Modal Logic and the Ontological Proof

The ontological proof for the existence of God (really “proofs” or perhaps “arguments,” as there are various versions) has popped up in the blogs a few times recently: e.g. Ophelia Benson, Josh Rosenau, Jerry Coyne. You’ve probably heard this one; it was most famously formulated by Saint Anselm, and most famously trashed by Immanuel “Existence is not a predicate” Kant. A cartoon version of it would be something like

  1. God is by definition a perfect being.
  2. It is more perfect to exist than to not exist.
  3. Therefore, God exists.

Now, this is a really cartoonish version of the argument — it’s not meant to be taken seriously. This kind of ontological proof is a favorite whipping-argument for atheists, just because it seems so prima facie silly. Just ask Jesus and Mo.

This kind of mockery is a little unfair (although only a little). What’s important to realize is that the ontological proof is perfectly logical — that is, the conclusions follow inevitably from the premises. It’s the premises that are a bit loopy.

It’s instructive and fun to see this in terms of formal logic, especially because the proof requires modal logic — an extension of standard logic that classifies propositions not only as “true” or “false,” but also as “necessarily true/false” and “possibly true/false.” That is, it’s a logic of hypotheticals.

So here is one formalization of the ontological argument, taken from a very nice exposition by Peter Suber. First we have to define some notation to deal with our modalities. We denote possibility and necessity via:

Just given these simple ideas, a few axioms, and a fondness for pushing around abstract symbols, we’re ready to go. Remember that “~” means “not,” a “v” means “or,” and the sideways U means “implies.” Take “p” to be the proposition “something perfect exists,” and we’re off:

There is something beautiful here, even if it’s somewhat silly as a proof for the existence of God. It’s silly in an illuminating way!

As Suber says, the argument is “valid but unsound.” He pinpoints three premises with which reasonable people might disagree: 1 (“if perfection exists, it necessarily exists”), 2 (“perfection possibly exists”), and 5 (“if something is necessarily true, then it is necessarily necessarily true”). That last one is not a typo.

For me, the crucial mistake is some mixture of 1 and 2, mostly 2. The basic problem is that our vague notion of “perfection” isn’t really coherent. Anselm assumes that perfection is possible, and that to exist necessarily is more perfect than to exist contingently. While superficially reasonable, these assumptions don’t really hold up to scrutiny. What exactly is this “perfection” whose existence and necessity we are debating? For example, is perfection blue? You might think not, since perfection doesn’t have any particular color. But aren’t colors good, and therefore the property of being colorless is an imperfection? Likewise, and somewhat more seriously, for questions about whether perfection is timeless, or unchanging, or symmetrical, and so on. Any good-sounding quality that we might be tempted to attribute to “perfection” requires the denial of some other good-sounding quality. At some point a Zen monk will come along and suggest that not existing is a higher perfection than existing.

We have an informal notion of one thing being “better” than another, and so we unthinkingly extrapolate to believe in something that is “the best,” or “perfect.” That’s about as logical as using the fact that there exist larger and larger real numbers to conclude that there must be some largest possible number. In fact the case of perfection is much worse, since there is not single ordering on the set of all possible qualities that might culminate in “perfection.” (Is perfection sweet, or savory?) The very first step in the ontological argument rests on a naive construal of ordinary language, and the chain is no stronger than its weakest link.

62 Comments

62 thoughts on “Modal Logic and the Ontological Proof”

  1. Sorry I came so late onto this.

    Based on 55 years of observational (if anecdotal) experience, I’ve never once encountered anything that existed that I could characterize as “perfect” or for which I could ever even remotely determine or confirm the presence of perfection. Never.

    Here is another ‘cartoon’ argument to refute those first three points Sean passes along, which are indeed, largely based on the tripe of Anselm which I have myself steamed over for decades, those purported “proofs” which so easily elevates premise to the status of self-evident fact.

    The following come about from my logical deductions based on what I’ve personally observed as well as what I’ve absorbed from formal theoretical AND observational work conducted by – and as far as I can determine – totally consistent work performed by distinguished physicists that do in fact make sense of the real world:

    Conjecture A (soft version): existence bears no relationship or obligation to our conceptual notions of the ideal, and since our ability to observe is (from many other lines of compelling observational evidence) anything but ideal, we can’t ever attribute perfection to anything that exists even if IT IS REALLY THERE, or merely if we suspect it is present based on preconceived premises of the ideal, however preciously we may harbor them. Nature works independently of our minds (which explains why we are so chronically if not obnoxiously in error).

    Conjecture B (hard version, encompassing the soft version): there is no such thing as perfection in anything that exists. Existence is by definition a state of imperfection definable as an inevitable consequence of order subject to changes of state or attrition through interactions with time (entropy inducing disorder according to the 2nd Law of Thermodynamics). The notion of existence being an attribute of perfection is therefore ridiculous.

    That last part is a conclusion of the “harder version” of the conjectures presented here. (But I’m not quite finished…bear with me).

    Since observational evidence seems in excellent agreement with one form of “idealistic” theoretical understanding available to us (quantum theory), it seems consistent to conclude that the existence of anything is predicated on its interactions with other things that too exist, rendering anything that exists in particular to inevitably depart from its ostensible state of perfection the instant we attempt to observe it or whenever anything else which happens to interact with it takes up the baton of ‘observership’ (Anthropic Principles thereby be damned and scrapped).

    It is reasonable to conclude that since perfect ‘isolation’ (in space) or ‘stasis’ (in time) defines non-existence by definition, the corollary – of constant potential exposure to interaction in spacetime full of other particular objects or things which co-define each other – necessarily demands the attribute of existence.

    So where is the putative “perfection” in existence?

    CHANGE or PROCESS is the fundamental barometer and attribute of existence, and the moment anything that ‘is’, once investigated to determine its integrity through observational bombardment, immediately becomes something that ‘was’, and therefore something completely different boggles the mind, not in terms of how nature does her thing, but in terms of stubbornly reluctant we are in taking her at her word at face value.

    Things get altered and ordered things fall apart all the time. One does not even have to observe a thing to ‘know’ (from our idealistic understanding of quantum theory) that there are other things in profusion throughout the universe that perform the task of ‘observation’ through interactions that are quite serviceable in our sentient stead. (So much for the specific CONCEIT associated with the Anthropic Principle).

    How can that be reconciled with “perfection”? If something – ANYTHING – was ever once perfect, it must perforce become imperfect the moment it gets ever-so-gently clobbered by any photon in the ambient cosmic medium that might have the wavelength-energy so much as to at least jostle it about. True, it COULD ‘change’ from one state of perfection to another, but the initial state of perfection is destroyed. How ‘perfect can anything possibly be – including any particular state – that is subject to modification, let alone destruction?

    There’s another way out of all this.

    Perfection may only be satisfied by the absence of change or process. In other words, only the absence of interaction can provide perfection. The absence of space and/or time satisfies the ‘condition’…or, more accurately – at the risk of introducing further semantic confusion – the LACK of condition satisfies ‘perfection’. To be ‘perfectly’ blunt about it, NOTHING is the ONLY POSSIBLE reservoir of ‘perfection’, as long as one does not place too high a priority on the term “reservoir” as a ‘something’ with the potential to exist.

    I’m talking about a null reservoir – a non-existent NON-PLACE – the NOTHING – which is not anything. It isn’t even ‘anything’ that may be described or referred to from our position of existence, which seeks meaning out of RELATION. There isn’t any relationship with NOTHING or NON-EXISTENCE to be had from our point of view. ‘It’ is a non-representational non-artifact unalloyed to any reference frame we might bring to bear, even in principle. ‘It’ isn’t even an ‘it’. That pesky ‘thing’ of NOTHING that does not (at least in principle) even exist is almost impossible to extricate from the arena of ‘thingness’ in our minds, because we have no recourse but to refer to something to RELATE to or otherwise semantically INTERACT with. But ‘it’ is not ‘there’. There isn’t even a ‘there’ there for ‘it’.

    But let’s amuse ourselves by assuming a reference anyway, and refer to it as something we can get a handle on: let’s just call it ‘non existence’. (Be forewarned that I speak in terms of quantum-scale entities, lest I be accused of hideous New Age tendencies):

    Conjecture C (ultra-hard version, although it is really just an addendum to the ordinary hard-version, with the extra baggage of explanation, because I’m by now bleary-eyed and this is a cartoon): Given the profuse and well-documented examples of IMPERFECTIONS in existence, and the theoretical bases outlined above that takes care of the rest even without observation (with some measure of uncertainty, to be sure, but that’s why this is called a ‘conjecture’ which may or may not be ‘proved’ in this cartoon) the only possible perfection that can possibly be had is in non-existence.

    If there is any meaning in the notion of a system’s state ‘in between’ successive interactions, that ‘state’ must be considered non-existent and may only be appreciated by another (‘observational’) interaction. There is NOTHING we can see of ANYTHING that exists unless we or something else in the universe bother it through some interaction, and what we do in the act is see something that isn’t what it used to be. Literally, in the meanwhile (so to speak) ‘between’ interactions, it doesn’t exist. There isn’t any ‘in between’. On the strictly quantum scale, there simply isn’t any existence of any particular state between one interaction and the next.

    What can we deduce from this so far?

    Conjecture D (the super-duper ultra-hard version, personally preferred by yours truly): Perfection being an attribute not only unknown in the realm of existence, but a requirement of NON-EXISTENCE, demands that the popular notion of THE perfect being – otherwise known as ‘God’ – is NOTHING. Non-Existent in purist perfection, unbridled by any issues presented by the constant turmoil of interaction existence demands.

    Conclusion: Prayer to Perfection is literally tantamount to praying at nothing. There isn’t even any interaction of any kind that can possibly take place, even in principle.

    Quite aside from the puerile pop-estimation of the profundity expended in making sure that the MOST POPULAR OMNISCIENT, OMNIPRESENT AND OMNIPOTENT BEING in all the world ought not properly to be given a name, according to the most excruciatingly fundamentalist. And yet they invariably go right ahead and point out one, as in “Yahweh” – notice the capitalization indicating a proper noun NAME…even though “God”, though likewise capitalized, ironically does a considerably better job of un-naming ‘Him’ through millennia of anonymity, in the collective population of generic “GODS” – and distinguishing THIS one simply by the expedient of singling out the particularity: it’s “The One and Only” and “The” god that NAMES the IDEA of it. Personally? I’d would find it more charming if He was referred to with a decent name like the rest of us, who are all supposedly Products of His Infinite Creative Ability that can only be ushered forth by such Perfection. Ralph might be a good name. So would Fido (as so many who invoke Him to terrorize their followers might as they would with an attack-dog). But any Perfect Creator who fashions, out of His Infinite Perfection of Talent, a world that produces the horror of the Japan earthquake and tsunami of March 11, and the ensuing horrific deaths and incomprehensible suffering, I would rather name something like, say, “Arschloch”. Monumentally so. Stinking tothe highest heavens so.

    There isn’t any “god” that isn’t an artifact of people’s minds. Like their ideals of what constitutes “perfection”. What possible harm could it inflict to apply a moniker to nothing at all?

    The IDEAL of the perfection of God is quite evidently EVERYTHING in the MINDS of the penitent. The REALITY is that perfection is incompatible if not impossible with existence, which is where we are, evidently, in spades (as I so clumsily relate above). The peculiarly funny thing is that we have cultivated cultures of people that find it so simple and easy to abrogate their own interactive responsibilities in favor of investing huge amounts of time, energy and resources in order to solicit assistance from NOTHING IN PERFECTION.

    Anybody want to wager that the same irrational mindset doesn’t rule the actions of people when they ostensibly act on BEHALF of God? Like obnoxiously wealthy people and corporate interests?

    Yeah, I’ll agree that “God” exists as long as it is understood it is NOTHING BUT an existing CONCEPT that infects the minds of a troublingly preponderant number of people. No, I will not entertain any notion of a perfection of being that exists anywhere in the actual reality of the universal realm of nature, which indubitably EXISTS…UNLESS one posits that non-existence (never mind ‘perfection’ for the moment) has some legitimately non-trivial bearing on existence besides its basic penchant for framing existence in the cockles of people’s minds…which seems to pose something of an impasse. In any case, it isn’t beyond the stretch of ordinary imagination to confine its trajectory to what is either demonstrably observable, or via theoretical excursions consistent not only with observational evidence, but with itself.

    [The above remarks are certainly contingent not only on a ‘premise’ that may or may not hold up to scrutiny, and of whether or not I can correctly identify things that exist, properly distinguish between the idealistic artifacts of the mind and the artifacts of the real world we acquire through our senses, and whether or not I understand what existence even means in terms of any workable definition that conforms to what nature screams at us through our well-documented scientific successes so far. But this cartoon certainly isn’t very much harder to appreciate than any ridiculous religion. Is it?]

  2. #49 Guido — How right you are. I believe the universe can be understood as a system for making sense. But physics has followed a path by which many physicists find the universe to be “pointless”, i.e. making no sense. Maybe wounds from the battle for free thought are not healed, or maybe the physical model is incomplete. But there is something pathological in using sense to make nonsense out of sense. (Physicist, heal thyself!)

  3. You can’t have consciousness in the universe without quantum mechanics, period. Who ordered that ? – quantum mechanics (not muons).
    Well it’s some topology of spaces just before the universe began so the multiverse goes. But doesn’t that beg the question – why a life-mind universe got selected. OK, you can say it was Martin Rees’s infinite coatshop, eventually you will find a coat that fits.

    But if this is true (I knew the some of the investigators, several distinguished Professors of physics, psychology and electrical engineering) – It was a three year international study:

    http://www.thescoleexperiment.com/

    and

    http://www.spr.ac.uk/main/publication/afterlife-investigations-has-science-proved-afterlife-dvd

    Doesn’t this mean life-mind universes are massively important, in that those are selected in which consciousness, or some kind of heightened awareness, continues after death? Inescapable. And this puts meaning back into life, notwithstanding Weinberg.

    It also arguably makes the universe very very God-like. Just as in Solaris (the recent film), when you die the universe looks after you – keeping your meaning going. Beautiful.

    All this occured in a bare stone cellar with other witnessess.

    It is from the 300 page Scole Report (1999)

    This is a report from an academic psychologist (a witness) – actually from The Scole Report – 1999:

    “The first phenomena that I saw were small points of golden light dancing in the corner of the room…They danced animatedly upwards and downwards…. Shortly following this, there appeared a ball of diffused light, which I estimated to have a diameter of about 20 cm, close to the ceiling in the same corner…as the lights. The ball had no physical boundary: it was simply a three-dimensional orb of diffused golden light. It hung suspended for a moment in the corner about 30 cm beneath the ceiling. Slowly the orb moved toward the centre of the room, pausing above the centre of the table round which we were all sitting. It lowered itself by about 17 cm, remained still, then retreated slowly upwards and backwards into the corner…There were no beams of light to the orb, and the light was not reflected onto a surface; it moved independently in space. This occurred twice in succession, and I became aware of an overwhelming feeling of gentleness and love which seemed to accompany this phenomenon or, more accurately, which this phenomenon seemed to embody.”

    Very, very challenging.

  4. Herman Newticks

    Alvin Plantinga made a nice modal ontological arg for the existence of God in The Nature of Necesssity. I recall that it started from the premise that God is the being such that none greater can be conceived. Then plowed into the same modal arg. Fun!

  5. Sean,

    I enjoyed reading this. Two things:

    1. I have been told the Godel’s version of this proof is more difficult to critique than Anselm’s. It would be fun to see you rip that version apart.

    2. Your numbers analogy gives theologians fuel for their cause. They might point out that there isn’t a largest finite number but that as numbers get bigger they approach infinity and then tie this into an argument that not only does a most perfect being exist but that, by analogy, he must be infinite. (By analogy)

  6. This whole argument rests on one major flaw. There is no such thing as perfect. Any mathematician will tell you that there are no perfectly straight lines in reality. There can be no perfect lines thus there can be no perfect objects or things with names. Perfection is an illusion just as measuring 5 inches is more accurately measured as 5.00000000000029. It depends on the perception and logic dictates that while there is assumed truth to things there can be no perfection.

  7. 1. I’m thinking of a spacecraft that travels faster than light and is blue.
    2. Things that are blue reflect blue photons.
    3. Things that don’t exist don’t reflect photons. (“It is more blue to exist than to not exist.”)
    4. Therefore anything blue exists.
    5. Therefore the blue faster-than-light spacecraft exists.
    I created something from nothing. Yay, me!

    Just because a combination of traits (including existence) is conceivable, doesn’t mean it is actual. It just means your brain is not restricted to conceiving true concepts.

  8. I think there’s a problem with treating existence as a characteristic. Any other characteristic of an entity (whether conceived or actual) might be alterable, but you can’t just add or subtract existence.

  9. I think all these reasoning about God are so non-sense; I don´t waste my time on that.
    That is so useless… It´s so wrong….

    Several non-sequitur arguments: in the argument, in the comments…

    Though there´re some comments containing quantum mechnics, topology and about inteligent consciousness. Hmm, MAYBE these are worth to be read until the end.

    P.S.: from someone who believes in God.

  10. Has anyone in this discussion read St. Anselm’s book from which the proof comes, “The Proslogion”, along with the Reply to Gaunilo? The second is an addendum which sketches out the ontological argument in its completeness. While many have disagreed with St. Anselm (including Gaunilo, to whom the aforementioned letter was addressed, as well as St. Thomas Aquinas), his argument is actually subtler than the way it’s portrayed here. Gaunilo had said what some say here, namely, that you can use can use the argument to prove that a perfect island (for example) exists. Anselm explains how his argument can only apply to God.

  11. Sean, Aquinas himself, whilst against this argument, unwittingly and surreptitiously, uses the notion of perfection in his fourth way-the degrees of perfection. Michael Scriven leaves to the reader to solve it as a silly argument in his “Primary Philosophy.”
    And the ” Star Trek’ argument is that as He is perfect and the Cosmos is imperfect, then He cannot exist!
    Plantinga the purveyor of solecistic, sophisticated sophistry of woeful, wiley woo- ignorant, complicated nonsense, claims that omni-God cam use flourishes-imperfections- in designng matters whilst limited God must use what little He has- the perfect in effect.
    He claims to have evisicerated the logical problem of evil, but he uses the unknown defence argument, which itself is the argument from ignorance and which makes topsy-turvy morality!
    No wonder then that Keith Parsons,except for stuff @ Secular Ouptpost, has gotten out of the philosophy of religion!
    Sir, please check out that blog of mine!

  12. Perfection emerges as an artifact of awareness within an organically individuating perceptual node. How might perfection thus be envisioned? It would lack nothing/include everything — eternal infinity. Coherence, often described as shared existence, may be the closest approach of science to such a conception. Indeed, Sean enlists ‘coherence’ as his criteria for judgment.

    Is perfection possible? Certainly not in the finite world of individuating perception, ubiquitously animated by awareness. But as sages have noted through the ages, awareness is everywhere capable of shucking its local trappings, thereby exposing underlying coherent superposition of boundlessly evolving probabilities. Shorthand for this might be coherent awareness.

    Note that ‘boundlessness’ acquires meaning only within the purview of a distinct perspective hosted by an evolving organism.

    Necessary versus contingent existence? This is a perversion of the very notion of existence. Clearly, only existants may be regarded as either necessary or contingent. Existence IS…period!

    In endlessly localizing perception, color, taste, smell and feel subsume organic domains, evoking recognition, anticipation, articulation and memorialization. The collective of all such local perceivables comprises a consensus universe, within which notions of creator and supreme-being inevitably percolate.

    Perfection is uniquely inclusive of all instances of existence. That’s a singularity. Inclusion of every instance of existence is essential to perfection, as any deficit is naught but imperfection.

    In the realm of scientific thought, it’s perhaps most eloquently articulated as an infinite and eternal multiverse, embodying all shimmering perspectives of fractally proliferating perception.

    Instances of existence boil down to awareness. Bacteria are less filtered from awareness than plants and animals. Humans deploy filters that subtly surf experience in domains of recognizing, anticipating and articulating — with memorializing as the ultimate articulation. Without situational recollection of present remembrance nothing can ever happen. That’s the organically based engine of time — present moment remembrance of experience coupled with situationally induced recollection. Even as experience pumps up remembrance, circumstantial recollection conditions recognition, anticipation and articulation.

    Symmetry is irrelevant to a non-dimensional singularity such as perfection.

    No-thing may be regarded as shorthand for all existants.

    Perfection isn’t a ‘largest possible number’ but infinity — again a singularity.

    So how might such an entity be described? I’d suggest labeling it ‘coherent awareness,’ as awareness — the animating force of perception — illuminates the meanings of local entities.

    Coherent awareness is non-dimensional; or, equivalently, omni-dimensional. It’s coincident with inflating probabilities entangling with boundlessly propagating perception. Coherent awareness is an eternal superposition condensing as endlessly individuating organisms on scales ranging from Planck to cosmic.

    The inevitable question of will arises here. Without feedback, all notions of anything are randomly generated artifacts of imagination.

    I’d propose that while coherent awareness encompasses all-that-is, it also occasions myriad singularities of local perception. The unique relationship of each such perceptual node to all others inflates its own bubble universe.

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