Well, except that it isn’t the logic itself that is presumed to apply to all possible worlds, but merely the conclusion. As Newton pointed out earlier on, Tisthammer actually takes a journey out of one world and into another (if we insist on applying both tenets of Becker’s Postulate) and then back.
Newton
I’m simply in awe of your clear and concise expository. Your ad hoc symbols are eminently identifiable, and the way you laid out your tableaux is as clean and readable as any chart I’ve seen. Please accept my sincere and humble thanks for your participation in this thread. I hope that my gushing has not embarrassed you, but dammit, good scholarship deserves praise.
I especially like this, though:
The funny thing is that there is nothing more futile than attempting prove your own existence!
That’s because, in order to do anything at all, including proving your existence, you must first exist. That makes your existence axiomatic. Therefore, because your conclusion — that you exist — will always be the same as your axiom — that you exist — your argument will always be circulus in demonstrando.
At the end of the day, all logic is tautological. And we believe in it on faith.
No, no, no! That’s Truth’s mistake. Look at Newtons charts above. All that has been proved is that the convergence of all maximums exists. No particular maximum has been shown to exist, and I’m not sure that it could be ontologically because defining it would be formidable.
Almost with me, Lib. Now: the perfect logic system as described above has inherent maximums/minimums that would go along with our intuitively described interpretation of “perfect”. Now we come to the point:
Does God know this system?
Since he knows everything, he must. But the description of a perfect logical system is logically impossible. Thus God cannot have a perfect logic system that is both complete and consistent. Thus God does not have perfection itself because, as I tried to mention before but only now have a footing on explaining, perfection itself along multiple qualities is itself contradictory.
Crap, I forgot to add my second point. This was that we still aren’t exactly clear on what these inherent maximums are. Tishammer’s proof could simply reduce to Aleister Crowley’s “there is no God but Man” after a rigorous epistemological examination.
And, shit, a third point. These three posts should really be considered as three seperate attacks on the same argument, not a continuation, ok?
Let us also remember that God’s explicit definition changes with respect to the world under consideration. For instance, in a world where identity is not a one-to-one function, God would simply be more than one thing, which doesn’t change our logic in any way. But we can also conceive of a world in which God’s only possible properties are that of our “perfect logic.” If Tishammer’s argument is true, then it should be a subset of a perfect logic. Thus, Tishammer’s proof should still hold in the postulated world, and G is equivalent to “the perfect logic system.”
Thus, the perfect logic system exists in all possible worlds. No?
Forgive me for being dense, but this modal logic stuff is new to me. I’m still struggling with the idea of a “possible world”. Is this the same as a “conceivable world”? Because I can certainly conceive of a world in which God doesn’t exist. Newton Meter already showed that if it is possible that God doesn’t exist, then God doesn’t exist. (Assuming the axiom “If god exists then he necessarily exists”.) Thus we have a contradiction.
Oops. God doesn’t know everything, at least not as defined. He knows only as much as can possibly be known. That which cannot possibly be known (such as inferences from inconsistent logic) is irrelevant with respect to God.
Actually, that’s not a problem. Where perfections might contradict, they will not converge, and thus again are not relevant to God. Toasters and circles are good examples of those.
As Tisthammer explained (and I think we might have mentioned earlier before you came into the thread), it really doesn’t matter. There’s no point in dwelling on the aspects of a broader metaphysical interpretation of a narrow ontological point. If you make an epistemological inference that you are God because you fit the definition, then that’s something that is outside the scope of this argument.
Although we don’t know what the inherent maximums are (the root of your epistemological concern), we do know the nature of their existence, namely that they are synthetic a priori. Therefore, you may conduct an epistemological examination of whatever might occur to you by asking yourself, “Will denial of this entity lead to a contradiction?”
No, the definition does not change. It is symmetric onto all possible worlds.
Consider, for example, whether God in His omnipotence can draw a circle defined as a set of points that are equidistant from a given point. In a world of 2 or more dimensions, He can draw the circle. But in a world of 0 dimensions, He can’t. But not because He’s impotent; rather, because drawing the circle in that nondimensional world is not possible (in fact, it can’t even be defined).
To be consistent, the definition need not (and does not) make Him a duplicated Holy Clone in every possible world. All that remains the same in all of them is that, whatever can be done there, He can do. Etc.
A possible world is a set of true statements. A statement is necessarily true if it is true in all possible worlds that are symmetric with respect to the world in which it is known to be true.
Well, he made that assertion under some controversy when he said that he thought that Becker’s Postulate, if applicable to P, ought to be applicable to ~P. As I think he later acknowledged, much to his credit as an intellect of impeccable honesty, that notion introduced a substantive denial of a positive ontological proposition. But I can’t speak for him, so if my statement contradicts what he thinks, then I retract it categorically. I would rather cut off my hand than offend him.
Apology accepted. I was rude to you in return, and shouldn’t have been.
I can’t accept (1) following from the definition. It says too much. It speaks not only to existence, but existence AS something in addition to what the definition says.
The flippant answer from a mathematician is that possible worlds are undefined primitives. Not very satisfying, so I’ll try to do better.
Consider these two statements:
Nine is greater than eight.
The number of planets in the solar system is greater than eight.
Both statements are true. The truth of statement 1 in seems stronger than the truth of statement 2. We would say that statement 1 is a necessary truth, whereas statement 2 is a contingent truth.
Aristotle tried to precisely say what we meant by necessary truths and contingent truths. Leibniz is generally credited with the idea of possible worlds, though he never used that term. Possible worlds (in the Leibnizian view) can be thought of as alternative ways that things might have gone, or alternative states of affairs.
A proposition is necessarily true if it is true in every possible world. Nine is greater than eight in every possible world. A proposition is merely contingent if it’s negation holds in some possible world. There are many ways that things might have gone such that the solar system didn’t have more than eight planets. Pluto could have never been captured, or it could have escaped. Mercury’s orbit could become unstable and it collided with the sun. A proposition is possible if it holds in some possible world.
There are some philosophical difficulties. We have different notions of necessity and possibility. For instance, when my fiance tells me that it’s necessary for me to take out the garbage, she doesn’t mean that I take out the garbage in every possible world (in fact, I might not even do it in this world!).
The fix is that we’re not just interested in possible worlds, but also in the relationships between them. Given a binary relation (called accessibility) on possible worlds, we can modify necessity and possibility. A proposition is necessarily true if it is true in all accessible possible worlds. A statement is possibly true if it is true in some accessible possible world.
All the above is necessary to answer your question about possible worlds. What we mean by possible worlds depends a lot on what our accessibility relation is. For instance, when it is necessary that I take out the garbage, that doesn’t mean that I actually do take out the garbage in every conceivable world. It means that in every conceivable world where I still get married next summer, I take out the garbage.
Part of the difficulty with the proof is figuring out what sort of possible worlds it (the proof) is about. The first assumption suggests that we’re talking about possible worlds that have similar realities. Sure, you can imagine that God doesn’t exist even if he does. But, if he exists in a world, then his existence isn’t accidental; there can be no accessible worlds that don’t contain him. And that fact itself is necessary. We’re not talking about conceiving of a world without God (it’s clear that many atheists do that–some of them think that the actual world is such a world). I must beg your indulgence though that some theists really do believe that if God exists, his existence is necessary for there to even be a possible world. You don’t have to accept his existence, just accept what we mean by him.
Next, God is possible. If you’re a non-theist, you might say “waitaminnit!”. If possible worlds are those that are sufficiently like our own to agree on the nature of reality, then why should we presume that God exists in any of those worlds? Good question. I think that pleading that God possibly exists just because we can conceive of it is weak in light of our justification for the first assumption.
There is a way out for the eager theist. Accessibility doesn’t have to be symmetric. If world B is sufficiently like world A that they’re related by accessibility, it is not required that world A is sufficiently like world B. Imagine that we were talking about “conceivable” worlds. We can conceive of a world exactly like ours except dogs have wings and all the denizens of that world are fundamentally unable to conceive that dogs might not have wings. It’s a strained example, but it illustrates that just because a world is accessible from here doesn’t mean here is accessible to that world. Thus, the theist could argue that you should accept both the axioms, because it’s not required that accessibility work symmetrically. It could be the case that in every world like our own, God’s existence necessarily implies that he necessarily exists; yet his possible non-existence doesn’t compel us to reject his possibility.
There is a twist in the proof. The proof requires that the accessibility relation is symmetric. It appears that a non-theist would only accept both assumptions if the accessibility relation were not symmetric.
The proof is valid, but I think it’s not compelling enough to get a non-theist to accept all three required assumptions. One who wishes to reject the conclusion that God exists in this world has three simple avenues (besides rejecting multiple assumptions and rejecting the entire philosophical enterprise):
Accept both axioms, but reject that the accessibility relation on worlds can be symmetric given the axioms
Accept the first (definitional) axiom and a symmetric accessibility relation, but reject that God is possible in any world sufficiently like our own
Accept that God is possible and that accessibility is symmetric, but disagree with the definition that requires that it’s tautological that God’s existence implies that God’s existence is necessary
erislover:
I misunderstood the point you were raising regarding inconsistent systems. Upon rereading your explanation, I understand better. I suspect that Lib would reject that it is necessary that the existence of a sound, complete, and powerful logic would entail the necessary existence of such a system.
Libertarian:
Thank you for the praise. I’ve enjoyed this thread. I’m pleased that you raised the initial question and that you have proven so able in defending your position that it has continued as long as it has.
Gah. But that’s my point! I’ve found something that should have intuitive maximums yet which cannot exist, even though Tishammer’s proof says it should.
Narrow ontological point? The existence of all perfection?
Not in an a priori proof. Existence is normally considered synthetic. But necessary existence is analytic by the very nature of the proof in question.
It changes. If, in one world, God is defined as MAX{x}{y}{z} but in another possible world the property {x} is undefined, then the definition of God can no longer be MAX{x}{y}{z}, but only MAX{y}{z} (assuming y and z still hold).
Thus, God[sub]1[/sub]!=God[sub]2[/sub]. The question: what inherently maximum qualities exist in all possible worlds? IOW, what must be defined in all possible worlds?
I haven’t time just now to get into this deeply again (I haven’t even had a chance to work through Newt’s latest posts.) However, I did want to comment on this as it is very similar to my orthogonality argument.
Suppose that in some possible world, x and y are not independent. Suppose, for the sake of argument, that y=1/x. y and x are both “optimized” when x=y=1. However, a being with x=10 has a “greater” x attribute than a being with x=1, even though this being’s y=1/10.
The point here is that since these attributes are not independent, one must make a subjective choice as to whether a being with infinite x and zero y is more or less perfect than a being with zero x and infinite y or whether a being with x=y=1 is “greater” than either.
Libertarian, you’re dismissing a lot of people who you say are less concerned with the technical aspect of the argument and more its meaning. You applaud those like Newton who attempt to attack the technicalities of the proof. And I agree that Newton has provided us with some excellent analytic posts. But ontology is part of the philosophical discipline, not pure mathematics. If it was pure mathematics, saying “look, I don’t care what 2 is or means or anything like that. If I apply these given rules, this is how 2 behaves. That’s all I care about!” might be valid. But even that would be formalism driven to its extreme.
Meaning is an integral part of logic. First order logic isn’t a bunch of random propositions that you toy around with to get some desired answer. They are meaningful propositions which we assign meaningful truth values to. Logic is a tool used in the search for wisdom. Modal logic extends this tool to include the concepts of possibility and necessity. Again meaningful concepts. If we don’t know what we’re searching for, you might as well pump the rules and propositions into a Turing machine and watch it crank out answers. Which would be about as interesting as watching an program execute on the Assembler level…
Aright, lets look at the proof technically. This, in a philosophical context, is really no different from looking at meaning. What we have in this argument are:
An infinite number of possible worlds we can access
Propositions that hold true in all accessible worlds
Propositions that hold true in some accessible worlds.
Some perfect thing defined as “the greatest possible being”
Or, more formally:
W^n
P == W --> P
<>P == W --> P or ~P
P
I have excluded basic rules of first-order logic and such. What’s really important is what item 4 essentially boils down to: P!!!
So, in the end, we have taken Anselm’s quest to root his belief in reason and turned it into something quite meaningless. We are no longer proving God. We are proving P. What has God in this context become to us? Any statement that by necessity is true in all possible worlds accesible to us. We don’t need any truth tables here. P is by necessity true in all worlds we have the capacity to extend our minds into. What we need to understand is the meaning of what we have said. And that certainly isn’t "God " as I phenomenologically apprehend the term.
Let’s again formally look at what the definition “greatest possible being”, implies. We have the axioms:
A. P -> P
B. ~~P
But given the definition of P (“greatest possible being”), P is indeed identical to P. So we substitute:
A. P -> P
B. ~~P
Now, if we go through the whole ontological argument, have we proven God? No, only that there are certain absolutes that must be true in all accessible worlds. But modal logic already presupposes this to be true. Therefore the argument is indeed pretty much a circulus in probando. The truth is, we need not prove that P -> P -> P because it is self-evident.
Or, put in other words, “the Greatest Possible Being” does not imply necessary existence. In fact axiom A is redundant because “the Greatest Possible Being” is “necessary existence” itself! This becomes even more evident when we substitute being with its synonym existence. The Greatest Possible Existence. So we are left with the next axiom:
~~P
Which again is a very redundant axiom because P by definition means that P is not unnecessary! So we can substitute the double negative “~~” with ". Now, what we’re left with is a single boring axiom, itself simply part of the modal universe:
P: There exist statements P such that P is true in all accesible worlds.
The ontological argument becomes redundant because modal logic tells us that:
P -> P: If P is true in all accessible worlds, P is true in this world.
But, again, what is P? Well, really (as previously mentioned) any given statement that has a in-an-of-itself a potential maximum (with other words is necessary in all to us accessible worlds). A few examples:
P IFF P
Every even n>2 is the sum of two primes
Summa:
God when defined as “the greatest possible being” becomes the set of all propositions that hold true in all possible worlds accessible to us, a Platonic universe of eternal ideals. Which I don’t believe is what Anselm was attempting to prove the existence of. If you, Libertarian, think that math and logic in-and-of-themselves must imply the existence of God, fine. But we don’t need the ontological argument for this. If “God” is where contemplating logic in-an-of-itself transcendentally leads you, so be it. What’s strange is that similar phenomenological contemplations don’t do the same for me. All I can say is there’s evidently an absolute that imposes itself on my way of being, an absolute I can know in the most intimate of ways. The term God, however, IMO stands for “all things maximally unknowable” or “all thing beyond my knowledge”. And not the Platonic universe of ideals I must accept when confronted with my mind in-and-of-itself! Hence, I’m afraid we’re back at square one:
I think it is important to mention, along with the explanation of contingency, that contingent attributes may not be assigned to God when God is defined as the most perfect possible being, thus exculpating whoever might form the definition the way Tisthammer did.
On what basis? Another synthetic existence attribute assignable to God is eternal existence, which gives God a sort of supersymmetry in that all temporal worlds (i.e., worlds with space and time) have, for Him, not yet been created, are ongoing, and have finished all at once. The modal symmetry axiom, p–><>p, holds only for temporal worlds because, by its very nature, an implication is one truth that is derived from another. A derivation cannot take place without the statement of one truth followed by the statement of another. Thus, any symmetrical relation restricts God only in temporal worlds, which happen to be worlds where He is unrestricted.
Again, why? By summarilly rejecting His possible existence in worlds like our own, we paint ourselves into a corner where we cannot disprove His nonexistence. A thing cannot be shown to be impossible before it has been conceded to be possible. If something is impossible a priori, then it cannot even be stated as a positive proposition and thus cannot be negated. “Not” by itself means nothing. And “not” something implies the possibility of “something”. This was what you and I agree was the most tenuous ground upon which a philosopher could stand.
I know that that’s controversial, and is usually what a philosopher will grasp at when he hasn’t been satisfied by his own objections to later premises. But as you pointed out, “[a] proposition is necessarily true if it is true in every possible world.” Clearly, we have a tautology if what we’re talking about is the convergence of all possible perfections from all possible worlds. When God exists AS the definition offered, then his existence is as necessary as necessary can be.
I’m just lucky that I happened to stumble onto Tisthammer et al. Having been out of the loop for so long, I was amazed at how much had been done in the 1990s. I think that, at least, it is as Triskadecamus has said elsewhere: if nothing else, the atheist can no longer accuse the theist of abandoning all reason for his faith. Wouldn’t you agree?
Eris
I’m glad we found a point of agreement, but I don’t know what you mean. What have you found, the perfect logic system? Not so. Logic systems are contingent on their own rules.
You said earlier, “Since we’ve already decided that God can be described logically by attempting to use Tishammer’s proof, we should state that describing the most perfect being is an inherent maximum.” And I’ve pointed out many times that nothing at all has been described about God other than the quality of His existence. Not Him — His existence. Ontology doesn’t deal with other kinds of descriptions.
You said also, “We can’t have less than no contradiction.” Set aside for the moment that we’re talking about maximums, and not minimums, here. You’re using an analytic, which is definitively contingent, to describe (presumably) some attribute of God. Zero is a number that we can reach by counting stones as we pull them out of a hat.
And in general, the whole effort by some to undermine the proof by substituting other words (someone suggested Grumpf, I think) for God is just simply silly. No one is proving that a combination of sounds exists. Calling God a toaster doesn’t change anything if you leave your toaster defined as the greatest possible perfection. I mean, a rose by any other name… It is what is defined that is proved.
Yes. By narrow, I mean that we are talking only about existence and its qualities. If we start speculating on how God uses His power and knowledge, or whether He has a big nose, or that we might ourselves be God, then we leave the confines of ontology, and step into broader metaphysical fields.
Descriptions of the rules under which the proof is bound are analytic, but the description of God is synthetic. What is there to deny about the convergence of all necessary existence that might lead to a contradiction? It’s an atomic statement, N. All you can posit is ~N.
But that’s not how it works. And I have labored extensively to get this point across, so I’m glad you gave me one more opportunity to make it. God is NOT defined as the maximum of any particulars, else His definition would be analytic. In other words, if you already know all the particulars of all possible worlds such that you can enumerate them with set notation, then you are indeed, by the definition given, God Himself.
The perfections denoted unto Him are not things that are perfect, but rather the perfections of things. All you’re showing is that there might exist differences in qualities among worlds, and no one would disagree with you. But the quality of God discussed here does NOT change among them. He can do whatever can be done and He can know whatever can be known in any possible world W{x, y, z …}.
Stay strictly within the bounds of ontology. It is not a matter of what He may do or know, but that He may do or know.
But no particular set is under investigation. All sets, infinite, countable, and empty converge into a superset of possible maximums that describe the nature of God’s existence.
Wow. What a great discussion. I wish we could all get together over some good beer and do this face-to-face.
Lib and Newt, thanks for your explanations.
eris and ethnic - great points!
I’m now struggling with “strict implication”. Am I correct to interpret as
“In every possible world it is the case that if God exists, then God exists necessarily”?
It seems to me that this is equivalent to:
“If god exists in any possible world, then god exists necessarily (i.e. in every possible world)”. Is this correct? If not, what the heck does mean?
It means that it is necessary that, if God exists, then He exists necessarily. And I agree with you that it is a great discussion, with many wonderful ideas coming from many wonderful people.
I didn’t mean to come across as dismissing people who did not argue the technical points, although, looking back at what I said, were I you, I would have taken it the same way. What I really intended was to corral some of the random assertions that were just being thrown out with no ties to anything: assertions that were the rhetorical equivalent of sticking out one’s tongue and making fart noises. I didn’t mean to include, and should have explicitly excluded, you from among such rabble.
Your latest objection fails because the proof is not a consideration of arbitrary propositions, but rather, of a particular proposition, one that carries through the table with its meaning still intact. What you’ve analyzed, to some extent at least, is modality itself, and you have found it to be tautological. But all logic is exactly that.
