Truth, Omniscience, and Cantorian Arguments: An Exchange Philosophical Studies 71: 267-306, 1993 Alvin Plantinga and Patrick Grim Introduction (Grim): In "Logic and Limits of Knowledge and Truth" (No–s 22 (1988), 341-367) I offered a Cantorian argument against a set of all truths, against an approach to possible worlds as maximal sets of propositions, and against omniscience.[1] The basic argument against a set of all truths is as follows: Suppose there were a set T of all truths, and consider all subsets of T --all members of the power set T. To each element of this power set will correspond a truth. To each set of the power set, for example, a particular truth T1 either will or will not belong as a member. In either case we will have a truth: that T1 is a member of that set, or that it is not. There will then be at least as many truths as there are elements of the power set T. But by Cantor's power set theorem we know that the power set of any set will be larger than the original. There will then be more truths than there are members of T, and for any set of truths T there will be some truth left out. There can be no set of all truths. One thing this gives us, I said, is "a short and sweet Cantorian argument against omniscience." Were there an omniscient being, what that being would know would constitute a set of all truths. But there can be no set of all truths, and so can be no omniscient being. Such is the setting for the following exchange.[2] 1. Plantinga to Grim My main puzzle is this: why do you think the notion of omniscience, or of knowledge having an intrinsic maximum, demands that there be a set of all truths? As you point out, it's plausible to think there is no such set. Still, there are truths of the sort: every proposition is true or false (or if you don't think that's a truth, every proposition is either true or not-true). This doesn't require that there be a set of all truths: why buy the dogma that quantification essentially involves sets? Perhaps it requires that there be a property had by all and only those propositions that are true; but so far as I can see there's no difficulty there. Similarly, then, we may suppose that an omniscient being like God (one that has the maximal degree of knowledge) knows every true proposition and believes no false ones. We must then concede that there is no set of all the propositions God knows. I can't see that there is a problem here for God's knowledge; in the same way, the fact that there is noset of all true propositions constitutes no problem, so far as I can see, for truth. So I'm inclined to agree that there is no set of all truths, and no recursively enumerable system of all truths. But how does that show that there is a problem for the notion of a being that knows all truths? 2. Grim to Plantinga Here are some further thoughts on the issues you raise: 1. The immediate target of the Cantorian argument in the No–s piece is of course a set of all truths, or a set of all that an omniscient being would have to know. I think the argument will also apply, however, against any class or collection of all truths as well. In the No–s piece the issue of classes was addressed by pointing out intuitive problems and chronic technical limitations that seem to plague formal class theories. But I also think the issue can be broached more directly--I think something like the Cantorian argument can be constructed against any class, collection, or totality of all truths, and that such an argument can be constructed without any explicit use of the notion of membership.... 2. I take your suggestion, however, to be more radical than simply an appeal to some other type of collection 'beyond' sets. What you seem to want to do is to appeal directly to propositional quantification, and of the options available in response to the Cantorian argument I think that is clearly the most plausible. In the final section of the No–s piece, however, I tried to hedge my claim here a bit: "Is omniscience impossible? Within any logic we have, I think, the answer is 'yes'." The immediate problem I see for any appeal to quantification as a way out--within any logic we have--is that the only semantics we have for quantification is in terms of sets. A set-theoretical semantics for any genuine quantification over all propositions, however, would demand a set of all propositions, and any such supposed set will fall victim to precisely the same type of argument levelled against a set of all truths. Within any logic we have there seems to be no place for any genuine quantification over 'all propositions', then, for precisely the same reasons that there is no place for a set of all truths. One might of course construct a class-theoretical semantics for quantification. But if I'm right that the same Cantorian problems face classes, that won't give us an acceptable semantics for quantification over 'all propositions' either. Given any available semantics for quantification, then--and in that sense 'within any logic we have'--it seems that even appeal to propositional quantification fails to give us an acceptable notion of omniscience. What is a defender of omniscience to do? I see two options here: (A) One might seriously try to introduce a new and better semantics for quantification. I think this is a genuine possibility, though what I've been able to do in the area so far seems to indicate that a semantics with the requisite features would have to be radically unfamiliar in a number of important ways. (I've talked to Christopher Menzel about this in terms of my notion of 'plenums', but further work remains to be done.) I would also want to emphasize that I think the onus here is on the defender of omniscience or similar notions to actually produce such a semantics--an offhand promissory note isn't enough. (B) One might, on the other hand, propose that we do without formal semantics as we know it. I take such a move to be characteristic of, for example, Boolos' direct appeal to plural noun phrases of our mother tongue in dealing with second-order quantifiers. But with an eye to omniscience I'd say something like this would be a proposal for a notion of omniscience 'without' any logic we have, rather than 'within'. I'm also unsure that even an appeal to quantification without standard semantics will work as a response to the Cantorian difficulties at issue regarding 'all truths'. Boolos' proposal seems to me to face some important difficulties, but they may not be relevant here. More relevant, I think, is the prospect 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. 3. Consider for example an argument along the following lines, with regard to your suggestion that there might be a property had by all and only those propositions that are true: Consider any property T which is proposed as applying to all and only truths. Without yet deciding whether T does in fact do what it is supposed to do, we'll call all those things to which T does apply t's. Consider further (1) a property which in fact applies to nothing, and (2) all properties that apply to one or more t's--to one or more of the things to which T in fact applies. [We could technically do without (1) here, but no matter.] We can now show that there are strictly more properties referred to in (1) and (2) above than there are t's to which our original property T applies. The argument might run as follows: Suppose any way g of mapping t's one-to-one to properties referred to in (1) and (2) above. Can any such mapping assign a t to every such property? No. For consider in particular the property D: D: the property of being a t to which g(t)--the property it is mapped onto by g--does not apply. What t could g map onto property D? None. For suppose D is g(t*) for some particular t*; does g(t*) apply to t* or not? If it does, since D applies to only those t for which g(t) does not apply, it does not apply to t*. If it doesn't, since D applies to all those t for which g(t) does not apply, it does apply. Either alternative, then, gives us a contradiction. There is no way of mapping t's one-to-one to properties referred to in (1) and (2) that doesn't leave some property out: there are more such properties than there are t's. Note that for each of the properties referred to in (1) and (2) above, however, there will be a distinct truth: a truth of the form 'property p is a property', for example, or 'property p is referred to in (1) or (2)'. There are as many truths as there are such properties, then, but we've also shown that there are more such properties than t's, and thus there must be more truths than there are t's--more truths than our property T, supposed to apply to all truths, in fact applies to. This form of the Cantorian argument, I think, relies in no way on sets or any other explicit notion of collections. It seems to be phrased entirely in terms of quantification and turns simply on notions of truths, of properties, and the fact that the hypothesis of a one-to-one mapping of a certain sort leads to contradiction. It is this type of argument that leads me to believe that Cantorian difficulties regarding 'all truths' go deeper than is sometimes supposed; that the argument applies not only to sets but to all types of collections and that ultimately even quantification fails to offer a way out. 4. Let me turn, however, to another passage in your response: "Still, there are truths of the sort every proposition is true or false..." What the type of argument offered above seems to suggest, of course, is that there can be no real quantification over 'all propositions'. One casualty of such an argument would be any quantificational outline of omniscience. It must be admitted that another casualty would be 'logical laws' of the form you indicate. 5. By the way, it's sometimes raised as a difficulty that an argument such as the one I've tried to sketch above itself involves what appear to be quantifications over all propositions. I think such an objection could be avoided, however, by judiciously employing scare quotes in order to phrase the entire argument in terms of mere mentions of supposed 'quantifications over all propositions', for example. 3. Plantinga to Grim Let me just say this much. Your argument seems to me to show, not that there is a paradox in the idea that there is some property had by all true propositions, but rather that the notion of quantification is not to be understood in terms of sets. Your argument proceeds in terms of mappings, 1-1 mappings, and the like ("Suppose any way g of mapping t's one-to-one to properties referred to in (1) and (2) above..."); but these notions are ordinarily thought of in terms of sets and functions. Furthermore, you invoke the notion of cardinality; you propose to argue that "there are more properties referred to in (1) and (2) than there are t's to which our original property T applies"; but cardinality too is ordinarily thought of in terms of sets. (And of course we are agreeing from the outset that there is no set of all truths). I don't see any way of stating your argument non set theoretically. If we think we have to employ the notion of set in order to explain or understand quantification, then some of the problems you mention do indeed arise; but why think that? The semantics ordinarily given for quantification already presupposes the notions of quantification; we speak of the domain D for the quantifier and then say that '(z) Az' is true just in case every member of D has (or is assigned to) A. So the semantics obviously doesn't tell us what quantification is. Further, it tells us falsehood: what it really tells us is that 'Everything is F' expresses the proposition that each of the things that actually exists is F (and is hence equivalent to a vast conjunction where for each thing in the domain, there is a conjunct to the effect that that thing is F). But that isn't in fact true. If I say 'All dogs are good-natured' the proposition I express could be false even if that conjunction were true. (Consider a state of affairs á in which everything that exists in (the actual world) exists, plus a few more objects that are evil-tempered dogs; in that state of affairs the proposition I express when I say 'All dogs are good-natured' is false, but the conjunction in question is true.) The proposition to which the semantics directs our attention is materially equivalent to the proposition expressed by 'All dogs are good-natured' but not equivalent to it in the broadly logical sense. So I don't think we need a set theoretical semantics for quantifiers; I don't think the ones we have actually help us understand quantifiers (they don't get things right with respect to the quantifiers); and if I have to choose between set-theoretical semantics for quantifiers and the notion that it makes perfectly good sense to say, for example, that every proposition is either true or not-true, I'll give up the former. 4. Grim to Plantinga You point out that the argument I offered in terms of properties is still phrased using mappings or functions, one-to-one correspondences, and a notion of cardinality, and that these are ordinarily thought of in terms of sets. "I don't see any way," you say, "of stating your argument non set theoretically." I do. In fact I don't consider the argument to be stated set-theoretically as it stands, strictly speaking; it's a philosophical rather than a formal argument. In order to escape any lingering suggestion of sets, however, we can also outline all of the notions you mention entirely in terms merely of relations--properties applying to pairs of things--and quantification. I don't see any reason for you to object to that; you seem quite happy with both properties and quantification over properties generally. 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: AxAy[P1x & P1y & Az(P2z & Rxz & Ryz) -> x = y] & Ax[P1x -> EyA z(P2z & Rxz <-> z = y)]. 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): AxAy[P1x & P1y & Az(P2z & Rxz & Ryz) -> x = y] & Ax[P1x -> EyAz(P2z & Rxz <-> z = y)] & Ay[P2y -> Ex(P1x & Rxy)]. [3] 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. I'm not sure that we might not be able to do without even that--I'm not sure we couldn't phrase the argument as a reductio on the assumption of a certain relation, for example, without using any notion of cardinality within the argument at all. I don't agree, then, that the argument depends on importing some kind of major and philosophically foreign set-theoretical machinery. Notions of functions or mappings and one-to-one correspondences are central to the argument, but in the sense that these are required they can be outlined purely in terms of relations--or properties applying to pairs of things--and quantification. Cardinality, if we need it at all, can be introduced in a similarly innocuous manner. You also suggest several other reasons to be unhappy with a set-theoretical semantics for quantification, and end by saying that "if I have to choose between set-theoretical semantics for quantifiers and the notion that it makes perfectly good sense to say, for example, that every proposition is either true or not-true, I'll give up the former." There may or may not be independent reasons to be unhappy with set-theoretical semantics for quantifiers--I think the points you raise are interesting ones, and I'll want to think about them further. My immediate reaction is that the first point you make does raise a very important question as to what formal semantics can honestly claim or be expected to do, and I'm very sympathetic to the notion that it has sometimes been treated as something that it neither is nor can be. My guess is that the second issue might be handled in a number of ways familiar from different approaches to possible worlds, without any deep threat to set-theoretical semantics. But I could be wrong about that--as I say, I'll want to think about these questions further. Even if there are independent reasons to be unhappy with set-theoretical semantics, however, I think your final characterization of available options is off the mark. For reasons indicated above, I think sets aren't essential to the type of Cantorian argument at issue--the argument can for example be phrased entirely in terms of properties, relations, and quantification. If that's right, however, the basic issue is not one to be settled by some choice between set-theoretical semantics and, say, 'all propositions'. The problems are deeper than that: even abandoning set-theoretical semantics entirely, it seems, wouldn't be enough to avoid basic Cantorian difficulties. 5. Plantinga to Grim Right: we can define mappings and cardinalities as you suggest, in terms of properties rather than sets. We can then develop the property analogue of Cantor's argument for the conclusion that for any set S, P(S) (the power set of S) > S as follows. Say that A* is a subproperty of a property A iff everything that has A* has A; and say that the power property P(A) of a property A is the property had by all and only the subproperties of A. Now suppose that for some A and its power property P(A), there is a mapping (1-1 function) f from A onto P(A). Let B be the property of A such that a thing x has B if and only if it does not have f(x). There must be an inverse image y of B under f; and y will have B iff y does not have B, which is too much to put up with. But if P(A) exceeds A in cardinality for any A, then there won't be a property A had by everything; for if there were, it would have a power property that exceeds it in cardinality, which is impossible. So there won't be a property had by every object, and there also won't be a property had by every proposition. Hence if we think quantifiers must range over something, either a set or a property, we won't be able to speak of all propositions or of all things. But of course the Cantorian property argument has premises, and it might be that some of the premises are such that one is less sure of them than of the proposition, e.g., that every proposition is either true or not true, or that everything has the property of self-identity. In particular, one premiss of the Cantorian argument as stated is (à) For any properties A and B and mapping f from A onto B, there exists the subproperty C of A such that for any x, x has C if and only if x has A and x does not have f(x). This doesn't seem at all obvious. In particular, suppose there are universal properties--not being a married bachelor, for example, and suppose the mapping is the identity mapping. Then there exists that subproperty C if and only if there is such a property as the property non-selfexemplification--which we already know is at best extremely problematic. So which is more likely: that we can speak of all propositions, properties and the like (and if we can't just how are we understanding (à)?), or that (à) is true? I think I can more easily get along without (à). One final note. These problems don't seem to me to have anything special to do with omniscience. One who wants to say what omniscience is will have difficulties, of course, in so doing without talking about all propositions. But the same goes for someone who wants to hold that there aren't any married bachelors, or that everything is self-identical. If we accept the Cantorian argument, we shall have to engage in uncomfortable circumlocutions in all these cases, circumlocutions such that it isn't at all clear that we can use them to say what we take to be the truth. But the problem won't be any worse in theology than anywhere else. The best course though (I think) is to reject (à). 6. Grim to Plantinga I think your response to the Cantorian property argument is an interesting one. Here however are some further thoughts on the issue. Let me start with a reminder as to where we stand. The Cantorian property argument as you present it is as follows: Say that A* is a subproperty of a property A iff everything that has A* has A; and say that the power property P(A) of a property A is the property had by all and only the subproperties of A. Now suppose that for some A and its power property P(A), there is a mapping (1-1 function) f from A onto P(A). Let B be the property of A such that a thing x has B if and only if it does not have f(x). There must be an inverse image y of B under f; and y will have B iff y does not have B, which is too much to put up with. As this stands, of course, it is merely an argument that the power property P(A) of any property A will have a wider extension than does A. But if we suppose A to be a property had by all properties, or a property had by all things, we will get a contradiction. There can be no such property...or so the argument seems to tell us. The escape you propose here is essentially a denial of the diagonal property required in the argument. Given some favored universal property A and a chosen function f, what the argument demands is a property B 'that is a subproperty of A such that a thing x has B if and only if it does not have f(x).' But there is no such property. The argument demands that there is, and so is unsound. Or so the strategy goes. Somewhat more generally, the strategy is to deny any principle such as (à) that tells us that there will be a property such as B: (à) For any properties A and B and mapping f from A onto B, there exists the subproperty C of A such that for any x, x has C if and only if x has A and x does not have f(x). (à) "doesn't seem at all obvious," you say. "I think we can easily get along without (à)." I don't believe that things are by any means that simple. Here I have two fairly informal comments to make, followed by some more formal considerations: 1. As phrased above, I agree, (à) is hardly so obvious as to compel immediate and unwavering assent. The diagonal properties demanded in forms of the argument similar to yours above--properties such as 'B, a subproperty of A, the property 'being a property', that applies to all and only those things which do not have the property f(x), mapped onto them by our chosen function f'--may similarly lack immediate intuitive appeal. I think this is largely an artifact of the particular form in which you've presented the Cantorian argument, however. Yours follows standard set-theoretical arguments very closely, complete for example with a notion of 'power property'. The argument becomes formally remote and symbolically prickly as a result, and the diagonal property called for is offered in terms which by their mere technical formality may dull relevant philosophical intuitions. But the Cantorian argument doesn't have to be presented that way. It can, for example, be phrased without any notion of power set or power property at all--on this see "On Sets and Worlds, a Reply to Menzel" ....[4] When the argument is more smoothly presented, moreover, the diagonal constructed in the argument becomes significantly harder to deny. Consider for example an extract from a form of the argument that appeared earlier in our correspondence: Consider any property T which is proposed as applying to all and only truths. Without yet deciding whether T does in fact do what it is supposed to do, we'll call all those things to which T does apply t's. Consider further (1) a property which in fact applies to nothing, and (2) all properties that apply to one or more t's--to one or more of the things to which T in fact applies... We can now show that there are strictly more properties referred to in (1) and (2) above than there are t's to which our original property T applies... Suppose any way g of mapping t's one-to-one to properties referred to in (1) and (2) above. Can any such mapping assign a t to every such property? No. For consider in particular the property D : D: the property of being a t to which g(t)--the property it is mapped onto by g--does not apply. What t could g map onto property D? None... Consider also the following Cantorian argument: Can there be a proposition which is genuinely about all propositions? No. For suppose any proposition P, and consider all propositions it is about. These we will term P- propositions. Were P genuinely about all propositions, of course, there would be a one-to-one mapping f from P-propositions onto propositions simpliciter: a mapping f which assigns P- propositions to propositions one-to-one and leaves no proposition without an assigned P-proposition. But there can be no such mapping. For suppose there were, and consider all P-propositions p such that the proposition to which they are assigned by our chosen mapping f--their f(p)--is not about them. Certainly we can form a proposition about precisely these--using propositional quantification and 'Ab' to represent 'about', a proposition of the following form: Ap((Pp & Ab(f(p))p -> ... p ...) Consider any such proposition Pd. What P-proposition could f map onto it? None... In the first argument, I think, we have an eminently intuitive property: the property of being a t to which a corresponding property we've imagined does not apply. In the second, we have an eminently intuitive proposition. There are, it seems clear, P-propositions which won't have a corresponding proposition that happens to be about them. Isn't that itself a proposition that is about them? The general point is this. In order for a strategy of denying the diagonal to prove effective against all offending forms of the Cantorian argument, one would have to deny an entire range of properties and propositions and conditions and truths liable to turn up in a diagonal role. Some of these, I think, will have an intuitive plausibility far stronger than that of the formal construction you offer in your more formal rendition of the argument above. The diagonals at issue will always involve a function f or a relation R supposed one-to-one from one batch of things onto another. Such functions or relations alone, we've agreed, seem entirely innocent. But passages such as the following, from other imaginable Cantorian arguments, seem intuitively innocent as well: f is proposed as a mapping between known truths (or truths known by some individual G) and all truths. Some known truths will have a corresponding f-truth on that mapping that is about them. Some won't. Surely there will be a truth about all those that don't--the truth that they all are truths, for example. f is proposed as a mapping between a group G of properties and all properties. Some G-properties will have corresponding properties by f that in fact apply to them. Some won't. Consider all those that don't, and consider the property they thereby share... f is proposed as a mapping between (i) the things a certain fact F is a fact about and (ii) all satisfiable conditions. Some things F is about will thereby be mapped onto conditions they themselves satisfy. Some won't. Consider the condition of being something that has an f-correlate it doesn't satisfy... Each of these is the diagonal core of a Cantorian argument: against the possibility of all truths being known truths, against any comprehensive grouping of all properties, and against any fact about all satisfiable conditions. When passages such as these are offered step by step and in full philosophical form, I think, the truth, property, and satisfiable condition they call for are very intuitive. How, one wants to ask, could there not be such a truth, or such a property, or such a condition? I don't believe, therefore, that the situation is one in which I have a formal argument on my side and you have the intuitions on yours. Although somewhat complex, the Cantorian argument can be presented as a fully philosophical argument with significant intuitive force. I'm also willing to admit that there are at least initial intuitions that somehow truths should collect into some totality, or that there should be an 'all' to the propositions. What we seem to face, then, is a clash of intuitions. But it is a genuine clash of intuitions, I think, with genuinely forceful intuitions on both sides. 2. There is also a further difficulty. Consider again the basic structure of our earlier argument against a property had by