The “your system” and "your logic"s were just meant to mean “the particular system you are using”, not “the system you developed”, of course. And as for the CD axiom (<>A -> A), I wasn’t saying that the CD axiom in general followed from your logic (since it doesn’t), just the particular instance saying that possible existence implies necessary existence.
My main question, which may well be moot depending on how much you actually care about the modal argument you gave for the necessary existence principle, is this, from my last post, concerning the usefulness of that argument:
This is the start of the main question
This is the end of the main question
But none of this actually affects your Modal Ontological Proof, which goes through just fine in all systems of S5, so it may not be of great concern to you. (Yes, I know, not “your” MOP; I appear to have a bad habit of using “your X” to mean “the X you presented”). My response to the MOP is to consider one of the premises as lacking any support, which is enough to make me an agnostic, I suppose, and, then, furthermore, to believe that same premise to be downright false, which allows me to be an atheist. (The specific premise which I most strongly reject is <>G; I don’t reject this for all definitions of G/God, but for the definition of G in play in the MOP (any definition which validates G -> G), I very much do).
Hello old chum - I realise I’ve arrived at this dinner party when everyone else is ready for bed, but I’d like to take you up on this. I have always tried to propose a physical mechanism for the act of conceiving, based on the cognitive science of memory. I define the universe as spacetime and all of the fields (and excitations of those fields, ie. particles) in it, which includes all the configurations of those fields, ie. objects and processes.
I suggest (but do not define) “conceiving” as a process, similar in some ways to the reactivation of a memory stored in hardware. So, for the string “horse”, a memory is triggered (perhaps being the “average” of many horses sensed over time). Similarly, for the string “horn”, a memory is triggered. But here’s the thing - the process can go the other way, rather like PhotoShopping stored images. The string “unicorn (or whatever arbitrary label is used) = horse + horn” can be input, triggering both images at the same time regardless of whether such a combination has ever been sensed! This is the process I call “conceiving things” and, though I know you disagree with every fibre of your being, I consider it as physical a process as star formation.
Now, other strings are just as easily input, such as “tall dwarf”. But here, there is a cognitive clash of gears. I cannot hold in my mind’s eye a tall dwarf, yet a horned horse is, to mangle my metaphors somewhat, a piece of cake. I won’t (and perhaps can’t) explain precisely why I struggle to build some combinations, but let us call such combinations inconceivable.
So, do I “define the universe as everything conceivable plus everything that isn’t”? Emphatically, No. I define the universe as everything physical, and then suggest how conceiving things might be a purely physical process. Tall dwarfs are strings of letters or phonemes, and only that. Unicorns are (currently) combined images of horns and horses, and only that. These are effectively computational entities, which do not have the characteristics of, say, mass, which other physical entities have.
Finally, to the OP. What does the string “God” trigger in my computational offal? I don’t think I can define it any better than a non-physical person or, perhaps, a supernatural agent (with “agency” suggesting an intentional stance in old Danny Dennett’s parlance). I’m afraid that my biological processor simply isn’t up to the task of conceiving a person which isn’t made of, you know, stuff. If yours can (as well as tall dwarves or whatever else), well, you are truly blessed. To me, God is but a string of letters whose meaning I’m afraid I just can’t mentally construct, try as I might.
As for “life”, I actually don’t think the definition really matters too much (though I’ll plump for Francisco Varela’s carbon chauvinism if pushed). What matters is whether science can explain how Life comes from a state we all agree is non-Life. Similarly, I think a great “intuition pump” (DD again!) is ultimately what one’s computer outputs for the 13.6 billion years before humans existed: There was simply nothing capable of conceiving then unless you wish to posit souls waiting to be born or such like. And if there was, well, I’d just be plain wrong…
I am tempted to ask “Would a world containing only the statement ‘This world contains a statement’ be a possible world?” I am tempted to ask “Who says a possible world must contain a truth bearer?” But I really don’t care. I can see that the sort of reasoning used in a ontological proof is not for me. It makes use of concepts rooted in human psychology like supreme and perfect and truth bearer and edification and possible and necessary which are subject to subtle shifts of meaning that invalidate deductions using them. If we want to understand the real world, we have to base it on observations. We cannot determine the nature of reality by manipulating symbols alone.
I’d like to belatedly compliment Begbert2 on his post #356. This was very well done. Having studied logic some time ago I had hoped I might be able to analyze Libs arguments for problems, but Begbert2 did this far better than I could have hoped to. I found his post valuable for refreshing my knowledge and it was clear and logical at each step, in contrast to many of Lib’s posts in which I tended to be scratching my head asking myself “where the heck did that come from?” Quite frankly, Lib’s flippant dismissal struck me as rude and inappropriate. Clearly in a two-valued system statements can be considered who’s value is unknown. (I have heard of systems with extra values for things like “don’t know” and “partially true”, but these allow for explicit handling of these values). Lib thereby ducked addressing important points about his nouned verb construction and how it was used. If indeed there was anything wrong with Begbert2s understanding of Lib’s argument, Lib chose not to address it. While Lib obviously has an extensive knowledge of various philosophical systems, from what I’ve seen here, Begbert2 is clearly the better logician.
First of all, let me apologize for disappearing. I haven’t had time in the past few days (and still don’t) to give proper attention to this thread. I’d rather not post than post garbage.
PC apeman, however, has stated my objection perfectly. The new proof on this page (not new, actually) starts again with the premise that God is possible, and the implied premise of supremacy. I think the last time we went through this you did not assume supremacy - assuming it now does indeed make the necessity argument work a lot better.
The problem with supremacy is that power is a partial ordering. You can define an entity who is maximally perfect in the sense of knowing every truth that can be known. You can also define an entity who is maximally powerful, able to do everything that is logically possible. But there is no entitity who can do both, since they are logically inconsistent. Even if there were two entities, A and B, one maximally truthful and one maximally powerful, how can you determine which is more perfect? I actually don’t think both can exist in the same universe, but haven’t tried to prove that. So, if we back off, and define two entities, one maximally powerful except in not being able to do things the other has seen, and the other maximally truthful and knowing, except not being able to see what the other will do, again we have the problem of determining which is more powerful. We have the same problem if the two entities are merged into one. Since there is no logical problem with omnipotence in the absence of omniscience, how can we say an omniscient but almost omnipotent entity is supreme over a truly omnipotent one?
Therefore supremacy fails, and therefore necessity fails. Thus ~<>G God cannot possibly exist.
Thank you very much; I like to see my work appreciated. As you have divined, I am proficient in formal and symbolic logic; however my training in this was primarily done through the math and computer science divisions of my college, and modal logic was never even mentioned. So I am certain that Liberal has a greater understanding of these modal arguments; I’ve never even seen the stuff before! (Though some of it seems on crack anyway. “Truth bearer”? Wtf?)
(Liberal seems to be the opposite side of the coin: he’s read extensively in philosophy but has little or no formal training in logic itself. While this in itself is not damning, his misconceptions about definitions and premises make him difficult to debate with, and leaves any unsound arguments of his unsupported since he relies on assumptions of imperviousness that do not, in fact, exist.)
As for possible worlds, if it helps any my limited understanding from the Stanford site (my “disreputable source” ) is that the whole “possible worlds” thing is nothing but an analogy. If something is possible (say, statement S), then either S is true, or it’s not. If it’s true, then that’s one possibility; if it’s not, that’s another. This divides the ‘probability plane’ into two parts; what could be the case if S is true, and what could be the case if S is not true. Two “possible worlds”. However no additional realities are created; we’re still just talking about possibilities. We are not somehow talking about parallel universes or anything like that, which might be empty or not, or contain Gods or not.
But then again, I didn’t notice the Stanford site mentioning anything at all about “actualization”, which Liberal seems to feel is necessary to apply anything to the real world. Since I seriously doubt he’s making all this stuff up, there must be some subtleties that are accepted in the official scholarly discussion of this that I did not divine from the Stanford site. I might feel inclined to ask Liberal about them, since he’s clearly done the reading, but I doubt he’s capable of holding a civil discussion with me. (Also I doubt he’s reading my posts anymore.)
But the particles aren’t real. They’re just mathematical constructs. Does it bother you that everything you observe is built from them? If not, why not?
Why does God get to be real, but not the particles? Are the particles possible? Suppose we prove mathematically that they’re necessary, in order not to contradict real-world observation?
I see. Thanks for explaining that. I’m sorry for my social density.
As I understand it, S5 is constructed with three axioms (and Barcan may be added): (1) K, (x -> y) -> (x -> y); (2) T, x -> x; and (3) 5, <>x -> <>x. I was not aware that any other form of S5 could be constructed. It’s true that this construct can be derived different ways. But all roads must come to the same place. If I’m misinformed, I’m open to instruction.
That’s fine. It is your prerogative to reject any premise, and I must respect your choice as I do The PC Apeman’s. I disagree, of course, and if you don’t mind a question now that our issues are settled, what do you think of G -> G when the definition is substituted for the term? We get G -> G, which is the base axiom for S4. Do you think it is unfair to make the substitution? If not, do you have a problem with the S4 axiom?
I’m relying on the observations of Werner Heisenberg, whom I trust as a reliable authority on quantum mechanics. He wrote, “The atoms or the elementary particles are not real; they form a world of potentialities and possibilities rather than one of things or facts.” His contemporary, Niels Bohr, seemed to concur. He wrote, “There is no quantum world. There is only an abstract quantum mechanical description.” And, “When it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images.”
Do you mean G -> G? Yes, that’s always a part of the MOP, and it follows from the coherent definition of God. Remember, G = G, and so substituting G for G, we have G -> G, which is an axiom subsumed by S5 (from S4).
I don’t agree that they are logically inconsistent. God’s sentience and potency, one would presume, follows the same sort of bounds as His existence. Therefore, He knows whatever it is possible for Him to know; and He can do whatever it is possible for Him to do. And so even if knowledge and power were to conflict, that contradiction itself excuses Him. Impossible things are not things.
Wow. First, your definition G = G is circular, and does not state what G means. (From your use of it in the proofs, it’s defined as “God exists”).
Second, has anybody actually accepted that what you call your definition, “G”, is true? If they haven’t, you can’t meaningfully substitute it into another statement and call the result ‘true’. If they have accepted it, then no convoluted proof is necessary; simply substituting G for the generic A in M (aka T) and applying modus ponens gives you G, “God exists”, your conclusion.
Third, G -> G is NOT the same as S4, “A->A”, since the G in the first is a specific statement and the A in the second is a generic statement. You might as well say that the statements “Everything is a dog” and “The cat is a dog” are the same.
Fourth, the S5 system doesn’t assume S4 to be true anyway; you say so yourself. So, one obvious problem with S4 is that it’s not in the logical system we’re using. That is, it’s not assumed to be true. Not that it would have anything to do with your ‘definition’ even if it did.
On preview: Holy Heisenburg! He read one of my posts! :eek: And reading what you’ve quoted, they seem to be saying that we don’t have full descriptions of atoms or subatomic particles, and only explain them by their behavior. Personally I’m fine with that, since the behavior is (afaik) consistent, especially at the atomic level. Mileage of others may vary, of course.
In all honesty, I posted too fast and dropped a couple of balls. I apologize to all for adding confuson where I was trying to alleviate it. I apologize to Liberal in particular for stating that there were four problems in his position, rather than merely two (and only one of them a serious failing). It was very nearly ad hominem, and factually incorrect.
Point 1 is weak, and I deserve to be smacked for calling premise 2 a definition again. :smack: (It is necessary to be clear on what G means, though. It’s actual definition, used for translating the terms back to english.)
Point 3 is a fine point. (And the english analogy was flat wrong before I edited it.) G -> G is not the same as A -> A, and it was inaccurate (that is, false) to say that it was. However, if we were accepting 4 as axiomatic (which we’re not) then a simple substitution would demonstrate that G -> G was true. That doesn’t imply that either G -> G or G is true, though, because substitutions and implications don’t work backwards.
Point 4 is merely pointing out the silliness of citing an explicitly unaccepted premise.
Point 2 is ironclad.
I freely invite all to eviscerate me for points 1 and 3, though. To construct critiques in haste is poor form always. :smack:
[QUOTE=begbert2]
Wow. First, your definition G = G is circular, and does not state what G means. (From your use of it in the proofs, it’s defined as “God exists”).
Second, has anybody actually accepted that what you call your definition, “G[]”, is true? If they haven’t, you can’t meaningfully substitute it into another statement and call the result ‘true’. If they have accepted it, then no convoluted proof is necessary; simply substituting G for the generic A in M (aka T) and applying modus ponens gives you G, “God exists”, your conclusion.
Begbert, congratulations! Finally somebody has pointed out the obvious flaw in Liberal’s logic in terms that maybe he can accept. I tried the same argument, in plain English, when Lib started in on modal logic a long time ago; Lib, of course, ignored my post. He has never, ever, tried to demonstrate that the god he’s trying to prove conforms to his “definition”, which is not a definition at all but a description of some of the supposed attributes of his god. He just says: “God is defined as the omni-whatzit, and the omni-whatzit wouldn’t be omni if it didn’t exist, so it must exist”. This ties somewhat to the question I have asked several times, which he has also ignored: supposing the god that exists is, in fact, the YHWH/Jesus of the Bible that several billion Xians have fervently believed in for two thousand years, who are demonstrably not omni-anything; doesn’t this mean, by Lib’s logic, that there must be another god above the Xian ones?
If I were Australian I’d say “Goodonyer, mate.” I’m not, so I’ll say “Andale, cuate.”
You clearly have some experience with symbolic logic, begbert2, and I agree with many of the things you say, but let me just pick some nits of yours.
Just a very technical digression to make here, and I don’t presume to speak for what exactly Liberal has in mind, but there’s a perfectly straightforward and natural way to accommodate certain nice recursive statements, such as this one. Define IsGod(x) as Ey(y = x). That is, x is God iff there is necessarily something which is x. Let G as Exists x . IsGod(x). That is, G is the statement that something exists which is God. Those are perfectly cromulent definitions, no?, and it isn’t hard to show with these that G is equivalent to G.
I think Liberal actually had in mind a different, stronger definition of what it means to be God, but it would come out to pretty much the same thing in terms of cromulence as a definition and entailing G = G.
The axioms of S4 are all implied by the axioms of S5. Assuming S5 to be true forces us to accept S4 as true. (Specifically, S5 is captured by the assertion that the accessibility relation on possible worlds is reflexive, transitive, and symmetric, while S4 is the weaker assertion that the accessibility relation on possible worlds is reflexive and transitive).
Er, I switched up in the above between using “E” and “Exists” for the existential quantifier, but it shouldn’t be too hard to understand. (Damn lack of post editing for guests…) The line “Let G as Exists x . IsGod(x)” should basically be “Let G be defined as Ex(IsGod(x))”.
To all readers: This is going to be a long post. The middle section is a lengthy explanation of the semantics of modal logic, designed for consumption by anyone who wants a better understanding of it, even if you have no background in modal logic at all. But if you don’t care about the semantics of modal logic, you should just skip to the last section instead.
Ok, here comes the post.
Hey, no problem. Occasional misinterpretation is just a fact of life.
Well, in short, all roads come to the same place for sentences without quantifiers (“for all” or “there exists”).
Specifically, if we restrict our attention to propositional logic (no “for all” or “there exists” quantifiers), then there is a unique system S5 and it is axiomatized as you say. However, once we start looking at first-order modal formulas (sentences that contain quantifiers as well as modal operators), there is more than one way to set it up and still be considered an S5 system. Some of these will prove the Barcan formula, etc., and some of these won’t. It may not matter so much for your MOP, since you seem to phrase it in such a way as that you don’t make any use of “for all” or “there exists” quantifiers (though, if you choose to adopt the approach I outlined before for making G = G definitionally true and thus not needing to take it as a premise, you will be bringing in those quantifiers, and the correctness of the approach will begin to depend on what particular way you handle them). It definitely matters for your argument concerning the validity of the necessary existence principle, which of course has to make use of existential quantification.
(Here starts the lengthy middle section)
It all basically comes down to this, much of which you already know, but but to spell it out in full will hopefully be useful for just explaining modal logic to everyone: a Kripke frame is a collection of what are known as possible worlds, along with some accessibility relation, saying which worlds can “see” which other worlds. At each world, some primitive facts are true, and some primitive facts are false, and the truth of more complex facts (e.g., A AND B) can be determined in the obvious way. A primitive fact can be true in one possible world and yet false in another. The particular rules of interest are the modal ones, dealing with A (which says A is necessarily true) and <>A (which says A is possibly true). The rules dealing with these are that A is true at the world W iff A is true at every world W can see, and that <>A is true at the world W iff A is true at some world that W can see. After we set this all up, the logically valid statements and acceptable rules of inference will be those which are guaranteed to work out at every world, no matter how the Kripke frame is set up.
(All of this I’m sure you already know, Liberal, but I’m hoping to educate the other posters about modal logic at the same time, so that they will better be able to take part in the discussion, and am also leading up to the points which you yourself seem unclear on; please don’t take any offense at my giving this explanation).
If we put different restrictions on how sight works, we get different systems of modal logic. With no restrictions, the system is called K. If we demand that every world be able to see itself (i.e., that accessibility is reflexive), the system is called T. If we further demand that W seeing X and X seeing Y implies that W sees Y (i.e., that accessibility is also transitive), we get the system S4. Finally, if we demand that W seeing X implies that X sees W (that accessibility is also symmetric), we get the system S5. We actually get the same collection of validities if we just demand that every world be able to see every other world, and basically do away with the accessibility relation altogether.
That hopefully brought everyone up to speed a bit, and was just a refresher for you, Liberal. Now, for the part that I think you may be unclear on.
The above was the end-all, be-all for the propositional aspect of modal logic, but it left unclear how to deal with the quantifiers. The basic idea for dealing with them is to add to the Kripke frame some “individuals” (things like God, the Eiffel Tower, my left hand, etc.), and have primitive facts at a possible world include relations between and properties of individuals (like “George Washington is Martha Washington’s husband” and “The Eiffel Tower is tall”). But there’s one major bit of trickiness here: how do we accommodate the intuition that some possible worlds don’t even have an Eiffel Tower, and thus shouldn’t be forced to assign a truth value to the primitive fact of its tallness, while other worlds would have to?
There are many different ways of dealing with this, and all of them are still considered S5 systems, as long as their “sight” rules are still just the S5 sight rules.
One way of dealing with this is to say that every world has to decide every primitive fact, no exceptions, and that the statement Ex(P(x)) is true at world W iff there is some individual z at all such that P(z) is true at world W. If we go with this set-up, the Barcan formula becomes logically valid, as does the necessary existence principle, and, basically, any individual that can be said to exist in one world will be said to exist in all worlds. This seemed to be the logic you were using, Liberal, with which my whole quibble started. It is particularly easy to give rules of inference for this logic, because not much care must be taken in terms of the quantifier rules, but it seems very unsuitable as a description of “the real world”, and what intuitively the modal operators and first-order quantifiers mean.
Another way of dealing with this, then, as outlined in the SEP article, is to say that each world still has to decide every primitive fact, no exceptions, but also associate with each world some particular subset of the individuals which are considered to exist in that world. A world still has to decide primitive facts about individuals who don’t exist in it, though. We will say that the statement Ex(P(x)) is true at world W iff there is some individual z such that P(z) is true at world W, AND such that z is part of the set associated with W. Now, the necessary existence principle is no longer logically valid (an individual can be associated with some worlds and not others, and thus exist in a world without having necessary existence across all worlds), nor is the Barcan formula, nor various other controversial propositions. This is Kripke’s quantified modal logic, and takes some more care to axiomatize, but is much more appealing as a description of reality.
There are many more ways of dealing with this, as well, some of which don’t require worlds to decide primitive facts about individuals not associated with them, and which thus require much more care to properly axiomatize, but seem even more appealing as descriptions of reality.
(Here ends the lengthy middle section)
Anyway… all that just arose from my taking exception to (what I perceived to be) your glibly considering the necessary existence principle an indisputable logical fact. But hopefully just spelling it all out was of some use.
I have truly no problem with S4, and am fully ready to accept G -> G, as S4 says I must. Furthermore, in fact, in the context of S4 alone, understanding the modal operators in an S4 sense, I would even be willing to grant both G -> G and <>G. But without the additional symmetry condition from S5, this would not be enough to derive G. And if I were told to understand the modal operators in an S5 sense, I would not be willing to grant both G -> G and <>G simultaneously (I might grant one or the other depending on what I was told to interpret G as).
I was trying to get at that type of God by describing one who is almost omniscient and almost omnipotent, so there would be no logical contradictions. And I agree that type of God is not logically impossible. However, there could be an entity that could do things that God cannot do, not being omniscient, and an entity that might know things God doesn’t know, not being omnipotent. To demonstrate that this disproves supremacy would require you to strictly define supremacy, which I don’t think you’ve ever done formally.
To get back to the stupid rock example, God not being to lift a rock he created is not an issue, not only because it is logically impossible, but also because no entity can do such a thing - thus God not being able to do so does not in anyway contradict a claim of supremacy. That is not the case for the things a bi-omni God can’t do.
) is that the whole “possible worlds” thing is nothing but an analogy. If something is possible (say, statement S), then either S is true, or it’s not. If it’s true, then that’s one possibility; if it’s not, that’s another. This divides the ‘probability plane’ into two parts; what could be the case if S is true, and what could be the case if S is not true. Two “possible worlds”. However no additional realities are created; we’re still just talking about possibilities. We are not somehow talking about parallel universes or anything like that, which might be empty or not, or contain Gods or not.
)