Saturday, September 24, 2005

Anselm and the Perfect Reductio

Anselm's infamous ontological argument effectively defines God into existence: We can conceive of the perfect being (that which none greater can be conceived) -- let's call it 'God'. Suppose (for reductio) that God does not exist. Then we can conceive of a greater being yet -- namely, one just like God but with the added virtue of existing. That would contradict our premise that we're conceiving of the greatest conceivable being (see our definition of 'God' above). So, on pain of contradiction, we must reject the supposition that God does not exist. Hence, God exists. So argues Anselm.

Now, this argument is obviously problematic. This is brought out by the fact that it would seem to commit us the existence of all sorts of perfect entities, as per the following schema:

1) We can conceive of the perfect X, for which no greater X can be conceived.
2) It is greater for an X to exist than not.
3) Suppose for reductio that the perfect X does not exist.
4) Then we can conceive of a greater X, namely, a twin of the perfect X that has the further virtue of existing.
5) This is a contradiction; thus (3) is false, i.e. the perfect X exists.

We can plug anything we want into the X placeholder. Anselm's argument goes through with X = 'being' (which then concludes with the existence of "the perfect being" = 'God'), but we might as well follow Gaunilo in putting X = 'island', and thus conclude, absurdly, that the perfect island must exist. So, we might think, the schema must fail, and Anselm's argument with it.

It isn't quite so simple, however. As was noted by Andrew in the comments at FQI, it isn't obvious that the (single, unique) perfect island is conceivable. Perhaps we can always imagine one slightly better, thus forming an infinite series with no upper bound. But that merely shows that 'island' is a poor choice for X -- it doesn't satisfy premise 1 -- but perhaps some other choice would do the trick. I'll return to this question in a moment.

First, I should explain what work this reductio is doing. Brandon has offered a couple of posts wherein he objects that Gaunilo's island objection (and my more general schema) fail to properly parallel Anselm's own argument. I think he's rather missing the point, and suggested as much in an unusually frustrating exchange in his comments section. Anyway, to clarify, here's my meta-argument:

(1') Absurd consequences follow from the conjunction of (1) and (2) in the schema above, for any X other than 'being'.
(2') Thus, for each such X, either (1) or (2) in the schema must be false.
(3') It seems implausibly strong to claim that 'the perfect X' is inconceivable for all such X.
(4') So it's most likely that (2) must be false instead, for some such X.
(5') If (4') is true, then it seems simplest and most plausible to take (2) as being universally false, i.e. false for all X.
(6') If (2) is false for all X, then Anselm's argument is unsound.
(C') Anselm's argument is probably unsound.

Put more loosely, the point of my reductio schema is to show that Anselm's argument can only survive if "the perfect X" is inconceivable for every X other than 'being'. (Okay, it's logically possible for (2) to be non-universally false, i.e. false for X = those conceivable perfect other-than-beings, and true for X = Anselm's perfect being. But this is implausibly ad hoc. Hence my premise (5').) That is a very strong claim, so Anselm's argument looks to be in trouble.

I should add that if you don't like (4') and (5'), we can replace them with the following:

(4") If (2) is true for X = 'being', then, given (3'), (2) is probably going to be true for some X other than 'being' for which 'the perfect X' is conceivable. By (1'), this will yield absurdities.
(5") Thus (2) is probably false for X = 'being'.
(6") Anselm's argument depends upon the truth of (2) when X = 'being'.
(C") Thus Anselm's argument is (probably) unsound.

Note that my meta-argument does not depend upon the sort of strict parallel that Brandon is criticising, so his objections are quite irrelevant. Indeed, the only point where I really depend upon the analogy is my premise (6'/"), but that one is surely uncontroversial, and unaffected by the sorts of nit-picky differences Brandon highlights. Anselm clearly relies on the claim that it's greater for a being to exist than not. Otherwise he wouldn't be able to reach the conclusion that the perfect being must exist.

Anyway, my main purpose here is to explain why the reductio has some significant rational force, contrary to Brandon's claim that it is merely "a clever bit of philosophical sleight-of-hand, useful for fooling those who don't take the trouble to analyze it, and nothing more."

It is illuminating because we can now see that the core issue is whether there is any other conceivable 'perfect X', for any X other than 'being'. We discussed this in the later comments over at FQI, and I presented some plausible candidates, e.g. X = "malevolent being". If the Anselmian responds by taking 'perfection' to be domain-general (so that the most perfect anything will always tend towards God's attributes: omnipotence, omniscience, and benevolence) then we can stipulatively define a domain-specific evaluative term to take its place in the reductio. Let "sperfect" =df "perfect according to the appropriate domain-specific criteria". Let us say that the evaluative criteria for malevolent beings are just the same as those for beings generally except that the criterion of goodness is replaced by that of evilness. It then follows, by the Anslemian logic of my reductio schema, that the perfectly evil being exists. This alteration won't affect my meta-argument, because Anselm still assumes that it's greater for a being to exist than not, and so it's sgreater for a being qua being to exist than not, and hence (2) must be true when X = 'being'. And the whole point of the reductio is to cast doubt on this latter claim.

As a final point, I'd note that you can have a lot of fun using Anselmian logic with domain-specific perfection. Over at FQI I discussed the conceivability of 'the perfect melody', evaluated against criteria which include its being actually accessible to me whenever I want to hear it. It follows from my reductio schema that the perfect melody really is actually accessible to me right now (given that I want to hear it)! If only...

Readers are welcome to leave a comment suggesting other interesting possibilities for defining perfection into existence. Extra credit if you use Anselm's own logic to prove that there is a perfect counterargument that will negate his own attempts! ;)


  1. You seem to be disputing the definition of perfect in a round about way. More precisely you are replacing their definition of perfect with another word - and if you can do that (why not I guess) then you can do anything for example

    A pink being for which no more pink exists.
    or a fat being for which one no more fat can be concieved.

    I like your last argument though - I will try with a fairly pure sort of a concept - "knowledge" (particularly knowledge in regards to anselm's arguments) excuse any gaping holes in my logic :)

    1) We can conceive of the perfect knowledge, for which no greater knowledge can be conceived.
    2) It is greater for us to possess this knowledge than not.
    3) Suppose for reductio that the perfect knowledge does not exist in us.
    4) Then we can conceive of a greater knowledge, namely, a twin of the perfect knowledge that has the further virtue of existing in us.
    5) This is a contradiction; thus (3) is false, i.e. You and I know all the answers already and dont need Anselm to help us out.

  2. Your argument can actually be put in a slightly stronger form: if the premises of Anslem's argument are true, then both a perfect being and a perfect malevolent being exist, both all-powerful, but absolute power isn't absolute power if it's shared. Therefore, the premises lead to self-contradiction and are not just probably, but definitely false.

  3. Thanks, Richard; this elucidates a great deal that wasn't clear in your previous responses, since, unless I simply missed it, this is the first time I've encountered where you've actually pointed out that you are trying to reduce (2) to absurdity rather than trying to run a reductio of the argument itself (as Gaunilo does).

    No absurd consequences follow from (1) and (2) as you have been defining 'perfect Y' (i.e., 'Perfect Y' is that Y than which no greater Y can be conceived). The only conclusion relevant to existence that can be derived from them is:

    (I) If any islands exist, that island than which no greater can be thought must be one of them.

    Even this only follows on the assumption that either existence is the only perfection or that existence is an overriding perfection, i.e., that no combination of other greatnesses are sufficient to override existence. (2) just tells us that at least one way for A to be greater than B is for it to exist; it doesn't tell us if there are other ways for A to be greater. That is, without the added assumption that existence is the only or overriding perfection, all it tells us about islands is that, of any two islands equal in all other respects, the greater island is the island that actually exists. When it is combined with (1), this doesn't give us anything without further assumptions about islands. For all it tells us, islands may have many ways of being greater than other islands, and that some islands that don't exist are greater than islands that do because their other ways of being greater outweigh their deficiency as to existence.

    If we add the assumption that existence is the only or overriding perfection, we get (I). (If no islands exist, and existence is simply an overriding perfection, existence wouldn't be required for any islands to be greater than any other islands. If no islands exist, and existence is the only perfection, no island is greater than any other.) But with this assumption (I) is trivially true: the assumption that existence is the only or overriding perfection means that all islands that actually exist are, ipso facto, the greatest islands that can possibly be thought.

    This applies generally to anything substituted for X.

    It was Gaunilo's reductio, which doesn't try to reduce (2) to absurdity, that I labeled a clever philosophical sleight of hand. Your objection, as you have clarified it here, just fails to be a reductio at all, because it doesn't produce any absurdities.

    I'm a little puzzled about your (4''), since just about any supporter of the ontological argument will deny it. What is the argument for it?

  4. On a more pragmatic note...

    Even if Anselm's argument is flawless, it doesn't really say much about God at all. Anselm shows the God exists, but only if we assume that "God"="the perfect being." What he really needs to prove, if his argument is to have any real significance, is that the Christian God exists ie. a good, benevolent male dressed in flowing robes, preferably white and preferably the Father of Jesus Christ, with one of those lovely gold haloes stuck to his head. Perhaps he went into this in more detail after he wrote the ontological proof.

  5. Brandon, what's wrong with the argument from (1) and (2), through (3) and (4), to the absurd conclusion of (5)? As before, you've merely asserted that the argument doesn't work, instead of actually engaging with it and showing what (if anything) is wrong with it. Maybe I'm missing something, but it seems pretty straightforward to me. If you're going to dispute this, then you really are obliged to engage with the argument as I've presented it. You seem to have completely ignored my steps (3) and (4).

    So, I've quite clearly shown how (1) and (2) together result in absurdity. All you've done is assert the opposite, without offering any relevant counterargument whatsoever.

    As for (4"), like (5') as explained in the main post, it would be implausibly ad hoc to deny. Do you really mean to suggest that it is greater for beings to exist than not, but not greater for any other entity (for which perfection is conceivable) to exist than not? That's just ridiculous. If it's greater for a being to exist than not, then it's surely also greater for, say, a malevolent being to exist than not.

  6. Richard,

    It's very clear that I didn't 'just assert' that the argument doesn't work; I went into some detail (more than two paragraphs worth). I suggested conclusions that follow and denied that other conclusions do, in a case for which you have provided no explicit argument here or (as far as I have been able to find) elsewhere. This is not 'just asserting' that the argument doesn't work; it is giving reasons for thinking that it wouldn't. If I'm really so off about these reasons, it should be child's play to show that the conclusions I'm suggesting follow don't follow, and that the conclusions I'm suggesting don't follow do.

    Contrary to (1'), (1) and (2) don't yield absurd consequences. Consider the following line of reasoning. (1) says 'That island than which no greater island is conceivable is itself conceivable'; (2) says that 'It is greater to exist than not to exist'. Together and alone they don't yield anything about islands of any sort; for all they tell us, as I noted, there might be other ways of being greater that a non-existing island would have (this is important for when we add the reductio supposition). If we add the reductio supposition that 'the island than which no greater island is conceivable' does not exist, the possible conclusions are (a) no islands exist; or (b) there is at least one island that exists that fits the description, 'that island than which no greater island is conceivable'; or (c) there is some other set of perfections besides existence, that in islands preclude existence, such that 'that island than which no greater island is conceivable' is greater than any island existing (because existence alone is not enough to overbalance the other set of perfections). For all the argument tells us, any of these three are possible. However, neither (b) nor (c) are absurd. So, quite clearly, you haven't shown anything to be absurd; indeed, it looks like all you've done is to assume that (1) and (2) will somehow run a parallel reductio to Anselm's. If that's the case, then, as I've pointed out, (1) is not parallel to Anselm's premise, and so one can't expect such a parallel. But you keep claiming that you are not trying to run such a parallel, so I don't know why you think (1) and (2) yield an absurdity. You haven't actually given any argument in which the conjunction of (1) and (2) clearly excludes both (b) and (c).

    As to (4''), this is a less important issue, since I was just curious about what sort of reasons are supporting it; but we know from fields like mathematics that unique solutions are in no way made less probable by the fact that they are unique; so there appears to be no a priori reason to accept (4''). This is particularly true given that one of the possibilities that have to be taken into account that no ontological argument can succeed except at the limit-case of things than which nothing greater can be thought. Your argument schema only gives this limit-case at Y='being'; you seem to be forgetting that there are two Y's in the formulation of (1), not one, and they are not performing the same function. One is identifying an object; the other is an integral part of a single unit that is performing the function of qualifying that object. When we remember that the two Y's are doing different things in the formulation, and that the second only achieves the Anselmian limit-case at Y = 'being', we can easily see that if it were to happen that (2) were only true at such limit-cases (when we are considering things than which nothing greater can be thought), your schema would yield one and only one such case, just from the way it is set up. In that case, the ad-hocness is built into the schema rather than the objection; it is the schema, not the objection, that creates the uniqueness.

    In any case the denial of (4'') doesn't commit anyone to saying either that it is only greater for beings (rather than other entities, whatever those might be) to exist than not, or that (2) is false in any case; it just commits one to skepticism about the rather odd attempt of (4'') to determine (however probabilistically) the truth value of (2) under Y= 'being' without any consideration of the actual case of (2) under Y = 'being'. Even someone who thought (2) is universally false could doubt the wisdom of denying a unique case in which (2) is true simply because of what is the case elsewhere. We know that the truth of some propositions is domain-relative; we know that the key formulation in this instance involves a limit-case; as far as I can see we have no clear reason to deny that some propositions may only be true of limit-cases. So, as a crude analogy, certain propositions may be true only for transfinites; one would be utterly unreasonable to say that such propositions are probably false because they are always false for finites. So what we would really need to know is whether this is a case of a proposition true only in a particular (relevant) kind of limit-case.

  7. "I suggested conclusions that follow and denied that other conclusions do, in a case for which you have provided no explicit argument here"

    I did provide an explicit argument, it's numbered from (1) to (5), and I'm still waiting for you to explain how anything you've written here applies to it. You deny that (1) and (2) entail the absurd conclusion of (5). To repeat, here is my argument for this entailment:

    3) Suppose for reductio that the perfect X does not exist.
    4) Then we can conceive of a greater X, namely, a twin of the perfect X that has the further virtue of existing.
    5) This is a contradiction; thus (3) is false, i.e. the perfect X exists.

    You keep ignoring this. Which step are you objecting to?

  8. To explain my "impatient tone", as you put it in the other comments thread, I'm frustrated with how you are approaching this dispute as a matter of form. I've provided a numbered premise-conclusion argument. The appropriate way for you to respond is either (a) explain which premise is false; or (b) explain which specific inference is logically invalid.

    It's that simple. You have not clearly done either of these. Until you engage with the specifics of my argument, I really don't see what else I can say to you.

  9. Richard:

    Since I have explicitly pointed out (twice) that the conjunction of (1), (2), and (3) doesn't yield a unique conclusion, but is able to yield different conclusions on different suppositions (which, if true, would, as you well know, mean that the argument is logically invalid with out adding further suppositions), I'm not sure I understand your complaint.

  10. Okay, re-reading your arguments, you suggest: "there is some other set of perfections besides existence, that in islands preclude existence, such that 'that island than which no greater island is conceivable' is greater than any island existing"

    This looks like a rebuttal of my premise (4). It would've been helpful to label it as such though, since I missed it on my first skim through. It certainly wasn't obvious to me how your comments were supposed to bear on my argument, hence the frustration.

    Anyway, back to the substantive debate, it seems pretty implausible that this could be the case in all the cases where you'd need it to be (i.e. not just 'islands' but all the other troublesome X's). I can't see why existence would necessarily preclude other positively-evaluable properties. So while my reductio might not be a strict logical refutation of Anselm's argument, I think it's still pretty strong.

  11. In a theoretical case Y=(100-x) lim x --> 0 % perfection
    where in some finite quantity Y/Z (where Z is a finite number >1) is dependant on being real (ie if it is not real then it disapears - the other component (Y-Y/Z) is things like the good (or perfection) god might create just as a result of being concievable as opposed to actually existing (logic suggests these are much smaller)
    Ie as x tends towards 0 (Z is static) then reality tends towards mattering more and more regardless of the example.

    now if there are only a small set of concievably real perfect islands this analysis may still fail BUT then if there is a non-existing island that is adding value in excess of a concievable existing one (for example lets say samoa) then we should know about it one would think otherwise it brings into question how it is more perfect(/good etc)

  12. I can just imagine some old collinist looking at black peopel and saying - they are not capable of developing a civilization even at the best of times.

    Also I am not sure your argumnt about xistance not overbalalcing the other areas works.
    lets say you have one island with 99.99 units of good/perfection and another with 99.98 (and it exists). it depends on what causes the perfection but the result is likely to be 0 and 99.98 because hte non existing island is unable to create perfect or good without existing.
    UNLESS you use a different definition of perfection wherein it is actualy more perfect to NOT exist because there wil almost certainly be a perfection/existing tradeoff and since value is carried across no existing always has the higher state of perfection.

  13. Actually, Anselm is easier to debunk than that :

    Consider 2:
    Then we can conceive of a greater being yet -- namely, one just like God but with the added virtue of existing.

    This contains the assertion that it is greater for God to exist than not.

    I disagree with this assertion, which can only be a value judgement : to me, and to many others, the existence of God would not be great, but terrible. The very idea gives me nausea.

    So, premise 2 is actually wish fulfillment, or circular argument : Anselm thinks it would be neat if God existed; therefore God exists for Anselm.

    Which is fine by me.

  14. Richard,

    Sorry about the lack of clarity on that point; I was mostly focusing on what one could draw from (1)-(3), so didn't think to use (4) explicitly as a way to focus the criticism. It might also help if I write shorter comments so the main point won't get lost in side issues!

    On the question of whether existence would often preclude other positively-evaluable properties, I rather suspect it's easy to show that cases are legion, although I could be wrong. Consider the following reasoning about islands:

    All islands, by nature, are such that they fall within the category of Effects: that is, they are necessarily effects of island-causing sets of causal factors (ICSCFs). Thus when we think of any island at all (including that island than which no greater island can be thought), we must implicitly be including in its conditions of possibility the existence of ICSCFs that are able to make that island actually possible. Thus, when we are considering whether there exists that island than which no greater island can be thought, we need to keep in mind the fact that the internal consistency of the claim, "That island than which no greater island can be thought actually exists" depends (given the nature of islands) on whether we can take for granted the existence of the ICSCFs that are entailed by the statement. [This is where (4) begins to be in question.] If the ICSCFs are not implicitly granted, the claim is inconsistent (and so the equally-great-island-except-also-existing is not, properly speaking, conceivable). If this reasoning is plausible for islands, it would be plausible for every Y that restricts the discussion to something that falls under the category of Effects. That would put most of the possible candidates for Y out of the contention.

    The Anselmian case, of course, doesn't make such a restriction, and so wouldn't be affected by the argument. (And the Anselmian could suggest that the reason why trying to run the argument with islands, etc., seems so straightforwardly absurd is precisely that it treats things that would have to be effects as if they weren't -- as if we could decide by reasoning that an effect must exist without considering whether the causes necessary for its existence do.)

  15. The silliness of this boggles me. There are just so many ways to show that it doesn't work.

    Start by substituting something we can prove does not exist for X, such as a largest integer (or largest prime.) This proof demonstrates that a largest integer exists, even though we could always add one. (Or add a pony.)

    The obvious fallacy is the erroneous parallelism of conceptions of things with the things themselves. For some reason (probably making the proof "work") the word conception is missing at the end, where the result should be that a conception of the perfect X exists.

    And as another commenter has suggested, "greater" and "perfect" are valuations by humans (outside of mathematics.) Which is greater or more perfect: a unitary god or a plural god? There's no real way to answer. It's not even obvious that existence is greater than nonexistence, except as we humans value it.

  16. Anselm's ontological argument is famous not for what it proves but how it fails. There is a whole range of suggestions as to why the argument doesn't work, but all this stuff about existence as a conceivable quality seems to me to miss the problem entirely.

    Allow me to explicate the point by inserting the word ‘idea’ into the appropriate places:

    We can conceive of an idea of the perfect X, for which no greater X can be conceived.
    2) It is greater for an X to exist than not.
    3) Suppose for reductio that the perfect X does not exist.
    4) Then we can conceive of an idea ofa greater X, namely, a twin of the perfect X that has the further virtue of existing.
    5) This is a contradiction; thus (3) is false, i.e. an idea of the perfect X exists.

    So Anselm may be correct in suggesting that between two ideas of the perfect island, the idea of the island as extant is a better island than the idea of island that doesn’t exist.

    Does this have any bearing on any actual islands or beings? None.

  17. hey mike and illusive mind, i got a "d" for a phil20something essay that more-or-less said that! never quite worked out why.... as this seems to cut to the nub of it for me.
    another way to "cut to the chase"-
    -however perfect an X we can conceive, we can ALWAYS conceive of that X NOT existing. ergo existance is not a necessary quality of perfection.

  18. Rob, If you can conceive of an object that necessarily exists, then it would be incoherent to conceive of this object as not existing.

    Brandon, even restricting ourselves to non-"effects", or uncaused beings, there stills seems plenty of fuel left for the reductio. I pointed to examples like "malevolent being" that seem more difficult to dismiss than the standard "island" example.

  19. How does one conceive of a non-extant object?
    If I am imagining a golden mountain, aren't I necessarily envisioning this mountain 'in existence' in some sense?

    The only way I can make sense of this distinction is to think of myself as an author writing a fictional story. I want there to be a perfect being such that no greater being can be conceived in the story. In one version, all his qualities are described in full detail but it turns out at the end of the story he didn't exist (wasn't real) at all. In the other version we find that he does in fact exist. Anselm is saying the later being is better than the former and only the extant being can be considered the greatest such that nothing greater can be conceived.

  20. Hi Richard- great website!
    And ok, offhand I can't....(conceive of an object that necessarily exists). Guess I'm a hopelessly contingent creature.
    But that's not quite the argument, is it? Anslem- and later Descartes et al - argue that existance is a necessary quality of perfection on the grounds that it is "better" (more perfect) to exist (for the "thing-in-its-own-perfection", I think, rather than better for us or anyone else). The strength of the ontological argument rests on the intuitive "rightness" of this- which is quite strong. Yet one can also- in a "flip-flop" on/off manner- conceive of a perfect X existing or not existing. And- importantly- this doesn't seem (intuitively to me) to do the sort of damage to that thing's conceptual "perfection" Anslem has to insist it does. So there are a couple of intuitions at war here... and I think your consideration of other perfect Xs does help elucidate this.

  21. Richard:

    If you can conceive of an object that necessarily exists, then it would be incoherent to conceive of this object as not existing.

    Why? If existence is just a value (a perfecting quality) that can be concieved about a deity, why can't it be concieved about anything else, existence or otherwise?

    If I can concieve of a pink flying unicorn, can't I also conceive of a pink flying unicorn that necessarily exists? In fact, by writing that sentence, I think I just conceived it. Let P be a pink flying unicorn that necessarily exists. There. Conceived. Now let P not exist. Viola.

    What is "conceiving" that it rules out the ability to imagine the absence of a necessary quality?

    (I'd also be interested to hear your thoughts on the argument I made in a comment earlier on this post, toward the top.)

  22. I see this blog post is out of date and the author has stopped rebutting counter arguments. But I'll ask this question anyways.

    In regards to the perfect island. Or apparently the lack of a perfect island. Since apparently we are not able to conceive of the perfect Island. Apparently we can always think of a more perfect island. That really boggles my mind. For one, there are quite a few instances where we need to be able to think of the perfect number in math. Since we are unable to realize these numbers but we have to be able to explain these number in some form. For instance, pi, infinity, -infinity, and e. All are representations of specific meanings. We have super computers crunching the numbers and after all these years we are still unable to find the exact number for pi. Yet we still need to use this perfect number.

    So I am wondering why we can not do the same for the perfect Island?


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