamp of technical disputes raised by talk of all of what an omniscient being is supposed to know. Theism need not be made hostage to the outcome of such arcane wrangling.

1. Gale (2003), Pruss (2003); Post (2003); Gale (1991).

2. Cf. Post (2002), which also considers cases where there is a tie between two or more survivors.

3. As is the aim according to Gale and Pruss (1999), 471.

4. Cf. Post (2003), 48 and n21 to p. 41.

5. Again cf. Post (2003), 48 and n21 to p. 41.

6. Cf. Post (1987), Ch. 8.

7. In Post (2003) I argue that sentences are at least theologically implausible for this role.

8. Pruss's (3a) actually reads, "There are more sentential properties than true sentences." But I assume this is a printing error, since it conflicts with his own replacement instruction and with what he says in his next paragraph but one. Cf. Pruss (2003), 52-53.

9. I am greatly indebted to Pruss for calling attention to an error in the draft passage replaced by the following paragraph. I have also made needed adjustments above.


REFERENCES

Gale, Richard M. (1991). On the Nature and Existence of God (Cambridge: Cambridge University Press).

Gale, Richard M. (2003). "A Response to My Critics," Philo, 6, 132-165.

Gale, Richard M., and Pruss, Alexander R. (1999). "A New Cosmological Argument," Religious Studies, 35, 461-476.

Grim, Patrick (1991). The Incomplete Universe: Totality, Knowledge, and Truth (Cambridge: MIT Press).

Grim, Patrick, and Plantinga, Alvin (1993). "Truth, Omniscience and Cantorian Arguments: An Exchange," Philosophical Studies, 71, 267-306.

Post, John F. (1987). The Faces of Existence: An Essay in Nonreductive Metaphysics (Ithaca: Cornell University Press, 1987).

Post, John F. (2002) Précis to Minimal Epistemology: Beyond Terminal Philosophy to Truth, available on line at Minimal Epistemology: Beyond Terminal Philosophy to Truth.

Post, John F. (2003). "Omniscience, Weak PSR, and Method" Philo, 6, 33-48. Preprint available at Omniscience, Weak PSR, and Method.

Pruss, Alexander R. (2003). "Post's Critiques of Omniscience and of Talk of All True Propositions," Philo, 6, 49-58.