Tuesday, October 5, 2010

Norman Malcolm's Ontological Argument

Apologies for not getting this up sooner. I've been unusually busy since getting back from my conference in Detroit, trying to balance my time between work on the new book and preparing a talk I'm giving this afternoon entitled "God and Gays: Rethinking the Traditional Condemnation of Homosexuality."

In any event, here is the promised outline of the ontological argument that Norman Malcolm develops in his essay "Anselm's Ontological Arguments."

What Malcolm discovered as he reread Anselm's Proslogion was this: what everyone seemed to take to be just a rewording of the argument Anselm is most famous for is actually a different argument. The first argument holds that existence is, in effect, a great-making property, and that therefore the greatest conceivable being must exist. Malcolm agrees that this argument is unsound, accepting Kant's contention that "existence is not a real predicate." But the second Anselmian argument, rather than focusing on existence, focuses on existing necessarily rather than contingently. Anselm argues, in effect, that it is greater to exist of necessity than to exist contingently. Hence, it is part of the very concept of a greatest conceivable being that this being exist necessarily.

What Malcolm notes is that the property of existing necessarily rather than contingently does meet the test of being a "real" predicate in Kant's sense. That is, it adds to our concept of the thing, describing what it is like rather than merely stating that the thing as described has an instance in the world.

For this very reason, of course, it remains an open question whether there actually exists an entity which possesses its existence in this unique way--necessarily rather than contingently. So we haven't defined God into existence by noting that the very idea of God presupposes necessary existence. We can still reasonably ask, "Does there actually exist a greatest-conceivable being?" If  the answer is yes, then that being does not exist merely contingently but necessarily. But Malcolm goes further than this. He argues (and here he is following in the footsteps not only of Anselm but of Leibniz) that the only reason why a greatest-conceivable being wouldn't exist would be because the concept named something whose existence was impossible.

In other words, a crucial feature of Malcolm's development of Anselm's argument is his insight that, as conceived, either "God" names an impossible being or a necessary being that actually exists. Put another way, if God is possible, God is actual.

This is a very interesting result in its own right, but Malcolm goes on to argue (in a manner reminiscent of at least some of Gödel's efforts to construct an ontological argument) that God's existence must be deemed possible.

Another feature of Malcolm's argument is that he sets aside Anselm's language of greatness (perhaps worried about this term being understood in subjectivist ways). Instead of defining God as the "greatest" conceivable being, he defines God as "an absolutely unlimited being." Now certain kinds of properties, he thinks, imply limitation (for example, having a shape--since a shape is defined by its outer boundaries). On this definition, then, God would not have a shape. More generally, physical existence in time and space seems to require boundaries or limits, and so God wouldn't have such spatio-temporal properties. God would be "eternal" and "transcendent." But the possession of power (capacity to do something) does not similarly imply limitation. Nor does the possession of knowledge. If this is right, then these are things an unlimited being would possess without limit.

What Malcolm argues, following Anselm, is that necessary rather than contingent existence is also something that would have to characterize an unlimited being. Here is an outline of his argument for that conclusion:

1. “God” means an absolutely unlimited being

2. Any being whose existence depended on something else, or which could be prevented from existing by something else, would be limited by something else and so would not be an unlimited being.

3. For every proposed being, B, its existence is either possible (but not necessary), necessary, or impossible

4. To say of B that its existence is possible but not necessary is to say that it exists in some possible world (call it PW1), but not in another (PW2)

5. If B existed in PW1 but not in PW2, then either (a) there is something that exists in PW2 that prevents B from existing, or (b) there is something missing from PW2 that B requires in order to exist.

6. Hence, if B’s existence is possible but not necessary, then (a) or (b) is true.

7. If (a) or (b) is true, then B is not an unlimited being.

8. Hence, if B is possible but not necessary, then B is not an unlimited being

9. Hence, if God is possible but not necessary, then God is not an unlimited being

10. Hence, it is not the case that God is possible but not necessary

11. Hence, God is either impossible or exists necessarily

At this point Malcolm takes up the question of whether God, conceived as an unlimited being, is possible. To make his argument here, he invokes two ideas: first, that an entity's existence is impossible only if it is characterized by contradictory properties (e.g., a round square); second, that such contradictions arise only when one property-attribution negates what is affirmed by another property attribution. But to negate what is posited elsewhere, a property attribution must embody, at least implicitly a limitation. Roundness negates squareness because it imposes boundaries or limits on the space occupied by the object precisely where squareness does not, and vice versa. These concepts, in other words, are partly "negative" concepts--they don't merely ascribe some property to an object, but deny something of it. But an absolutely unlimited being would be such that no real "positive" attribute could be denied of it (we could only deny "negative" properties of it--that is, properties which ascribe absence or limit). As such, Malcolm concludes that an unlimited being cannot embody a contradiction, since that would require the possession of both a positive property and a negative property that denies the former. But an unlimited being would only possess positive properties. This part of his argument can be outlined as follows:

12. In order for the existence of some proposed being B to be impossible, the concept of B must imply, with respect to at least one positive property P, each of the contradictory claims “B has property P” and “B lacks property P.”

13. To lack a positive property is to be limited.

14. If 13, then the conception of an unlimited being cannot include or imply anything of the form “B lacks property P.”

15. Hence, God is not impossible.

16. Hence, God exists necessarily.

Critics of this argument are often skeptical of the idea that the positive and negative property distinction is a meaningful one. If it's not, then we might be forced to say that the failure to possess certain properties is a limitation, whereas their possession would contradict the possession of the opposing property (which one could not deny possession of without imposing limitation). But while it might be difficult to offer definitions of "positive property" and "negative property," there does seem to be an intuitive distinction here. Some property attributions assert that an object lacks something ("ignorant" or "impotent" or "empty"), while others that it has something ("knowledgable" or "capable" or "full"). Others are mixed, in that they assert than an object has this but lacks that ("round" or "green"). Nevertheless, especially when we get to the idea of mixed properties we wade into thorny territory. To use what is perhaps a silly example: Is heat the presence of something and cold the lack of it, such that we must say of God that God is infinitely hot? Or does being the sort of thing that's subject to heat and cold imply a limitation, such that anything of any heat is limited, and an unlimited being would therefore have to be something to which the categories of hot and cold simply do not apply?

Other questions arise, of course, when we consider moral properties. Can there be such a thing as unlimited goodness? And if so, what does that look like? To adhere with traditional theology, we'd need to hold that evil is a lack (something Augustine affirmed), such that any presence of evil implies limitation. If we think of evil as something positive, we get a God who must embody unlimited evil (as well as unlimited good--and hence must embody a contradiction). As such, Malcolm's argument leads us directly into a consideration of the nature of good and evil.


  1. Thanks, Eric. I studied this version of the ontological argument in my medieval philosophy class when we were reading Anselm. It's fascinating, but maddening... something feels wrong about it, and yet the logic seems impeccable. I guess that's why the ontological argument is still around in one form or another centuries after Anselm first explicitly formulated it.

    Do you have a text for your talk you are giving, or could you post here some of the subjects or arguments you intend to cover? I'd love to read what you have to say.

  2. Hi Eric

    I've had a brief think about this and have concluded I don't understand the way the word possible is being used here at all.

    In everyday usage I use possible to mean 'I don't have enough information about this event to predict its outcome with any certainty.' So, I roll a die and I say there are six possible outcomes, this is a useful manner of speaking. In fact, only one of them is possible, the one that is certain given the way I will throw the die, and five are impossible, given the forces, angles, surfaces etc involved. But because I can't say which is which, the situation is beyond my modeling capacity, I use the language of possibility as a way of at helping to quantify and deal with the unknowns..

    Possibility used in this argument though seems to refer to something different, things that could happen but do not necessarily have to happen, and I can make no sense of this idea. Certainly we make folk reference to this concept (damn, if only I'd thought to check the oven), it's a useful way of psychologically constructing our narrative, giving us regret, determination to do better next time and so forth. But is that secure enough a foundation to build an argument for God upon?

    Intuitively, it seems to me that anything that exists must exist, and in this sense is necessary, and anything that does not exist does not exist and is therefore, at this place and time, impossible.

    Possibility then seems to change shape a little, referring variously to things we can imagine, things that are consistent with the rules of the world as we understand them, things that do not produce contradictions, things that may have happened differently if we rewound the clock and so forth. But I can't make any of these definitions hold much water. Some physical phenomena appear to defy logic (wave functions), often I feel I can imagine something but can not describe its workings satisfactorally (zombies), the laws as we understand them are themselves dependent upon our observations of the world, and I'm not convinced the idea of turning back the clock to rerun an experiment is coherent (because if things turn out differently, it's not the same time is it?)...

    So, if you have a chance, I'd be interested to know a little more explicitly what a philosopher means when they say something that is could have been otherwise.


  3. Hi, Bernard-

    I think you misapprehend this mode of thought. It is not really an "argument" per se, or at least not about actual existence or non-existence. Rather, it is a working-through of the concept of god, saying that if one begins by thinking that it is a "limitless" "being", then ... it has to be necessary, existent, etc.

    But this thread has no empirical power to say whether this imagined concept, with all these "deduced" properties, actually does exist or not. That is an entirely different matter.

  4. Hi Eric,

    The first thing that strikes me when I read these arguments is how vague and imprecise the language is. At least, the more I try to make sense of the words, the more complicated it seems.

    In this context, what does “unlimited” mean? I suppose it refers to the idea that a given property may come in a (possibly infinite) number of “values” and that one of them is an absolute maximum (it does not mean that properties are quantifiable). Not all properties admit a “maximum” - shape is an example you mention. If we call these properties having a maximum value “positive” I suppose we can define an unlimited entity as one that has all these properties (and only these) and, furthermore, it has them set at their maximum value.

    Now, this is only a reformulation of what Malcolm is saying (is it?), in terms I can understand better. But now I am stuck. What are these elusive “properties”? Is there any way to define them in some kind of objective manner?

    Take any quantifiable property (size). They clearly cannot qualify: there is no “maximum” number (infinity is NOT a number). Even allowing for transfinite cardinals (as we do in set theory) doesn't help: there is NO maximum cardinal number. But all this is very arbitrary. We could as well define our “maximum” as a size of zero (things like this work both ways). We can order values any way we choose – there is no “better” way.

    What is left? Knowledge? Power? You mention good and evil. If we assume we can define them in some absolute sense, I don't think we can have one without the other (one is the opposite of the other).

    But this leads us nicely to (12). It ignores the possibility that we can have two positive properties P1 and P2 that are mutually exclusive. Good versus evil is one such pair but there are others. You had a post on the paradox of the stone which describes another. Absolute knowledge also contradicts absolute power (the fable in the twilight zone episode I mentioned in a comment on the stone post has the mathematician asks the all-knowing daemon to get lost, which of course he cannot do).

    The only way out is to explicitly exclude these mutually exclusive properties. But at this point the exercise becomes extremely complicated.

    Bernard mentions a real problem with the notion of “possible”. I would add that the idea of “possible world” seems very tricky. If we mean by “world” the whole of reality (transcendent and all), well, there is only one. In this context, I don't understand statements like (4).


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  6. Eric,

    I don’t at all agree with premise (5). Consider a world where throwing dice is a truly random process. I throw a six, so in that word the event that I throw of a five does not exist. But there is nothing in that world that prevented that event from existing, nor is there anything missing for it to exist.

    What worse, it seems to me that proposition (11) "God is either impossible or necessary" if true would have appalling implications:

    1. God is either impossible or necessary.
    2. If God is necessary then God exists in all possible worlds.
    3. A world in which nothing exists is possible.
    4. Therefore there is a possible world in which God does not exist. (from 3)
    5. Therefore God is not necessary. (from 2 and 4)
    6. Therefore God is impossible. (from 1 and 5)
    7. If God is impossible then God does not exist in any possible world.
    8. Therefore God does not exist in any possible world. (form 6 and 7)
    9. The real world in which we exist is possible.
    10. Therefore God does not exist in the real world in which we exist. (from 8 and 9)

    Do you see anything wrong in this argument? Premise (3) strikes me as obviously true.

  7. Dianelos,

    I believe there is something wrong with premise 3
    3. A world in which nothing exists is possible

    For nothing to exist 'in something' thee must be a 'place' for this 'nothingness' to reside.
    If this world of nothingness is nowhere, then it does not exist anywhere, so its existence is ruled out.
    If there only was this nothingness existing, then it still would possess the quality of 'existing' as opposed to not existing. If it possessed this quality, then it would be something and not nothing, if it truly was nothingness, then it could not exist.

    In what way would one say this is possible? Because, and I may be mistaken here, just by stating this world of nothingness exists in X way. You have already attributed the nothingness world to possess the quality of X, thus stating that it is something and not pure nothingness.

  8. Doug,

    To my third premise “A world in which nothing exists is possible” you write: “ For nothing to exist 'in something' thee must be a 'place' for this 'nothingness' to reside.

    No, not at all. The linguistic form “in which” used in that premise does not entail the existence of space. I could have written the same premise without using any linguistic form one might associate with the idea of space, for example (3*) “There is a possible world such that the set of its existents is empty”.

    If this world of nothingness is nowhere, then it does not exist anywhere, so its existence is ruled out.

    Space is not required for existence. For example numbers do not exist somewhere in space. Neither do beliefs. Neither does God.

    But perhaps you don’t mean physical space, but rather space in its logical sense, the way one says “numbers exist in the space of mathematical objects”. Even if it is the case that for X to exist the respective logical space(X) must exist also, it would not affect premise (3), because in the world referred by that premise there are no existents at all and therefore no need for the existence of the respective logical space either.

    If there only was this nothingness existing, then it still would possess the quality of 'existing' as opposed to not existing.

    When one says “nothing exists” one does no mean “one thing exists, namely a so-called ‘nothing’”. Rather one refers to there not being any existents.

    You have already attributed the nothingness world to possess the quality of X, thus stating that it is something and not pure nothingness.

    If I understand you correctly you are saying this: A world where nothing exists has a property, namely that nothing exists in it. Therefore it has a property. Therefore it has something that exists, namely that property.

    I define the world which premise (3) refers to by stating that the set of its existents is empty. Other sets related to that world may well not be empty. So, for example, the set of properties that that world possesses is not empty, and might be: {‘there is by definition nothing in that world’, ‘Dianelos believes that that world is possible’, ‘Doug believes that that world is impossible’, …}. Now philosophers distinguish between intrinsic properties and extrinsic properties. The intrinsic properties of X are the essential properties of X, i.e. those properties that are necessarily there (if any of them isn’t then neither is X). For example, an intrinsic property of a circle is existing on a plane (but the slope of that plane is not), an intrinsic property of a solid is having spatial extension (but its color is not), an intrinsic property of an electron is having electrical charge (but its position is not), etc.

    Now it seems to me that the world to which premise (3) refers has only one intrinsic property, namely its defining property of being a world lacking any existents. Which gives me an idea of how to justify premise (3):

    A1. For X to be impossible there must be at least two intrinsic properties of X that contradict each other.
    A2. A world where nothing exists has only one intrinsic property (namely the defining property that it is a world where nothing exists).
    A3. Therefore a world where nothing exists has not two intrinsic properties that contradict each other. (from A2)
    A4. Therefore a world where nothing exists is not impossible. (from A1 and A3)

  9. Dianelos,

    Thank you for your response,
    I am still having trouble grasping this concept of a wold of nothingness as you've described.
    Could you explain what is meant by this statement,

    "I define the world which premise (3) refers to by stating that the set of its existents is empty. Other sets related to that world may well not be empty."

    I am having trouble understanding this statement, mainly because I cannot wrap my mind around a set, like the purposed 'set of existants', which contains things other than 'existants.'
    I am assuming, i believe, that all sets, as i understand you to be referring to the concept of 'set's', contain existants.
    That is to say all set's that exist.

  10. A couple of points. First, the following propositions contradict each other: “There is a being that exists necessarily” and “It is possible that nothing exist, that is, that there be nothing rather than something.”

    Put in “possible worlds” language (however problematic that language might be), the former proposition translates as “There is a being that exists in every possible world” while the second translates as “There is a possible world in which no beings exits.” The former clearly implies the negation of the latter.

    (“Possible world” means, roughly, a TOTAL state of affairs, or a complete way that things might be, and not just at a given moment).

    Dianelos' first argument therefore starts out with contradictory premises--and so it is not surprising that a contradiction is thereby deduced.

    But Dianelos’s argument in his last post raises an interesting (and I think important) challenge to Malcolm's ontological argument. Malcolm’s case for the possibility (and hence actuality) of an infinite being rests on the claim that no contradiction can arise among properties that are “purely” positive—since a contradiction only exists when you affirm and deny something at the same time. But a purely positive property contains no denial of anything, and so no conjunction of purely positive properties can generate a contradiction. (He'd reject JP's notion that positive properties might be mutually exclusive on the grounds that mutual exclusivity of properties requires that one or both are implicitly affirming limits).

    Suppose that Malcolm is right about all of this. What Dianelos maintains, in effect, is that the same line of argument can be made with respect to purely negative properties--they cannot contradict each other. "There being nothing rather than something” describes a state in which nothing is affirmed and everything is negated. And if Malcolm’s view about possibility is sound, then such a state is also possible, insofar as it contains no contradiction.

    But if an infinite being is shown to be possible on the basis of this reasoning—and if Malcolm is right that an infinite being is either impossible or necessary—then, on the basis of this reasoning we’ve reached BOTH the conclusion that an infinite being exists necessarily AND the conclusion that a world in which nothing exists in possible. But these conclusions, as we'e already seen, contradict one another.


  11. Adequately addressing Bernard's question about what we mean by "possible" would take a post in its own right to do it justice. But a brief run-down of different senses of "possible" might be of use. Letting "S" refer to a state of affairs, here is a quick run-down on some senses of possibility:

    1. S is epistemically possible = For all we know, S might obtain (we lack any good reason to think that S does not obtain).

    2. S is logically possible = There is no internal inconsistency in S

    3. S is possible relative to state of affairs T = There is no inconsistency between S obtaining and T obtaining.

    4. S is possible on R, where "R" names a set of rules governing how things may stand in relation to one another = S obtaining is not inconsistent with rules R holding.

    5. S is possible relative to T on R = Given R, there is no contradiction between S obtaining and T obtaining.

    Epistemic possibility can be correlated with each of the other forms. Consider a SUDOKU puzzle (look it up if you don't know what it is). Such a puzzle in effect specifies rules for filling in a grid with numbers ("R") and places some initial numbers in the grid ("T"). Let "S" stand for "There is a 7 in the upper left hand square of the grid." If we say S is possible, we could have either sense 1 or sense 5 in mind. If we have sense 1 in mind, then what we say is consistent with it being the case that only a 9 in the upper left hand corner is consistent with T on R.

    Take away any initial numbers from a SUDOKU grid and there are all sorts of ways the grid could be filled out consistent with R which are all possible on R, and on many of those arrangements S would be true. Here, then, S is possible in sense 4.

    These various ways of filling out the grid on R are only a subset of all the ways the grid might be filled in--and we could imagine many different sets of rules (R', R'') which might be followed. If we take "One number per square" as analogous to the rules of logic, then S would be logically impossible if there were already a number other than 7 in the upper left, and logically possible under any other condition T. Each arrangement of numbers, regardless of whether a set of rules is followed, would be analogous to a "possible world." Obviously, the rules R would be analogous to the physical laws which regulate interactions in the real world (whereas R', R'', etc., represent alternative physical laws that might prevail in a different possible world). But there are many possible world which share the same physical laws as ours, just as there are many different arrangements of numbers that are consistent with the rules of a SUDOKU puzzle.

    I could go on, but I need to get home.

  12. Thanks Eric

    I remain uncertain about this, and will need to think about it more. I am wary, I think, of setting up the rules of a proof in such a way as the result is guaranteed only because those rules were chosen.

    Intuitively, and this is fairly undigested at this stage, the sudoko analogy begs the question. At some point, as the numbers fall, the solution becomes necessary. So there is a minimum amount of information that takes all options from that point off the table, and this is how sudoko puzzles are created. So, how far do the physical rules of existence close off possible worlds? Maybe entirely. It seems to me we don't yet understand the physical rules well enough to talk usefully about what types of possibility are consistent with this world. To say there are many possible worlds that share our physical rules seems premature to me.

    The other approach, to say something is possible if it doesn't contain a contradiction, requires us to define contradiction and I wonder if this isn't also empirically tethered. For example, many things that in the past would have appeared contradictory (design without a designer, the creation of time itself, time passing at different rates for different observers, the impossibility of simultaneous events, the ultimate indivisibility of time, wave particle duality, Hesienberg uncertainty) are now considered part of the intellectual superstructure. When evidence comes in that challenges our sense of reason, we tend to adjust our reasoning don't we?

    It is one thing to reason our way towards a testable hypothesis, where amongst other things the reasoning itself can be scrutinised, but to use reason alone to reach an untestable hypothesis is trickier I think.

    So, all this talk of necessary versus contingent and possible worlds seems at first blush to contain hidden assumptions about the way the world is, and if we tweak these appropriately should we be surprised when proof for (or in the case Dianelos proposes) again God pops out? The ability to imagine an alternative world without apparent contradiction is not in itself enough for me, because I suspect with a little effort our imagination can lead us wherever we so please.

    This may be what Burk had in mind when he suggested I had misapprehended the nature of the ontological proof.


  13. I agree with Bernard, somewhat

    I find these concepts of 'possible worlds' to be highly speculative. By this i mean they grant too much to the term possible.
    I define what is possible as what is existent. Therefore I do not believe a unicorn is possible, because it does not exist. One could say that something can become possible, but it isn't actually possible, until it is.

    Furthermore, I find the concept of 'a world of nothingness' to be somewhat nonsensical. As far as I can tell, Dianelos has defined a 'world of nothingness' as simply containing the intrinsic property that is has no existants in it, yet it itself exist.
    Which means if i open my hand, cupped, and look at the 'space' within my hand I can see that there this universe is within my hand.
    No shape, size, temporal reality, nothing that would constitute an existant, except for the fact that it has this intrinsic property of existing, as defined. In truth I cannot find a difference in saying these statements
    1. There is this 'world of nothingness' in my hands, which only exists in the sense that it possesses the intrinsic property of containing a set of no existants.
    2. There exists no 'world of nothingness' in my hand, but just this intrinsic property of set containing no existants
    3. There is no 'world of nothingness' in my hand

    Am I missing something here?

  14. Eric,

    Malcolm’s contention that a conjunction of purely positive properties can generate no contradiction sounds OK at first, but on further thought becomes kind of shaky. Atheologians often device arguments for the incoherence of God by arguing that divine attributes X and Y (which some theists think are implied by St Anselm’s definition) contradict each other, even though X and Y appear to be purely positive ones. In comparison my contention that a null world is possible rests, I think, on must more solid ground. In such a world there exists only one true belief (namely that anything does not exist in it) and therefore a contradiction is objectively impossible.

    If the modal version of the ontological argument entails a contradiction (namely that God exists in all possible worlds and also that there is a possible world where nothing exists), then there must be some erroneous premise in that argument.

    One possibility is that God, defined as the greatest conceivable being, does not exist in any possible world. Or, in other words, that the concept of “greatest conceivable being” is inherently incoherent. I think we can safely exclude this possibility because the idea of the “greatest conceivable being” entails coherence. Logical coherence is certainly a real predicate. (Thus the above mentioned atheologians are only helping clarify what God is.)

    The only other possibility I see is that the premise that if the greatest conceivable being exists in one possible world then it exists in all possible worlds. (In his version of the argument Plantinga puts it thus: “It is proposed that a being has maximal greatness if it has maximal excellence in every possible world.”) Or, in other words, that the concept of the greatest conceivable being entails necessary existence. Perhaps we can thus use the contradiction we found to positively demonstrate that the greatest conceivable being need not exist in all possible worlds.

    The most charitable interpretation I can find for the traditional belief that God not only exists but exists necessarily is this: God is that through which all exists, thus the very concept of existence entails God. Therefore, trivially, in all possible worlds where something exists there is God. The confusion appears to reside in using the expression “God exists” as being a claim about God when in fact it is a claim about existence. After all, clearly, the greatest conceivable being is not one that exists as anything else exists, but a being on which all existence depends. On the other hand the naturalist may reasonably claim that there are logically possible worlds where existence is not God-based, e.g. in a materialistic world, where God doesn’t exist. In response the theist may try to argue that all such materialistic worlds are logically impossible, but I think they will not succeed. To even hope to succeed in such a project reveals in my judgment intellectual pride which is unbecoming.

    Actually the idea that God exists in all logically possible worlds did always strike me as kind of excessive not to say wasteful. I don’t see why the most perfect being, or the most loving being, or the most beautiful being, or the most creative being, or the being most giving in joy – should exist in all possible worlds, i.e. exist not only in ours but also in all worlds that might have been without contradiction.

    In general I think there has been an unfortunate tendency in theology to kind of go overboard when thinking about God. There is a famous ancient Greek saying “ouk en to polo to eu” meaning “not in quantity lies the good”. Many theologians have ignored that true principle of greatness with unfortunate consequences. In my mind all the talk of God being infinite, having infinite power, having infinite knowledge, and so on, are a residue of the unfortunate tendency to conceive God as an earthly king only grown to the infinite. Hence all the talk of God’s “holy” wrath, which, not surprisingly, is also supposed to be infinite in quantity.

  15. Eric,

    According to your definitions of possible, Dianelos and Malcom both argue from a case of logical possibilities correct?

    If so, within the context of logical possibilities, i agree that a world of nothingness is possible, as an object of logic. However, I have trouble agreeing with its possibility as existing apart from logic.

    Suppose I have a cup of water, I fill it all the way to the top, such that there isn't room for anymore water. Water fills every bit of space within this cup that there is. Except that, there is this spot within the cup. This spot is unique. All this spot is, is that which contains the intrinsic property of no space. It is the spot where this intrinsic property manifests; it contains no space, no water. Thus, the spot in no way limits the amount of water within the cup. There could be infinite amounts of these spots, and the amount of water in this cup would never change because of them.

    My question is, while this spot is logically possible, is it actually possible or relevant; or is it an object of logic superimposed onto reality....or something else?

  16. Doug: My friend and co-author John sketched an argument over the phone for the impossibility of a world with no existents, based on the premises that there are necessary truths (propositions which would be true in any possible world) and that all truths need a truth-maker. I think the argument here is for a kind of metaphysical impossibility as opposed to a purely logical impossibility--and I think you are also trying to make a case for the metaphysical impossibility of there being nothing rather than something. I think thinking about the concept of metaphysical possibility (which I take to be narrower than logical possibility but broader than "physical possibility," by which I mean what is consistent with the physical laws the obtain in the real world) is important. Metaphysical possibility might be roughly characterized as what is consistent with the nature of being (and nonbeing).

    Your original argument, by the way, is very reminiscent of Parmenides, who basically argues that nonbeing can't be said to be, and so it is incoherent to posit there BEING nothing rather than something. We might put it this way: While there is no contradiction in a complete set of negations of all positive properties (which we might call the negative set), what it means to say, "This is a way things might be" precludes saying this of the negative set. So the negative set is logically possible but not metaphysically possible (and as such doesn't name a possible world).

  17. In any event, it seems to me that an adequate critical assessment of either Malcolm's argument or Dianelos's parallel case for the possibility of nothing existing depends upon developing an adequate notion not merely of logical possibility but of metaphysical possibility (since a possible world--a possible way things might be in totality--invokes the concept of "being" in such a way that the nature of what it is for something to BE the case has to come into play).

  18. Bernard,

    Take an empty sudoku grid. Write in a number between 1 and 9 somewhere on the grid. At this point, given the rules and that one number as the "initial condition," there are multiple "solutions" to the puzzle--but some arrangements of numbers have been excluded by that first number. If we keep adding numbers (if we make the initial conditions more complex), eventually we will hit a point where the value of every other number on the grid is determined by the initial conditions and the rules--at which point it is either not possible that the upper left number be 7, or necessary that it be 7, given the rules and the initial conditions. In that case, if we say it's possible that the number is 7, we mean it in a purely epistemological sense. But there are at least some initial conditions such that, given those initial conditions and the rules, there is more than one solution to the puzzle and one but not both solutions has a 7 in the upper left. In that case, if we say that a 7 is possible in that space, we could have in mind not merely epistemic possibility but this other kind.

    Making an analogy to the real world, we might think that the laws of physics are like the rules, and the conditions that prevail at the big bang are like the initial numbers on the grid. And the question then becomes whether those rules and initial conditions necessitate everything else that is true of that world throughout its history or not. If they do, then all possibility within this framework is merely epistemic. If they don't, then other possibility assertions might be true.

    Now one question you in effect raise is whether the rules themselves, apart from any initial conditions, might imply all else that is true of the actual world. In other words, we can ask whether the actual world is more like a sudoku puzzle (where you need initial conditions in addition to the rules in order to define a unique solution) or more like some other kind of logic puzzle which sets out a set of rules for how to arrange numbers on a grid such that there is only one unique arrangement consistent with those rules. You're right that we can't rule that out, and that I was too quick on that point.

    But all of these questions are about the actual world--in which there are at least a given set of rules and maybe also a set of initial conditions relative to which we can ask what is possible, what is impossible, and what is necessary (determinism being the theory that relative to these things every claim that can be made about this world is either impossible or necessary).

    But with respect to that world we might ask if entirely different rules and/or initial conditions are possible. In this case, we are not asking about epistemic possibility (for all we know...), nor about possibility relative to a framework of rules and initial conditions. We are at least asking a question about abstract logical possibility (is a different set of rules/initial conditions logically coherent?), but we might have in mind some via media between logical possibility and physical possibility, which is sometimes called metaphysical possibility (assuming one can make sense of such a notion).

  19. I will try to formalize Malcolm's argument and see where it leads me.

    Let's say a possible “world” is a set populated by unspecified “beings” (entities) having “properties”. A property is similar to a mathematical function in that it associates a being to some “value” (not necessarily numerical). It can be simple (size) or quite complex as the property of “being dependent on another being” used in step 2. We define an ordering relation on some of these properties and we are interested in those properties for which there exist a maximum value (in some sense). What we're looking for is a being for which each of its properties has the maximum possible value (so, we don't use properties that don't have a maximum, like “shape”).

    In order for this to work, we must carefully limit the properties we consider to a set of mutually independent properties. For example, assuming that “good” is well defined and that “evil” is its opposite, we cannot consider both properties at the same time (although either one will do). I think that other pairs like knowledge and power are also exclusive (for instance, absolute power allows you to forget, thus making you less than all knowable). This leaves us with a rather diluted (and largely artificial) set of properties (and, besides, quite difficult to figure out).

    In any case, this description above seems to leave room for all kinds of possibilities, including the empty world of Dianelos and, indeed, of a being achieving the maximum value for each member of our undetermined (and diluted) property set. But Malcolm's argument introduces meta-properties, which considerably complicate the issue.

    First, in step 5, a notion of inter-world dependency. The situation is this: we assume that being B exists in world W1 but not in W2 because of some characteristic of W2. The property we are interested in is then “being dependent on some characteristic of some world” (the “maximum” is reached when there are no such dependencies). But this implies a very strong claim on possibilities: a being B exists in world W if nothing prevents it from existing. Why should that be? Isn't being forced to exist when nothing prevents it some form of limitation? In fact, (5) does not appear to hold at all: we can perfectly conceive of worlds when some entity does not exist even if nothing prevents it from existing.

    Then, in step 14, we enter the realm of self-referentiality with the property “list of properties for which the maximum is not attained”.

    In any case, what about these two points: (a) how are we to decide on the the set of mutually independent properties? And (b) do (5) holds or not?


  20. Eric,

    [...] there are necessary truths (propositions which would be true in any possible world) and [...] all truths need a truth-maker.

    This does not seem right. If a truth is necessary then why does it need a truth-maker? The statement seems to mean that the truth-maker somehow determines the truth of propositions (by its existence or whatever). If so, no truth would be necessary – because dependent on the truth-maker.

  21. JP

    I think your point about the need for mutually independent properties is a good one. If we adhere to it we appear to dilute the definition of the God being established almost to the point of meaninglessness don't we? For example, one might argue that goodness itself, as understood by humans, requires that the thing being good has limitations, because a thing that exists without limitation may not be able to be thought of as making choices at all. A being with unlimited time is not in the position of making the morally good choice of sharing some of its precious, limited resource in the service of another. So do we subject it to a finite existence, or take out the requirement that it has this type of generosity?

    Dianelos also queried the validity of 5, as in a truly random world such uncaused and unconstrained non-existence is part of the package.

    You said in an earlier post that these types of proofs are more enjoyable intellectual exercises that serious attempts at establishing descriptions of reality and in this case that seems a fair observation perhaps.


  22. «But if an infinite being is shown to be possible on the basis of this reasoning—and if Malcolm is right that an infinite being is either impossible or necessary—then, on the basis of this reasoning we’ve reached BOTH the conclusion that an infinite being exists necessarily AND the conclusion that a world in which nothing exists in possible. But these conclusions, as we'e already seen, contradict one another.»

    To argue against the possibility of a world in which nothing exists based on the possibility of an unlimited being must at elast be considered arbitrary. One could just as well argue that the existence of God (as an unlimited being) is impossible given the initial possibility of a world in which nothing exists. Given Malcolm's definition of what is possible or not, such a world would a priori be just as possible as the existence of God. There is no apparent reason why the possibility of God could be introduced to argue that such a world is not possible after all rather than the other way around. If Malcom's definition of what is and isn't possible is exhaustive we have a paradox on our hands, but even a modification of the definition gets us no further towards a resolution (as far as I can tell, but please prove me wrong). One could formulate the problem in this way:

    1. Something can be considered impossible only if a) it posesses properties contradicting each other or b) it's possibility is contradicted by a neccessary truth (something that must be true of or hold true in every possible world) or c) it's possibility is inconsistent with a given fact or sound premise.
    2. The existence of an unlimited being is either necessary or impossible (assumed for now based on Malcolm's previous argument,
    3. A world in which nothing exists is possible in terms of 1a, but could be impossible in terms of 1b or c.
    4. An unlimited being is possible in terms of 1a, but could be impossible in terms of 1b or c.
    5. The possibility of a world in which nothing exists would make an unlimited being impossible (given 1c and 2)
    6. The possibility of an unlimited being (which entails the neccessity of such a being) would make a world in which nothing existed impossible (given 1b and 2)

    Since there is no way of knowing from these conditions alone whether a world in which nothing exists or an unlimited being is possible (all we know is that they cannot both be possible) there is no way to resolve this problem.

    Malcolm's argument is nonetheless, I must say, quite ingenious and of profound philosophical interest and I suspect that much more can be said about this. Yet I remain suspicious on any argument that try to solve such fundamental questions by recourse to pure logical thinking. Einstein might have been right when he said that all knowledge of reality starts with experience and ends in it.

    Øystein Evensen

  23. Forgive me my academic bad manners. My argument was a reformulation of Dianelos' argument, but I forgot to give him credit for it.

  24. Hi Bernard,

    Yes, the resulting God becomes quite diluted. I suspect that if we could pursue the analysis far enough (which would include the substantially difficult task of defining precisely all these “properties” - goodness and so on) we might end up with only one or two maximized properties. Maybe they are all mutually exclusive in the end.

    I think these arguments are in a sense like magic tricks. (Not only that of course: I entirely agree that they are useful in many ways, not the least being to force us to think about the relationships between the concepts involved.)

    They are like magic tricks in the sense that we know something is wrong (the trick) but we can't figure out what. That's what makes then so maddening, as Evan said above in one of the first comments.

    In this case the trick seems to hinge on two things. First, pairs of mutually exclusive properties are implicitly excluded (goodness versus evil, goodness versus freedom as you point out, etc.). This seems to be what Eric means when he has Malcolm say that “mutual exclusivity of properties requires that one or both are implicitly affirming limits”. This conveniently takes care of the impossibility question (the searched for being cannot be impossible because the argument has a built-in feature excluding possible contradictions).

    As for the necessity, we can read (5) as saying that if a being is possible then it exists. Of course (5) says does not say this directly. Rather, and this is a nice trick, it states the equivalent statement: if a being does not exist in a world, then it is impossible in this world (of course this is false - the being may not exist in this world simply because nothing forces it to). Thus, question begging it seems to be.

    Is there is a way to reformulate the argument around these difficulties?

  25. Eric,

    You write: “My friend and co-author John sketched an argument over the phone for the impossibility of a world with no existents, based on the premises that there are necessary truths (propositions which would be true in any possible world) and that all truths need a truth-maker.

    An example of a necessary truth would be, I presume, a mathematical truth such as 2+2=4. I agree that this is a necessary truth in our world, for given the meaning of that proposition in our world there can’t be evidence that would defeat it, but why should one think that this truth obtains in all possible worlds? In the null world there are no numbers, and thus all propositions about numbers are meaningless. Or perhaps the idea is that numbers exist on all possible worlds. But why should one think that? In the null world, by definition, there are no numbers. I fail to see any contradiction in that being the case.

    Metaphysical possibility might be roughly characterized as what is consistent with the nature of being (and nonbeing).

    If we focus on the set of all metaphysically possible worlds, defined as those that are consistent with the nature of being, then it seems to me we are talking about the “nature of being” in the real world in which we exist, for we know of no other. If then the nature of being in the real world is God-based, then God exists in all metaphysically possible worlds, because given God’s goodness we can see that God would not want to delete Him/Herself out of the nature of being. If, on the other hand, the nature of being in the real world is not God-based, and if we further assume that the nature of being is a fixed thing which cannot be re-engineered, then God will not exist in any metaphysically possible world. So far then we get to the familiar “God either exists in all metaphysically possible worlds or in none”, or to put the same in less ambiguous terms: “The nature of being in all metaphysically possible worlds is either God-based or not God-based”. Which is a truism, for it states, twice, that all words which possess the same nature of being do possess it.

  26. Eric,

    You write:
    "Making an analogy to the real world, we might think that the laws of physics are like the rules, and the conditions that prevail at the big bang are like the initial numbers on the grid. And the question then becomes whether those rules and initial conditions necessitate everything else that is true of that world throughout its history or not. If they do, then all possibility within this framework is merely epistemic. If they don't, then other possibility assertions might be true."

    This at least seems to be the case with our world that much I believe.
    Given that that which 'caused' the Big Bang, existed outside of the framework of time. Thus this 'cause' has a timeless, unchanging, effect upon our universe. Therefore, whatever rules and conditional framework that the Big Bang existed within, the universe will always exist in; which is why the laws of physics are unchanging and balance out the way they do.

    To say that the 'cause' of the universe was 'caused' by something else, seems to posit some sort of temporal framework for these pre Big Bang, and hence pre-time cause and effect events, to work within.

    So, if we speak of a world like our own, whose existence was 'caused' pre-time, then the pre-time conditions that where existents at the moment of the worlds 'cause' will exert the same 'effect' upon said world eternally; since they exist outside the framework of time and therefore do not exist within a framework where 'change' is possible.

    Thus, if a world has the same temporal framework that ours does, then all possibilities, if I understand your logic correctly, are epistemic. If it does not, then they may not be.

  27. 1. What is the source of this particular Malcolm argument?

    2. Note that (5)--"If B existed in PW1 but not in PW2, then either (a) there is something that exists in PW2 that prevents B from existing, or (b) there is something missing from PW2 that B requires in order to exist"--looks very much like a form of the Principle of Sufficient Reason, and a dubious form at that. Why shouldn't something whose existence is possible simply exist at a possible world without requiring a reason for its existence? Why shouldn't something whose nonexistence is possible simply fail to exist at a possible world without requiring a reason for its nonexistence?

    3. "Possible worlds" are possible *descriptions*. (If a description of the way the world is or could be is not self-contradictory, then it is possible. A description on which nothing exists, if self-consistent, just *is* the null world.) If an object X's existence or a property F's exemplification is not logically impossible, then the statement "X exists" or the statement "F is exemplified" is true at some possible world without in any way implying a reason for X's existence or for F's exemplification. It is not as though we were surrounded by metaphysically real possible worlds; rather, we are concerned with consistent descriptions. The use of the PSR in (5) seems illegitimate.

  28. I don't know exactly why Malcolm's argument fails, but it certainly does fail. We know it fails because it can be used to prove the non-existence of God.

    I think I am right to think that these two statements are equivalent: 1) "God's existence is impossible", and 2) "God's non-existence is necessary".

    In the same way, the following two statements seem to mean the very same thing: 1) "God's existence is necessary", and 2) "God's non-existence is impossible".

    Norman Malcolm's premise that "God is either necessary or impossible" can be rephrased in this way without any change in meaning. Something interesting happens when you state Malcolm's argument using the alternative phrasing.

    1. The non-existence of God is either impossible of necessary (logically equivalent to Malcolm's premise)

    2. The non-existence of God is not impossible (since the statement of His non-existence is not self-contradictory).

    3. Therefore, the non-existence of God is necessary.

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