Three ontological arguments
April 2, 2012 — 8:44

Author: Alexander Pruss  Category: Existence of God  Tags: , , , ,   Comments: 32

The first and third arguments use S5. I will leave filling in the steps in the arguments as an exercise (maybe not so easy in the case of A) for the reader, though I can help out as needed.

Argument A (in a paper I have in Szatkowski’s forthcoming anthology on ontological arguments):

  1. Necessarily, if a property B is limiting, so is any property A that entails B.
  2. Necessarily, if a property B is limiting, its negation is not limiting.
  3. Possibly lacking existence is limiting.
  4. Possibly lacking omniscience is limiting.
  5. Possibly lacking omnipotence is limiting.
  6. Possibly lacking perfect goodness is limiting.
  7. Possibly not being creator of everything else is limiting.
  8. It is not possible that x is a creator of y while y is a creator of x.
  9. So, there exists a necessary being that is essentially omniscient, omnipotent, perfectly good and creator of everything else. This being has every property that it would be limiting to possibly-lack.

Argument B:

  1. Every first-order truth is knowable.
  2. The conjunction of all basic first-order truths exists and is a first-order truth.
  3. If all the basic first-order truths of a world w1 hold at a world w2, then w2=w1.
  4. Necessarily, if someone knows p, then p is true.
  5. So, there actually is a being that knows the conjunction of all basic first-order truths.

I don’t have an account of “basic”. Perhaps fundamental will do. I am thinking of “basic” here as a placeholder for a notion that makes (11) and (12) true.

Argument C:

  1. Possibly, an unlimited being exists.
  2. Necessarily, for every proposition q that is possibly true, there is a state of affairs p(q) such that p(q) grounds the possibility of q.
  3. Necessarily, if s grounds the possibility of x not existing or the possibility of x being limited, then s limits x.
  4. Necessarily, nothing limits an unlimited being.
  5. So, there is an unlimited being.
  • I have a question about argument B, I feel like I am missing something. I don’t have a deep background on formal logic, so it’s possible I am missing the implication of one of the premises.
    How is 14 not affirming the consequent on 13? It appears that 10 and 11 show that the conjunction of all first order facts is knowable, and 12 shows that worlds with the same first order facts are equivalent. 13 is true necessarily, but it appears that 14 requires p to be known, not just to be knowable.

    April 2, 2012 — 17:19
  • While I listed all the premises, there were a number of intermediate steps left as an exercise to the reader.
    Here they are. Let w1 be the actual world. Let p be the conjunction of all basic first-order truths; this exists and is a first-order truth (11). There is a possible world, w2, where someone knows p (10). p is true at w2 (13). w2 = w1 (12). Since at w2, someone knows p, and w2 = w1, and w1 is the actual world, it follows that actually someone knows p. QED

    April 2, 2012 — 20:00
  • Clayton Littlejohn

    Hi Alex,
    A question about B and about (12). Not sure this is totally worked out in my head yet, but I had a vague worry that you might help me put to rest.
    Suppose there are two worlds w1 and w2 where all the basic first-order truths are the same. Could there be some truth that distinguishes these worlds?
    In keeping with (12) you might say ‘No’. Perhaps we should say that the rest of the truths should strongly supervene upon these and we’ll accept the identity of indiscernibles when it comes to worlds.
    Fair enough. But what about the following truth?
    (O*) There exists an omniscient being.
    This doesn’t seem to be the kind of thing that would supervene upon the basic first-order truths (unless (O*) is itself among the basic first-order truths, in which case (O*) supervenes upon itself). Intuitively, I wouldn’t have thought that (O*) would count as a basic first-order truth. Moreover, it doesn’t seem to be a derivative truth (given some plausible assumptions about what an omniscient mind has to be like). If (O*) could be contingently true (why not?), then perhaps (O*) is a counterexample to (12). There could be two worlds perfectly alike in terms of their basic first-order truths that differ wrt to the existence of an omniscient being.
    On the other hand, if you’re up front about the fact that (O*) is one of the truths you’re including in the lot of basic-first order truths, the proof is less surprising.

    April 3, 2012 — 5:14
  • Clayton Littlejohn

    Sorry, one more question. Maybe other readers all know the answer to this question, but can you say a little something about what you take ‘unlimited’ to mean in A and C? I don’t understand why lacking perfect goodness would be limiting and have a hard time understanding why a being that is only contingently omnipotent is actually limited.

    April 3, 2012 — 5:20
  • Since we move from basic truths to a total description of reality wouldn’t basic truths be truths that are not (cannot?) be formed by a conjunction?
    This, though, reminds me of David W Miller’s arguments surrounding language dependency in theories of verisimilitude. Pavel Tichy sketched a very simple theory; two prisoners are guessing at the weather. One says “it’s dry,still and cool”, the other says “it’s rainy, windy and cool”. As it’s rainy, windy and hot it looks like the first chap has scored a blank whislt the second is almost there. “Not so”, says Miller all we have to do is translate the statements into another language and both men can say the same thing and the second draws a blank. Miller’s language has the predicates “Minnesotan” (hot iff rainy) and Arizonan (hot iff windy). The actual weather is hot, Arizonan and Minnesotan, the second prisoners guess is “Cool, not Minnesotan and not Arizonan.”
    What’s all this verisimilitude **** got to do with the matter at hand? Well statements can be made in Millerese that are conjnctions in Tichyoisse and vice versa. Maybe you’re having trouble with an account of “basic” because the concept of a basic truth is a bit dodgy.
    (If anyone fancies frying their brain with versimilitude, and it does fry the brain, there’s a good article on it in the Stanford Encyclopedia. Tichy’s theory and Miller’s reply were in the same issue of British Journal for the Philosophy of Science- 25 (2))

    April 3, 2012 — 7:34
  • Marcin

    Argument D:
    1. Possibly, UB doesn’t exist.
    2. If UB exists, it necessarily exists (possibly lacking existence is limiting)
    3. So, UB doesn’t exist.

    April 3, 2012 — 8:18
  • Clayton:
    As for limiting, I don’t really have much more to say on it. Not having much to say on the primitive notions is, as Graham Oppy has pointed out, one of the weaknesses of Goedelian arguments. Lacking perfect goodness, though, does seem to limit one morally–it limits one away from perfection. I don’t know if “limits away from perfection” is a gloss that helps.
    I like your objection to (12). Now, I think it’s plausible that knowledge facts supervene on first-order facts and mental states. If naturalism holds, then mental states supervene on first-order facts (since all physical facts are first-order). And I want the “basic” first-order facts to be such that all first-order facts supervene on them. So I can save the argument by assuming naturalism. 🙂 Maybe the argument does something interesting ad hominem then, but I don’t want to go that route.
    But instead of naturalism, I could just suppose directly that mental states supervene on first-order facts. Is such a supervenience claim true? I am not sure. So that’s a definite weakness of the argument. I worried about this when I wrote the argument, and I am more worried now.

    April 3, 2012 — 8:41
  • Clayton Littlejohn

    Hi Alex,
    This is all great fun, so thanks for putting these up (and for putting up with my questions).
    Here’s a question about limiting that troubles me. Let’s say that x is necessarily F iff x is F in all possible worlds. Let’s say that x is safely F iff x is F in all nearby worlds. If I’m evaluating an agent’s moral character, it’s not obvious to me that it’s better from the moral point of view to be necessarily honest than to be safely honest. A creature whose very essence precludes lying doesn’t seem much better (morally speaking) than a creature who could lie but has forged the kind of character such that they wouldn’t lie except in far off possibilities where they have a very different character.
    If safely having some superlative feature isn’t worse (on the relevant scale) than having that feature essentially (and, indeed, I would say in the moral case, being safely honest strikes me as _better_ than being necessarily honest), I fear that the argument is unsound. Possibly lacking omniscience, omnipotence, perfect-goodness, etc. wouldn’t be limiting per se.
    Would a modified version of the argument go through if a being is no more limited for being safely F than it would be for being necessarily F?

    April 3, 2012 — 15:17
  • Clayton:
    I do agree that this is fun. I don’t take ontological arguments too seriously, though I do think they raise the credence of their conclusions a little.
    Safety seems to come in degrees, depending on how close “close” is. Maybe greater safety is better? Sam would remain honest despite any physically possible torture. Sally would remain honest even given physically impossible tortures. Maybe Sally’s honest is better?
    In your examples you mention creatures. Maybe intuitions are different, though, in the case of beings that have aseity? My intuitions about free will become different in the case of beings with aseity.
    So in a creature, it might be better to be accidentally rather than essentially honest. But that’s compatible with a lack of essential honesty being limiting. Given that one has a limiting property P, there will be cases where it is better to also have another limiting property Q rather than to have the negation of Q instead. For instance, ignorance of mathematics is limiting. By 1, it follows that knowing that you are ignorant of mathematics is limiting, since it entails a limit. But if you have ignorance of mathematics, it is better that you know that you are ignorant of mathematics.
    That said, I do find 3-5 and 7 to be more plausible than 6. Without 6, the argument still gives an essentially omniscient, omnipotent creator.
    Here’s a reason why possibly lacking omniscience is limiting: If x possibly lacks omniscience, then either x could be bad at logic or there is something that is possibly true but that x couldn’t possibly know. For suppose that at world w, x lacks omniscience. There are two ways that x could lack omniscience at w: x could fail to know a truth p or x could believe a falsehood q. Suppose first that there is a truth p that x fails to know. Then either x cannot know (p is true and w is actual) or x can know it. If x cannot know it, then there is a possible truth that x can’t possibly know. If x can know it, then x does know it at w, and so x is bad at logic at w, since he knows (p is true and w is actual) but doesn’t know p. Suppose that at w, x knows every truth but isn’t omniscient. Then x believes some falsehood q. But then x knows ~q. So x believes q though he knows ~q. So x is bad at logic at w.
    Here is a reason why possibly lacking omnipotence is limiting. Suppose x lacks omnipotence at w, so there is some action A that x can’t do at w, and the action is the sort of action we’d expect an omnipotent being to be able to do. But then x at our world lacks some kind of a power, too, namely the power to do A after actualizing w.
    Here is a reason why possibly not being creator of something is limiting. For essentiality of origins is true. 🙂 So if at w, x isn’t creator of y, then x can’t be creator of y. But being unable to be creator of y is limiting.
    You could run an argument with safety in place of necessity, provided that we had a collection W of worlds that includes the actual world and is such that (a) safety is truth at all the worlds in some collection W and (b) we get to say that any property that W-entails a limiting property is limiting. (P W-entails Q iff any entity x in a world w in W that has P at w also has Q at w.)

    April 3, 2012 — 20:47
  • Oops: revision to last argument. This would work only if (a) was safely true, i.e., true at all worlds in W. That’s needed for safety to satisfy S5. But I doubt that that would in fact be the case. I imagine that safety in a world “on the edge” of W would be something other than safety at all worlds in W.

    April 3, 2012 — 20:49
  • Dianelos Georgoudis

    It seems to me that when thinking about God’s omniscience, or in general when thinking about free agents, one should also take into account what a person wants. So, for example, instead of “If x can know it, then x does know it” one should use “If x can know it, and x wants to know it, then x does know it”.
    I have two general questions about modal arguments. The first concerns free agents again. Consider the proposition “Necessarily, Obama (i.e. the current US President) is married to Michelle”. This would appear to be a false statement; after all that Obama is married to Michelle appears to be a contingent fact. On further thought though, in any possible world in which Obama did not choose to marry Michelle, that person wouldn’t be Obama. A free agent is partially defined by their choices. Thus it appears to be a necessary truth that Obama is married to Michelle. To put it even more starkly, an atheist Julian of Norwich is not Julian of Norwich. An Albert Schweitzer who did not go to work in Africa is not Albert Schweitzer. If this is right then it greatly complicates the use of modal logic in philosophical arguments about persons.
    The second question concerns an ambiguity about the meaning of modal terms, for one can understand them both logically and metaphysically. If one understands them logically then it seems to me that modal logic is useless in philosophy, because since the empty world (a world in which nothing exists) is logically possible all necessary propositions are not true. For example “necessarily, 2+2=4” is not true, for there exist no numbers in the empty world and thus the proposition “2+2=4” is rendered meaningless in it. But if one understands modal terms metaphysically, i.e. as referring to possible states of the actual world, then it would seem that ontological arguments for God become impossible. For suppose that the actual world is naturalistic. Then all metaphysically possible worlds are naturalistic too, and therefore naturalism is necessarily true. So it would seem that for ontological arguments to get off the ground one should assume that the actual world is not naturalistic, thus begging the question.

    April 4, 2012 — 0:08
  • John Alexandder

    HI Alex
    Been a while:-)
    Does not 3 imply that existence is a predicate? If so, can’t one employ Moorean type arguments and maintain that existence is not a predicate and therefore not limiting? How can something that does not exist be limited – there is nothing to limit.
    If 3 is not warraned then it seems that 9 is also not warranted.

    April 5, 2012 — 17:29
  • I think 3 only implies that “possibly lacks existence” is a predicate. Which it is. 🙂

    April 5, 2012 — 19:40
  • OEP

    It seems to me that Argument A and C beg the question of existence by defining non-existence as a limit.
    Also Argument B affirms the consequent.
    Even assuming that Necessarily, if someone knows p, then p is true, it does not follow that if p is true, someone knows p.

    April 7, 2012 — 13:34
  • OEP:
    How do you understand “begging the question”? Apart from the obvious case where the conclusion is a premise in the argument, it is far from clear what “begging the question” means. The best account I’ve seen is that an argument begs the question provided that it includes a premise whose justification, for all/most/typical reasonable agents, depends on the conclusion. But I don’t see why thinking that possibility-of-not-existing is limiting depends on theism for its justification.
    As for the affirming of the consequent, that would certainly be a fair accusation if 13 were the only premise in the argument. But once you add 10-12, the argument becomes valid, as you can see from my first comment in this thread.

    April 7, 2012 — 14:30
  • OEP

    Mr. Alexander Pruss:
    My understanding is that begging the question is the fallacy of using some property of the conclusion as a premise to prove the conclusion.
    My objection to both A and C are as follows.
    Argument A:
    Possibly lacking existence is limiting.
    Therefore an unlimited being exists.
    Argument C:
    Not existing is a limit
    Therefore an unlimited being must exist.
    Don’t both arguments then presume that unlimited conceptually encompasses existence as an assumed property?
    I could also dispute the premist that not existing is a limit. I think given the binary possibility space of existing/non-existing, assigning only one of the two to a being essentially is a limit.
    As to Argument B:
    Thank you for the clarification.
    There is a possible world, w2, where someone knows p (10). p is true at w2 (13). w2 = w1 (12). Since at w2, someone knows p, and w2 = w1, and w1 is the actual world, it follows that actually someone knows p.
    Wouldn’t the definite existence of a being that knows p itself be a first order truth and thus w2 (with a being knows p) is not equivalent to w1?

    April 7, 2012 — 22:18
  • 1. No, that’s not how arguments A and C work. There is a lot more relevant detail. One cannot logically go directly from “Possibly lacking existence is limiting” to “Therefore an unlimited being exists”. You’d need an additional premise, like “Possibly an unlimited being exists”.
    2. As for Argument B, actually that x knows p is a second-order truth (because it’s a truth about attitudes towards propositions), but I am not sure it matters.

    April 8, 2012 — 8:19
  • OEP

    Thank you for the clarification. I am enjoying this a lot although I have little experience with this, as must be apparent.
    Regarding Argument A and C
    It seems possible to attack premises 3 through 7 then.
    Possibly lacking existence is limiting.
    Existence and nonexistence are binary states. Being confined to either is limiting.
    Possibly lacking omniscience is limiting.
    Being unable to not know is limiting
    Possibly lacking omnipotence is limiting.
    omnipotence and impotence and the variable degrees of potence are all possibilities. Being limited to one is limiting
    Possibly lacking perfect goodness is limiting.
    Perfect goodness limits behavior.
    Possibly not being creator of everything else is limiting.
    Does not creating everything else limit you from not creating everything else?
    Fundamentally doesn’t any attempt at definition inherently impose limits? It seems that the only possibility for an unlimited being is to be undefined.
    Regarding Argument B
    x knows p may be a second order truth, but isn’t the existence of x a first order truth?

    April 8, 2012 — 12:17
  • Re A and C:
    Yes, of course, the other premises are subject to discussion. I don’t share your intuition that perfection is limiting, though.
    Re B:
    Presumably, yes. But it wasn’t assumed that this being does not exist at w1.

    April 8, 2012 — 13:41
  • Marcin

    Hi, Alex
    1. But being unable to eat a banana is limiting, isn’t it? If so, an unlimited being cannot possibly be essentially nonphysical or metaphysically simple.
    2. Is there any compelling reason to think that “Possibly, an unlimited being doesn’t exist” is conceptually/metaphysically more problematic than “Possibly, an unlimited being exists”? I can quite easily conceive of a world containing only limited beings (e. g. abstract objects or persons limited in their powers). And if (D-1) is question-begging, (C-1) is surely also question-begging 🙂

    April 8, 2012 — 16:39
  • CliveStaples

    Isn’t there trivially a possible world not containing an unlimited being, viz. the possible world containing all and only a sphere? Or the empty possible world?
    Are these not valid possible worlds?

    April 8, 2012 — 18:52
  • CliveStaples

    Isn’t there trivially a possible world that contains no unlimited beings, viz. the possible world containing all and only a sphere? Or the empty possible world?
    Are these not valid possible worlds?

    April 8, 2012 — 18:57
  • CS:
    That an empty world is possible is dubious. After all, plausibly, if there were nothing, it would be true that there is nothing, and hence the proposition that there is nothing would still exist.
    As to the sphere, it is far from clear that uncaused contingent beings are possible.
    1a. Answer 1: An unlimited being can eat a banana. It’s no harder to eat a banana than to eat a piece of fish, and he actually ate a piece of fish. Of course, God can only eat a banana qua human (or qua having some other nature that he can take on and that is capable of banana consumption). But that’s no limitation–when we eat bananas, we eat them qua humans.
    1b. Answer 2: Maybe being capable of eating qua having one’s primary nature is limiting, since eating is tied to nutrition, and being the sort of being that is capable of nutrition is limiting.
    2. Yes, you can run this parallel argument. I think claims of the possibility of existence are less problematic than claims of the possibility of non-existence. See this paper and section 2.3.1 of my “Leibnizian cosmological arguments” paper (that section is also relevant to CS’s query).

    April 9, 2012 — 14:21
  • Helen De Cruz

    Hi Alex, w/r/t the banana issue. I would like to think an ontological argument would not have to rely on the Incarnation as an auxiliary hypothesis (suppose one were to formulate the argument within kalam Islamic philosophical theology). Within such theological frameworks, God could in principle take on any nature, since if he could not, that would be a clear limitation. (A developmental psychologist friend of mine once asked as part of her unpublished fieldwork to young children if there was anything God couldn’t do. Many of them (I think they were Jewish kids) replied “he can’t do cartwheels and stuff like that.

    April 9, 2012 — 16:18
  • Marcin

    Thanks for your response, Alex
    1. It seems to me that if x can eat a banana, it is possible for x to be physical or to have a physical proper part. If this is correct, no essentially nonphysical, metaphysically simple being can eat a banana (according to some prominent theistic philosophers like R. G. Swinburne, God is essentially nonphysical and metaphysically simple).
    2. So “Possibly, an uncreated non-person doesn’t exist” is more problematic than “Possibly, an uncreated non-person exists” 🙂

    April 9, 2012 — 20:31
  • Marcin:
    1. A divine person qua God cannot be physical, but a divine person can additionally have a non-physical nature. Swinburne does believe in the Incarnation, so he has to say something like that. I haven’t looked at what Swinburne says about the Incarnation, though. So a divine person qua God is necessarily non-physical and metaphysically simple.
    Now, when we use “God” as the subject in a sentence, we can read that as including an implicit “qua God” qualifier, just as when we say “The basketball player is good”, we mean that she is good qua basketball player. So we can say that God is necessarily nonphysical. But this is elliptical for a qua claim.
    I work out a semantics for these qua claims in a paper I am working on. Email me if you’d like a copy.
    2. Well, there is that negation in the “non” which I think makes both a bit more problematic. I think negative qualities, like negative existentials, are more problematic.

    April 10, 2012 — 8:37
  • Helen:
    I am not relying on the Incarnation having occurred, but on the metaphysical possibility of an incarnation. Of course, Jews and Muslims are very likely to deny not just the actuality but also the possibility of an incarnation, but there is a difference here.
    Alternately, one could take this as a positive argument for the possibility of an incarnation. Jews and Muslims agree that God is an unlimited being. But an unlimited being can eat a banana. That’s only possible if an incarnation is possible. So an incarnation is possible. 🙂
    Or one can disjoin my 1a with my 1b.
    Plus, it’s not like my argument relies on the possibility of an incarnation as a premise. I just used the possibility of an incarnation as a response to an objection. It may well be that although arguments for the existence of God can avoid relying on theological premises, responses to atheistic objections perhaps sometimes do need to rely on theological premises or at least on their possibility. For instance, a response to the problem of evil might require relying on the possibility of a great good like Christ’s sacrifice for our sins that logically presupposes evil.

    April 10, 2012 — 8:43
  • Dianelos Georgoudis

    “That an empty world is possible is dubious. After all, plausibly, if there were nothing, it would be true that there is nothing, and hence the proposition that there is nothing would still exist.”
    I do not hold that a proposition which is true in a possible world exists in that world. Perhaps your sense of “existence” is different than mine.
    In any case please consider this: For a proposition to be true in a possible world, the concepts used by that proposition must first be meaningful in that world. But since nothing exists and nothing can potentially exist in an empty world, the concept of “existence” makes no sense in it. Therefore the proposition “nothing exists” is meaningless (and thus neither true nor false) in the empty world. That proposition is only true in the actual world where we discuss the empty world.
    Or let me put this way: With Plantinga I hold that a possible world is defined by its “book”, i.e. the set of propositions that are true in that world. The empty world is defined by the empty set. Since the empty set of propositions exists and is provably free of internal contradictions, the empty world is provably possible in the logical sense.

    April 10, 2012 — 10:07
  • Joshua Rasmussen

    These are clever arguments. I found A to be the most plausible, B next, and C last. The crucial (least plausible/obvious) premise of each argument, as I see it, are these: A.3, B.12, C.15 (or C.16).
    Actually, A.3 strikes me as quite plausible; I only looked at it more critically once I saw that it (in combination with (1) and (2)) is the key to deriving the possibility of a necessary being. (I found (1) and (2) utterly obvious, but if there’s doubt, you could treat them as stipulations on the term ‘limiting’.)
    To get around Clayton’s (very insightful) worries about B.12, you could treat (11) and (12) as implicit definitions of the term ‘first order truth’. Then the argument will hinge upon B.10. I worry that B.10 can be defeated by the prima facie equally plausible premise that every first-order truth is such that it could fail to be known. This is the modal parity problem.
    C.15 seems to me to be vulnerable to defeat by Martin’s Argument D above. If an unlimited being seems possible to one, it might seem equally possible that there is no unlimited being. But if argument C is valid, then you can’t have BOTH possibilities–hence the we should be agnostic about both (without independence evidence). This is the modal parity problem again.
    Argument A doesn’t face the modal parity problem, and I confess that each of it’s premises did strike me as prima facie plausible (perhaps even obvious) when I read it. It took me a little while to see how the conclusion follows from the premises, but I think that is actually an advantage of the argument (because the argument is not likely going to be question-begging). So, of the 3 arguments, I suggest that A has the most going for it, and that the most is more than nothing.

    April 16, 2012 — 10:21
  • Hi Alex,
    Interesting arguments! The first premise of argument B (10) is closely related to the first premisse of the alternative rendering of my modal-epistemic argument for the existence of God that was presented and discussed some time ago on this forum. Therefore, I’m quite interested in how you would defend your premise that every first-order truth is knowable. Further, with respect to argument A, I didn’t succeed yet to arrive at an adequate derivation of the conclusion (9) from your premises (1)-(8), so perhaps you can help me out here? Thanks!

    April 22, 2012 — 10:30
  • Emanuel,
    Argument A is in a forthcoming paper. Email me for a copy. It’s not that easy.
    In Argument B, all I have in favor of 10 is that it seems plausible.

    April 22, 2012 — 16:18
  • Josh:
    Regarding C, we can at least use it to give a prior of at least 1/2 to the existence of an unlimited being (that’s an Al Plantinga point in the setting of his ontological argument): That there could be no unlimited being is no more plausible than that there could be an unlimited being. (Actually, generally, I take possibility claims about non-existence to be less probable than similar possibility claims about existence, because non-existence is harder to imagine than existence.)
    But once we have a prior of 1/2, we can’t help but take Pascal’s argument seriously. Moreover, fairly modest probability-raising theistic arguments can then add up to a significant probability of theism.

    April 25, 2012 — 11:40