Tyrrell McAllister’s thread renewed my interest in some of the classical ontological God arguments. I’ve been googling for the past couple of days. I started with Anselm of Canterbury and ended up with Tisthammer of Minnesota. Who? Wade Andrew Tisthammer, a Phi Theta Kappa second year junior college student who wants to major in computer science.
Tisthammer’s presentation is by far the clearest, most succinct, levelheaded, and dispassionate I’ve seen by anyone anywhere. He defines God as the “greatest possible perfection”, introduces the reader to even the most mundane details of concepts he will use, and then presents his argument in an elegantly simple modal symbology reduction.
1. G ⊃ []G If God (on the definition above) exists, He exists necessarily.
2. ~[]~G This states that G is not impossible, or <>G
Those are the premises. The conclusion logically follows as demonstrated:
3. []G ⊃ G If God necessarily exists, then he exists.
4. []G ∨ ~ []G Law of Excluded Middle
5. ~[]G ⊃ []~[]G Becker’s postulate applied to ~[]G
6. []G ∨ []~[]G 4,5 substitution
7. []~[]G ⊃ []~G 1, modal modus tollens
8. []G ∨ []~G 6,7 substitution
9. []G 7,2 disjunctive syllogism
———————————
∴ G 9, 3 modus ponens
(Rats. On preview, some of the symbols aren’t showing up here. But I think you can tell by implication what they are, and you can click the link to his actual paper.)
I take the position at least at this point in the debate that Tisthammer’s argument is sound. And I think I would enjoy arguing against an opposing view. Please deal in this thread only with Tisthammer’s treatment, and not any of the many others.
This isn’t an original argument (or even an original treatment of the Ontological argument, actually). I think his modal argument is due to Charles Hartshorne.
A description can be found on Peter Suber’s web pages. Suber’s conclusion is correct, the argument is valid but not necessarily sound. Becker’s postulate is not valid in all modal systems; and we’re not sure which modal system is best to describe everything we mean about necessity and possibility.
Yes, I know that the modal sequence is not original. It is merely elegant. I also saw Suber’s work in my surfing. But Tisthammer does a much better job in his post-proof analysis, directly addressing such objections as Suber only describes, e.g., the notion of the argument applying to the universal set of propositions.
Regarding criticisms of Becker’s postulate, you can argue this way, and circumvent it:
Ahh. My criticism was that Mr. Tisthammer has taken Mr. Suber’s (properly attributed) reformulation of Hartshorne’s modal argument, without any attribution.
Did McHugh avoid Becker’s postulate? It looks like he avoided assuming ~~p. He has avoided assuming that God is possible, by assuming “No positive descriptive term can be applied to God, by the definition of God”, “If no positive descriptive term can be applied to God, then the concept of God is free from contradiction”, and “if the concept of God is free from contradiction, then God is possible”.
He replaces one assumption with three that are equivalent. Are the three new assumptions more palatable than “God is possible”? I don’t know. I find his statement that “The issue of proving the logical possibility of the existence of God is no longer a difficulty since the consistency of the God concept is guaranteed by the self-evident truth of Axiom 1.” to be mildly offensive. Is “If no positive descriptive term can be applied to God, then the concept of God is free from contradiction” self-evident? What say you of this claim?
I still question Becker’s postulate here (and in general).
In both the Hartshorne proof and the “New Argument”, it is used to get from the assumption of ~p to the conclusion ~p (which directly contradicts one of Anselm’s assumptions), in order to use disjunction elimination and conclude p. Besides questioning the two assumptions due to Anselm, we can clip out this bit of proof and ask about it.
Does it make sense that if perfection (or God) is not necessarily true, then perfection (or God) is necessarily not true? That is: if there is a possible world where perfection does not exist, is it the case that perfection does not exist in any possible world? We can derive this in certain modal systems, but again, it’s not clear that this says anything about truth in the actual world.
In particular, it appears to me that the accessibility relation on possibile worlds has to be transitive: xRy AND yRz -> xRz; and also have this property: xRy AND xRz -> yRz. I think these two properties are necessary for the argument, but I’m not sure if they’re sufficient to ensure that ~p->~p.
As for McHugh, I do not accept that “No positive descriptive term can be applied to God”; and I think that even so, Objections IV and V to his “new argument” are damning enough, and that he does not adequately address them in his responses (follow Lib’s link to read the details).
I won’t argue with his logic, which seems fine, but rather with his premises, which seem flawed.
Consider first his definition of God as the “being of maximal perfection”. What is this supposed to mean? How does one go about ranking different beings in terms of relative perfection in order to determine who is more perfect than who? “Perfection” is an ill-defined term in this argument.
He then uses this fuzzy definition to establish premise #1: “If the most perfect being exists he must exist in all possible universes.”
He uses a different form of the definition to establish premise #2: “The existence of the greatest possible being is not impossible.”
The problem is that the beings in #1 and #2 are not necessarily the same. Because he’s using a fuzzy definition of “perfection” and calls the entity in question “God” (with the weath of symbolic baggage that carries) he’s making unwarrented assumptions.
In order to tie these two different definitions of “God” together you need another assertion:
“There must exist an entity that exists in all possible universes.”
I think he misses his own response to an interesting point.
But is this really the case? For example, couldn’t we think that a God that could do anything could also make it so he is not the most perfect being, instantly removing a quality of existence? And could we not conceive that a God that exercized its power to be more perfect than one that did not since we already decided that actuality was more perfect than possibility (since we decided that the most perfect being would necessarily have the quality of being an actual being)?
It is not clear what “perfect” means as a logical construct itself… that is, his G premise is not obviously a logical construct. Can the most perfect being create a rock so heavy etc? Is a god that can contradict his own power more powerful than one that cannot? Is a God that can permantently limit his power more powerful than one than cannot?
Yes, but what is that? I don’t feel it is as obvious as he might make it seem. In addition to that, his quip about “the greatest possible island is impossible because that’s like the greatest possible integer” is probably a decent response, but why isn’t “the greatest possible power” subject to the same thinking? Why isn’t “the most possible knowledge” subject to the same thinking?
He remarks that we may not enumerate all the possibilities for the most perfect island because we can always imagine an island that is just a little bit greater, but I don’t see why this is so. Consider that God has properties X, Y, and Z. Those three properties (whatever they may be) are what defines a being as a god. An island has these three properties: A, B, and C. Now, the most perfect being is one in which X, Y, and Z are at their so-called (hypothesized) intrinsic maximums. Why cannot the perfect island have instrinsic maximums? What distinguishes X, Y, and Z from A, B, and C?
It is not obvious to me. The most perfect island has the perfect temperature, the perfect size, and the perfect amount of resources. Does it matter if those perfections are derived from my own preferences? Again: that isn’t clear from the proposition. “The most perfect” is not a clear logical construct since logic doesn’t seem to deal with values that vary, only to things which the law of the excluded middle applies.
Could we not use the proof several times by restating G to be “A exists” where A is one of the most perfect properties of islands, and instead of “exists” being interpreted as normal we instead say “A describes a property of some object”? Then the same for B? Then the same for C? Then we have instrinsic limits defined, and we can merrily prove the existence of the most perfect anything. Why couldn’t we do this for A?
Since there are intrinsic maximums, by hypothesis, to the qualities of god, then these, too, must exist necessarily. Then any being which can possess the same properties as god has a “most perfect” form which necessarily exists. So what distinguishes God from Man? Nothing that I’ve ever heard.
A great many troubles with this, IMO, though I’ll be interested to see what you have to say, Lib, before I go emial this guy.
This Christopher McHugh attempts are merely a poor phenomenological too too dance around the intuited axioms q -> Nq (Perfect Being implies Necessarily Perfect Being, i.e. a Perfect Being is not contingent) and ~N~q ( not Necessarily not Perfect Being, i.e. a Perfect Being is possible). For those that didn’t read it, McHugh’s essential point is that:
God can’t be explained in positive terms because God is what we cannot describe.
Hence, God is free of contradiction. A contradiction can only take place when juxtaposing a negative with its positive. Since the Perfect Being can’t be described in terms of positives, we can’t threaten the Perfect Being with any negatives. Ergo: no contradiction can occur.
This is where he goes wrong. The concept of contradiction can only exist in a binary world. To say that the Perfect Being is free from contradiction is perhaps valid, but to assume that this means q->Nq is non sequitur. Contradiction becomes meaningless outside a non-binary world. It would be more accurate to say that God is beyond contradiction. Therefore, it makes no sense to apply a modal necessity in thinking of Her.
It’s a bit of an oxymoron: how can we clearly (in a non-contradictory fashion) think of what we cannot think of except in terms of what we cannot think off ???
A possible world is a set of true propositions. Truth in modal logic is always relative to a possible world. We say something is true in a world exactly when it is contained in the set of true propositions for that world. Something is necessarily true in a world if it is true in all possible worlds accessible to that world. Something is possibly true in a world if it is true in some possible world accessible to that world. Here is the best treatment of modal logic and accessibility I could find online.
The modal version of the ontological argument asks us to accept the two assumptions due to Anselm (or their equivalents). The first of these is if God exists (in the actual world), then God exists in all possible worlds accessible to the actual world. This is the claim that necessity is a property of God. Is there a possible world where God does not exist? Is that world accessible to the actual world? We must ask you to admit that if God exists in this world, then it would not be possible to imagine a world without Him. If God exists in this world, then his existence is intrinsic to all worlds.
The second of these is that God exists in some possible world. This is the claim that God is at least possible. We must ask you to accept that even if God doesn’t exist in this world, it is impossible to imagine that he fails to exist in every world. Why? It seems that the spirit that caused you to accept the first assumption works against this one. If God doesn’t exist in this world, then why is his non-existence not intrinsic to all possible worlds?
Together, these two assumptions amount to the claimant (the theist) getting the benefit of the doubt in two (opposing) cases. If God exists, then he’s necessary. If God doesn’t exist, then he’s at least possible.
Then, the modal argument performs a dirty little trick. We have that God’s existence in the actual world would compel God’s existence in every world; and that God exists in some possible world. We use a modal “trick” (Becker’s postulate) to identify the possible world where God exists, and make that serve as the “actual” world causing God to exist in every world.
There are two ways for one to argue for acceptance of Becker’s postulate here. We could argue for the validity of the inference rule. We really only need <>A-><>A. This axiom is known as 5, and S5 is the (simplest) modal system that contains it. The question should be: if something is possible, does that mean it’s necessarily possible? S5 is equivalent to adding the axiom B to the system S4. Suffice it to say that S4 isn’t all that controversial, but B is. The axiom B is A-><>A. If something is true, then it is necessarily possible. This might seem like something we’d accept, but it’s equivalent to <>A->A. If something is possibly necessary, then it is the case. Do we believe that? Details can be found here. One last word: B is equivalent to requiring the accessibility relation on possible worlds to be symmetric; 4 requires transitivity, and 5 requires it to be transitive and symmetric.
The other way is to push for the acceptance of this particular instance of Becker’s postulate as an axiom. To state as an axiom that if it is possible for God to not exist, then it is necessarily possible for God to not exist. That is, if God does not exist in some accessible world, then there is a world accessible to every accessible world where God does not exist. I think we’re too far into the accessibility relation to claim that anything is “self-evident” here.
Neither the inference rule or the assumption seems self-evident. Since the proof rests on this trick to cause the possible world where God does exist to be used compel His necessity; the acceptance of this rule requires some justification.
An awful lot of people have looked at Anselm’s Proslogion and discerned modality. Probably the first was Norman Malcolm, who published “Anselm’s Ontological Arguments” in the Philosophical Review in 1960. Hawthorne was one of the people to rediscover Anselm (and probably the most popular because he practically stood alone in the twentieth century in defense of theism), but he certainly wasn’t the only one. Robert M. Adams, Alvin Plantinga, Paul Oppenheimer, and Edward Zalta were others. Hawthorne’s treatment, in fact, was highly specialized. He insisted that God’s nature had to be panentheistic.
I’d say that McHugh gets off-base a bit, unnecessarily, in my opinion, skirting Becker’s postulate. But his point about God being free from contradiction is consistent with his definition (2) of a positive descriptive term as being positively conceived. The perfect thing is indescribable except in very informal ways, such as we describe God here at Straight Dope, and how He is described in the Bible. Even Jesus Himself made it clear that, although when we see Him we see God, God is even greater than He. This was because His manifestation presented God through the dim filter of our senses.
I don’t know why you would question Becker’s Postulate and not question the Induction Axiom (assuming you don’t).
I disagree.
In what other way can God be described? If you aren’t careful, you’ll end up examining a strawman of the actual argument. By definition, there can be one and only one manifestion of any greatest thing.
Because of the type of thing that islands and integers are. But certain other things do have intrinsic maximums, like angles, for instance. There is no greater angle than 360 degreees. But for every integer (assuming you buy the Induction Axiom), there is a successor.
As Tisthammer explained in some detail, “It would seem that there are intrinsic maximums for the qualities of God. For instance, God’s omniscience can be defined as knowing everything that can be known, and God’s omnipotence can be defined as being able to do anything that can be done. There aren’t any greater possible conditions for those aspects of knowledge and power. They are intrinsic maximums.”
That’s what he did.
It is irrelevant whether our thoughts are confused (or contradictory). The question is whether the argument is consistent. And it is.
You try to imply a contradiction where none exists. These do not oppose. If it is possible for something to exist, then it cannot be impossible. And if something is perfect and exists, then its existence is necessary because of its definition. The greatest possible thing must have the greatest possible form of existence.
Dirty little trick? Is completing the square to solve a quadratic equation a dirty little trick? You had a broad choice of descriptives, and selected one that is pejorative and misleading to the extreme. As your source explains, “One simple way to protect ourselves is to formulate B in an equivalent way using the axiom: <>A->A, where these ambiguities of scope do not arise.”
If I’ve figured out your syntax right, I’ll respond this way: the greatest possible being must of necessity exist in at least one possible world, and that is all the proof attempts to demonstrate. Remember that an ontological argument is a priori.
—d = No positive descriptive term can be applied to God.—
Then how can it meaningfully be called “God,” or anything? This quite litterally an admission that the thing we are describing is in every way, shape, and form, indistinguishable from non-existence… with the additional meaningless stipulation that it is perfect (whatever THAT means)… a perfect WHAT?
—In what other way can God be described?—
Well, hopefully there is some other way, because I don’t see how the statement in question successfully describes anything. First of all, the word perfect requires a modifier, as my exclaimation above demanded: perfect in which respect? All of them, including a perfect jerkwad? Or just those that one thinks might be nice for a god to have?
But second of all, if used as it seems to be, this amounts to a yet another via negativa definition: in other words talking about what god doesn’t have (any imperfection, any limit on its knowledge, any limit on its power) instead of what it actually is. How can one logically discuss an entity for which no actual characteristics can be ascribed? Unless positive characteristics are added, the entity remains indistinguishable from non-existence (which also lacks imperfections, limits on power, limits on knowledge, etc.)
You’re like a slippery eel, Libertarian. You don’t even comment on the essentials of my argument against this McHugh, mainly that going from his definition of God to q->Nq is non sequitur.
McHugh tries to convince us of the self-evidence of q->Nq. He tries to prove that “God is not contingent” follows phenomenologically from the very definition of God (i.e. that which is outside the realm of description).
I don’t question his argument once we’re off the ground (which is little different from Anselm’s original one). It’s the axiom q->Nq which is in question. Basically, the plain to heaven remains firmly stranded on the tarmac.
McHugh claims God is not contradictory. But how can we be sure when we can’t even conceive of a contradiction in the absence of positives? Contradiction has no apparent meaning outside the descriptive world. Therefore whether the possibility of God implies a necessity or not must still be judged prima facie, just like the idea that a being can’t be an not be must be judged prima facie.
Formally McHugh’s premise is:
G: God
A: All Descriptives
G = ~A
Ergo: ~A <–> ~A
Ergo: ~A --> ~A
Que??? I must have this massive blind spot in my brain. I guess it’s my missing esoteric eye…
Well, I don’t think you’re a slippery eel, Libertarian. I’m really glad you kept this thread alive; and I’m preparing a response. It’ll take a while, so stay tuned.
Your “exclaimation” notwithstanding, Apos, a positive descriptive term is a formal construct of complexity theory. A positive property is simply one that is observable. To say that no positive descriptive term can be applied to God is merely to say that He is not, for example, a dodecahedron.
I presume, Rsa, that you’re being humorous, since I seem to recall that you have a better grounding than that in logic theory. For the benefit of lay onlookers, however, I’d like to stress that it is NOT a non-description. It is simply not a description of something observed. Please kindly limit discussion to serious comments on the formal logic aspects of Tisthammer’s argument.
I’m not trying to claim an actual contradiction. I’m claiming that those who are likely to accept the first assumption will likely reject the second and vice versa. Or at least they will insist on the adoption of other reasonable assumptions that are damning to the proof.
The first assumption insists that if God exists, then he has the property of existence in all relevant possible worlds. Then he is necessary. Worlds not containing God, while imaginable, are not sufficiently like our own to be useful for the argument.
The second assumption is, regardless of whether God exists or not, he must be at least possible. First, we begged that if God existed, worlds without God were not relevant. Now we beg that even if God doesn’t exist (here), there is some relevant world where he does. Before, we narrowed relevance to be “sufficiently like the actual world to agree on God’s existence”, now we widen it to be “anything we can imagine that is similar to the actual world”.
This is my criticism. On the one hand, if God exists he exists necessarily. On the other, his non-existence can only be contingent (not necessary). I am offended by this asymmetry of the argument.
I think that if one accepts p->p: “if God exists, he is necessary”, then one ought to also accept ~p->~<>p: “if God does not exist, then he is not possible”. If he actually does not exist, then there is no purpose for him, no reason for him, and nowhere for him to hide. The same insistence that if he existed, only worlds containing him would be relevant should lead us to conclude that if he doesn’t exist, only worlds not containing him would be relevant.
Likewise, if one accepts <>p: “God is possible”, then one ought to accept <>~p: “it is possible God does not exist”. The insistence that God’s possibility is imaginable should to compel us to admit that his impossibility is also imaginable.
The refusal to admit these two new axioms (while insisting on the first two) is a problem. I’m not claiming that one must accept the new axioms, but I think elucidating reasons to reject them is instructive, since they are the modal duals of the original axioms. If we do accept the new axioms, we have the following valid proof that God cannot exist (the modal dual of the other proof, and I carefully verified each step):
1. ~p -> ~<>p Assume: if God doesn't exist, he isn't possible
2. <>~p Assume: it's possible God doesn't exist
3. ~<>p -> ~p Modal axiom: what's impossible cannot be
4. <>p V ~<>p Excluded middle: God is either possible or impossible
5. <>p -> ~<>~<>p Becker's postulate, modal status is necessary:
If God is possible, then it is not possible he is
impossible
6. ~<>p V ~<>~<>p Substitution(4,5): God is either impossible or not
possibly impossible
7. ~<>~<>p -> ~<>~p Modal modus tollens(1): if god is not possibly
impossible, then it's impossible he doesn't exist
(i.e., he is necessary)
8. ~<>p V ~<>~p Substitution(6,7): God is either impossible or
necessary
9. ~<>p Disjunction elimination(8,2): God is impossible
10. ~p Modus ponens(9,3): God does not exist
Pejorative? Yes, intentionally so. Misleading? Obviously, I don’t think so. I can summarize the modal argument:
If God exists in a world, then he must exist in every world knowable from that world
God must exist in some world knowable from this world
If a world is knowable from this world, then this world is knowable from that world
Therefore, God exists in this world
First, one must insist that if a world contains God, then only worlds that also contain God are knowable. Next, whether a world contains God or not, there is always at least one knowable world containing God. Now knowability must also be symmetric? This notion of knowability (relevance, accessibility) is a moving target, and I think a significant flaw. Step 3 is the trick.
It is not obvious to me that when we speak of possibility, necessity, and actuality, we are governed by a symmetric relevance relation. That insistence on such a relation is required to prove the existence of God is suspicious. I call it a “dirty trick”, because by agreeing to two “reasonable” axioms about what we mean by God, we are suddenly and unexpectedly warped into a possible world and then back in to the actual world. I’m not entirely convinced this is a reasonable round-trip.
By the way, adopting <>A->A as B resolves the ambiguity of scope, but does not resolve the problems with B itself. In fact, it makes the problem explicit. Do we believe that whatever is possibly necessary is actually true? I can be quite specific with my problem: I question whether the relevance relation on possible worlds is symmetric. I accept reflexivity (we can know this world). I accept transitivity (we can know worlds which are knowable from other knowable worlds). I question whether this world is knowable from every knowable world.
I think McHugh’s expansion of “God is possible” is instructive, because we can see precisely where he goes (might have gone?) wrong. My claim is that we do make positive claims of God: he is loving, personal, triune, etc. It is hard to imagine that “God is triune” is a claim that he is “not monoune, not diune, not quadriune, not quintiune, …”. I maintain that we do describe God through “acts of direct conceptualization”. Perhaps McHugh would consider any positive conclusion about God (other than his existence) to be unfounded, but I’m not willing to admit that.
I also maintain that if we had two descriptive terms that were each other’s negations, then they cannot both be negative terms. We cannot simultaneously insist that “invisible” is the negation of the positive term “visible”, and “visible” is the negation of the positive term “invisible”.
The proof contains exactly such a contradictory pair of descriptive terms. On the one hand, God is possible (by assumption). How is this not a positive descriptive term? Only if you insist that possible is defined as “not necessarily false”. Fine. On the other hand, God is necessary (the penultimate conclusion). How can this not be a positive descriptive term? Only if necessity is defined as “not possibly false”. So possibility and necessity are each other’s negations, and neither of them is defined positively? I don’t understand. It seems to me that one of them must be taken as a primitive, positive modality.
I further object to McHugh’s axiom 2 (that allows him to conclude that God is free of contradiction). If God is the most perfect thing that we can conceive; then isn’t a “new” God (with all the old one’s powers), except not governed by the rules of classical two-valued Aristotelian logic even more perfect? Is a God that can embody a contradiction more perfect than a God in whom contradictions are impossible? I would claim that he is (I think God quite happily embodies contradictions, making any proof that uses excluded middles and non-contradiction suspect).
I don’t even see how this gets out the gate. Existence is not a predicate, and all that. But this seems to go even farther down a twisted road.
—d = No positive descriptive term can be applied to God.—
Then how can it meaningfully be called “God,” or anything at all? This quite litterally an admission that the thing we are describing is in every way, shape, and form, indistinguishable from non-existence… with the additional meaningless stipulation that it is perfect (whatever THAT means)… a perfect WHAT?
—In what other way can God be described?—
Well, hopefully there is some other way, because I don’t see how the statement in question successfully describes anything. First of all, the word perfect requires a modifier, as my exclaimation above demanded: perfect in which respect? In all respects, including a perfect jerkwad, a perfect coward, a perfect sadist? Or just those that one thinks might be nice for a god to have?
The concept of perfect is only meaningful when it describes the degree of similarity of a certain entity to a certain ideal. Perfect is not a term meaningfully applied to “existence,” because existence is not, say it with me, a predicate. Existence is not an extra characteristic of some beings: it is, rather, what allows a being to have real characteristics in the first place.
And second of all, if used as it seems to be, this amounts to a yet another via negativa definition: in other words talking about what god doesn’t have (any imperfection, any limit on its knowledge, any limit on its power) instead of what it actually is. How can one logically discuss an entity for which no actual characteristics can be ascribed? Unless positive characteristics are added, the entity still seems to remain indistinguishable from non-existence (which also lacks imperfections, limits on power, limits on knowledge, etc.)
I screwed up, and illustrated the difficulty of reasoning about complicated modal utterances. What I said was:
What I should have said is:
Likewise, if one accepts <>p: “God is possible”, then one ought to accept <>~p: “it is possible God does not exist”. The insistence that God’s existence is imaginable should to compel us to admit that his non-existence is also imaginable.
I was using “possibility” and “impossibility” in a sense different from the modal diamond quantifier.
I disagree. Existence is a predicate in modal discourse. In a possible worlds framework, it is meaningful to talk about the property of existence.
There are two main ways to interpret possible worlds (besides outright rejection). The possibilist believes that possible worlds exist as surely as the actual world. “Actuality” merely identifies the world where an utterance occurs. In this view, it is meaningful to talk of objects that exist in some worlds and not in others.
The actualist believes that possible worlds are merely abstract imaginings about the way things could be (or could have been) in the actual world. In this case, existence is the property that discriminates between things that are actual and things which merely inhabit imagined possible worlds.
In either case, once we agree to engage in modal discourse, it becomes meaningful to talk about whether an object exists, and to talk about properties of non-existent objects.
I think agree with your other objections, but I also agree with Lib: talking about McHugh’s argument leads us too far astray from what’s interesting about the modal ontological argument.