Here’s another way to think of this, Lib. Consider that you start with any modal logic that has a symmetric accessibility relation (so that the proof is valid). Then, rather than stating as an assumption of the proof, just add it to the system of logic as an axiom. (Not as an axiom scheme; we are saying that this is true of a particular proposition, not generally true of propositions).
This has the consequence that G is either true in all possible worlds or false in all possible worlds or the system is inconsistent.
Now, rather than stating <>G (for the same proposition G) as an assumption of the proof, lets add it to the system of logic as an axiom (not axiom scheme).
This has the consequence that G is true in all possible worlds or the system is inconsistent.
Now, the proof merely says that if we adopt a consistent logical system where G is true in all worlds, we can prove G is true in any world. Is this interesting? Is there any way we can say that this logical system has anything to do with the actual world?
And I’m starting to agree with erislover and Truth Seeker. Is there any way we can say that the only way to formulate G is “God exists”? Couldn’t we use something like the negation of Goldbach’s conjecture? Certainly, if the negation holds then it (the negation) is analytically true of number theory and necessarily holds in all possible worlds. And certainly, it is possible that it (the negation) is true (since we don’t know for certain that it is false). Did I just disprove Goldbach’s conjecture?
I’d never throw out logic, Lib, but when we assign English words to logic symbols we create a situation where the inherent ambiguity of English definitions and implications come along with it.
Note how many times people mentioned that it was possible that we’ve only proved that the concept of God exists (at least twice that I can see, maybe more, I ain’t rereading this whole thread again). Concepts exist, do they not? If we replaced G with “The concept of the most perfect being etc” how would the proof change? The symbology wouldn’t.
Yes, so here we are: the empirical nature of G is contingent upon the properties of the world in which it resides. Thus, there is no general G which can logically encapsulate {G[sub]1[/sub], G[sub]2[/sub], G[sub]3[/sub], …, G[sub]n[/sub]} because the elements of each G[sub]n[/sub] are contradictory with other G[sub]k[/sub].
“But erl,” you might say, “We simply forget the actual properties and stick to their perfection.” But perfection in world {1, 2, 3, …, n} doesn’t mean the same thing, either! Not all possible worlds are symmetric to the point where we may abstract out a universal (quality that applies to many discrete objects) that exists in all of them… except one. Can you think what property all these worlds must have in common?†
[ul][li]God cannot be the superset of all sets because then God doesn’t exist in any possible world, but in the union of all possible worlds. But such a union would require that all possible worlds obey the same principles of joining sets which isn’t itself necessary.[]God cannot be the intersection of all these sets because, again, that would mean that God doesn’t exist in any of them but in the intersection of all worlds, and again, this requires that all worlds obey the same principles of intersection. This, too, is not necessary.[]God cannot be “the most perfect possible being” and exist necessarily in all possible worlds because the mapping of the English word “perfection” doesn’t mean the same thing in all possible worlds.[/ul]God can exist in all possible worlds similar to our own in that the words and symbols we use to represent concepts are symmetric with concepts and words in the other worlds. This is part of what Newton was saying. and this is not all possible worlds.[/li]
No big deal, really, this alone doesn’t damage the proof. It just says that in all worlds that have symmetry with respect to the qualities we describe God with, God exists in this worlds (and this one).
Now that we have our symmetry, the problem of contradictory contingency disappears. In fact, the whole problem of contingency disappears because we are only considering possible worlds where our definitions are able to be mapped. This makes god to have the qualities {a, b, … n}. A formal definition exists, then, whether or not we can enumerate all the possibilites, so long as we restrict the worlds under consideration.
Now we state our axioms and conclusion in such terms.
I. If God exists in any of the worlds similar enough to our own such that all the properties we ascribe to God have corresponding properties in those worlds, then God exists in all of these worlds under consideration. II. It is not impossible that god exists in one of these worlds. III. Because it is not impossible that God exists in at least one world, then God must exist in one of these worlds. But because of (I), we know that if god exists in any world he exists in all of them. Thus, god exists in all worlds.
Notice anything funny there? Probably not, because we left out the definition of god.
G: A being whose properties are the set of all possible properties {a, b, c, …, n} in the worlds under consideration.
Now, when we look at Tisthammer’s rejection of the “<>~G” as a substitute for (II) we see that it leads to a contradiction: the greatest possible being must be possible, by definition! But if that is the case, and we know that we can only be discussing worlds sufficiently like our own, then we have simply defined God into existence! The most assured case of question begging I’ve seen.
We define god as possible; this forces us to reject any formulation of <>~G—in effect defining God into existence somewhere. We then say, “If god exists, then he exists necessarily,” thus making our assertion of over one world apply to all worlds.
There are two problems here, IMO.
[ol][li]Definitional problem. I reject the notion that “possible” in the definition of god means <> in the symbolic proof. They are not the same kind of possible (unless we simply want to assert that god exists in one world, period, which sort of makes the idea of proving anything superfluous when we can simply assert that god is possible in our world and leave it at that). I would rather say that God is the being with qualities {a, b, c, …, n} which are not contradictory in all worlds “like” our own. This sets the scope, and this sets us up without asserting existence—what we’re trying to prove![/li] That being the case, making (II) as <>~G doesn’t make step 8 a contradiction. This is a problem.
[li]A problem with (I). We want to justify the acceptence of one by saying that surely necessary existence is more perfect than possible existence. This being the case, our set of symmetric properties (with respect to the worlds in question) necessarily includes necessary existence as a member! Forget possibility… if we are saying that existence is a predicable property, that existence as a predicable property has a “most perfect form”, and that god’s definition is the possession of all most perfect properties in the worlds under consideration, then the whole proof is superfluous: necessary existence e is an element of {[set of all properties of god]} already! We’ve not done anything different from Anslem at all![/li]
This is not a problem, it is simply reduced to the “standard” ontological argument all over again.[/ol]What do you think, Lib and Wade?
†[sub]Possibility, by definition[/sub]
I know you didn’t ask me, but I agree and I think it was extremely well put. And I think it’s cool that you worked a dagger in there. 10.0 from the American judge.
Portions of this will address you both, due to the considerable overlap in your positions, so please consider this as a response to both of you, with the initial quotation being merely something to give us a place to start.
At this point, you and I agree. Controversy arises after we pass it. From this agreement, I go one way and you go the other. But before I go anywhere, I think personally that it is important to stop and deal with exactly this clause: “there must be an object called “God” that is in some sense the same object across worlds”.
I have labored to express exactly in what way He is the same object across worlds for some time now, but either no one is listening or else everyone thinks that it’s too absurd to address. Once I have heard the definition of God, it is abundantly clear to me in what way He is the same across worlds; namely, He is the greatest possible perfection in all of them.
NB: The modal symmetry in this case is applicable — not to any particulars, attributes, or aspects of His perfection — but merely to His perfection itself.
The definition of God here succeeds because it leaves Him as analytic. Attempts at synthesizing God, notoriously futile as they are, fail simply because He is not a perceivable being, at least not in His fullness. We might, as Wade and many others have done, talk of particulars about God for purposes of illustration only. When we say that God is omniscient, we certainly aren’t saying that that is God in His totality. And frankly, we aren’t really even saying, at least not with any confidence or ability to prove it, that God is omniscient. We are merely saying that omniscience is the kind of attribute that might be considered a perfection in a world that is epistemic in nature. There might possibly be worlds where no knowledge exists. In those worlds, God would know nothing — not because He is not still omniscient, but rather because there is nothing to know.
And this is where, it seems to me, that our paths diverge. You seem to want to put God in the box of possibility of some world or set of worlds such that that box must fit into other worlds as well, but that’s the wrong approach for the reasons I’ve explained. The commonality that is God among the worlds is not His omniscience, nor His omnipotence, nor His eternality, nor any other attribute in particular that may be listed out in set notation as elements, the aggregation of which is assigned wrongly to Him. The commonality that is God among the worlds is perfection. Not this perfection or that perfection, but simply perfection.
God is not the union of anything, nor the intersection of anything. He is the convergence (there’s that word again, another of my points that no one has addressed) of perfection from every world. The only world that will not contribute to this convergence is a world that consists of nothing, a world that is definitively nonexistent.
You complain that God is defined into existence. But that is not the case. God is defined into conceptualization. The proof of His existence is nowhere to be found until G. Neither the definition, nor any of the axioms or premises that follow say anything about God’s existence outside some arbitrary contingency that is introduced because of the nature of logic. This sort of contingency (if God exists, then…) is the only way we have to “hook into” a concept and say anything at all about it. But that doesn’t place contingency on what is represented by the concept; rather, it places contingency on the “hook”. What is represented by the concept did not come into existence just to accomodate our hook; rather, our hook found it, and it will be there when our hook is removed.
And that brings me to the complaint that the way God is defined implies contingencies that restrict God. Not so! (Surpised? ;)) Another undaddressed statement that I’ve made is that the restrictions of the type found in the observation that God is not a shoe tier in worlds where shoes aren’t tied is not a restriction of God but of the worlds. God isn’t restricted by this in the least. By the way He is defined, and once He is proven to exist, the contingency of shoe tying (again, a contingency of the world) does not limit Him whatsoever because if shoes could be tied in that world, then that possibility would converge into Godness. He still would be able to do no more and no less than He could do before. Before, He could tie shoes in every world in which it is possible to tie shoes. And now, the same.
What you seem to want to do is somehow enumerate, list, or otherwise quantify the worlds in which God can tie shoes. But that effort bears no relevance whatsoever to this argument, or, I dare, say to ontology itself. You seem to want to establish a symmetry among set elements, along these lines: God can do {x, y, z} here, but only {v, w} there. But when you go there, you not only leap outside ontology into much broader metaphysical speculations, you also speculate about things that are epistemologically vague. How do you know whether any synthetic {x} from this world exists in any other? You don’t. You can’t. You have no epistemological access to any world other than this one where synthesis of your {x} might bear no resemblance to synthesis of the same {x} elsewhere. In fact, we cannot even share them here! If we could, we would all agree about this argument one way or the other.
You must therefore speak only of analytic {x}s when speaking of {x}s that cross from one world into another. So long as your {x} is something that is attributable tautologically to any world, you are safe to talk about your {x}, but not otherwise. We can’t say what God knows, but merely that, if it is knowable, He can know it. That’s what possibility means, not just generally, but in modal logic. Possibility is itself contingent on existence. If something exists, then generally, there is at least one possibility with respect to it. Thus, God is just as omniscient in a world that consists of nothing more than a singularity as He is in a world that is defined as the most complex possible world. He knows all there is to know about the point in Singularia, and He knows all there is to know about the infinitely many points in Manifoldia.
That which is the same is knowing all there is to know.
And with respect to the matter of whether G can be defined some other way and brought through the same proof, no, it can’t. Why not? Because then you lose the symmetry of maximum possibility for each world, and descend to establishing synthetic attributes that are definitively arbitrary for any world but this one.
I’m going to stop here for two reasons: (1) my wife will be home soon, and I have to have her supper ready, and (2) my train of thought has derailed. But I would greatly appreciate anyone who might specifically address the items that I’ve described as unaddressed.
I’m not an expert on this type of logic, but if you can throw out the premise “it is possible for god not to exist”, because it leads to the conclusion that god cannot exist, which is contradictory because we can easily conceive of a world where god exists, then why can’t we throw out the premise “it is possible for god to exist” because it leads to the conclusion that god necessarily exists, which is contradictory because we can easily conceive of a world where god does not exist?
The god box as valid: first of all, if the definition of God is valid in all possible worlds, then it is valid in any subset of possible worlds. Thus we choose a subset where the properties of god are {a, b, c, …, n}. These are worlds sufficiently similar to our own. In principle, the worlds only need to include the possibility of modal logic existing for them to be accessible to this proof, and by extension, us. But we may take any subset of these worlds and the relationship between them must hold. Some properties may only make sense in our worlds, some might not. In either case, we aren’t actually concerned with what any particular element is, only that it exists (narrow ontological point? “Et tu” says Lib).
Secondly, defining God implicitly like this is not forbidded by the definition nor the proof, nor modal logic in general.
I appreciate this point, but that’s like saying modal logic exists in the minds of all humans, even if they can’t understand it. I’ve mentioned several times that the explicit definition of God must be contingent upon the reality, and I can supply quotes where you make such statements yourself (apart from the one above!). I am simply, for the sake of convenience, choosing to remoev the messiness of contingency by restricting the set of possible worlds. Since I am not stating what {a, b, c, …, n} is comprised of, there is no obvious flaw.
Convergence is not a logical operation
I agree he is not the union or intersection of possibilities. I’m glad we agree on this point, because it necessarily follows from contingency inherent in his definitions.
Stating that he is the convergence of these worlds doesn’t mean anything to me seperate from union or intersection, and seems to contradict the very nature of modal logic considering accesible (that is, knowable) but distinct and seperated possible worlds.
If, out of worlds {1, 2, 3, …, n}, worlds {i, j, k} were not only possible but existent, then would three gods or one god exist? Accepting contingency, three must exist, each perfect relative to the world in which it finds itself. To say anything else requires that god is defined as the intersection of all properties a being can have in all possible worlds, which would lead to a contradiction in any particular world where an element of {a, b, c, …, n} had to be left out because of the intersection operation.
I don’t want to do it, it is in the definition of god! Really, this is trivial.
A being god is defined. For that definition to apply across multiple possible worlds there must be a function f[sub]12[/sub] such that if G[sub]1[/sub] = {a[sub]1[/sub], b[sub]1[/sub], c[sub]1[/sub], …, n[sub]1[/sub]}, and G[sub]2[/sub] = {a[sub]2[/sub], b[sub]2[/sub], c[sub]2[/sub], …, n[sub]2[/sub]}, then f[sub]12[/sub]({a[sub]1[/sub], b[sub]1[/sub], c[sub]1[/sub], …, n[sub]1[/sub]}) = G[sub]2[/sub]. In general, for all G[sub]n[/sub], there must exist a function f[sub]i i+1[/sub] such that (the above holds true, christ I’m not typing that again!)
What this function actually does can be just as contingent on the worlds involved. So long as these worlds are possible and a set of perfect properties exist in them, the definition of god must remain either remain constant (intersection) or it must be symmetric (able to be converted in some consistent manner).
If there is no definition that can do this, the concept of “perfect” cannot be used in the proof, and each god set is far more seperate than merely having different elements, but they aren’t even the same kind of thing! If you have sameness, you must have a sense in which there is something to be similar. I am not necessarily forcing each element n[sub]i[/sub] to be equal to n[sub]i+1[/sub], but that there exists a relationship between them—a function—which can relate the two. This has to be or we haven’t defined anything in all worlds!
More on contingency
Yes, so. It is a flat contradiction to say anything otherwise. The most perfect being in world 1 will have specific properties allowed by this world, and those properties depend on that world. This is contingency. In some worlds the most perfect being will, for instance, know everything. In others there is nothing to know (that is, knowledge k is not a member of {a, b, c, …, n}). The properties of god are contingent.
To make this contingency not an issue we must create the functions as above. To do anything else removes the universiality of the definition and destroys the proof, because the proof requires that we are discussing the same being across different possible worlds.
That is not what Tisthammer said. He said he rejected the premise “it is possible for god not to exist” because the conclusion he got from using that premise contradicts not itself, but because it contradicts the definition! (greatest POSSIBLE being) Therefore the definition is already defining god to be possible.
Tisthammer:
Basically, the proof uses two different meanings of the word possible. It first uses the word possible to mean that god is possible, therefore a conclusion that he is not contradicts the definition. It is possible that god exists.
It then uses the word possible to mean what we can imagine. God is the greatest possible being; the greatest being we can conceive of.
Only one of these uses can be justified in the same proof, but the proof fails no matter which of the uses you choose to get rid of. Eris and Newton have explained in greater detail, I’m just trying to understand it more clearly.
Apparently, you feel like you understand it just fine inasmuch as you’ve already decided that the proof fails categorically — unless, of course, you just accept things that you don’t understand as true.
You’ve mixed up Anselm’s definition with Tisthammer’s. There is no equivocation with respect to the word “possible”. As Tisthammer himself said, his interpretation — “the greatest possible being” — of Anselm’s definition is one way to interpret the phrase “which no greater can be conceived”. He was adapting Anselm’s definition to modality, which deals with such things as possibility and necessity. He does not use Anselm’s definition for his proof.
There is nothing wrong with defining God to be possible. Is the Induction Axiom invalid because it defines every number to have a succesor? If so, then you’ve shot down Peano’s proof that 2 + 2 = 4. You’ve also shot down the whole of mathematics. God is possible a priori because you cannot say that something is impossible without first acknowledging that it is possible. Otherwise, you find youself negating something that isn’t there. Technically, it’s called a substantive denial of a positive ontological proposition and is a famous fallacy.
Let’s set aside for the moment that modal logic doesn’t “exist” in the same sense that you or I exist. Tisthammer’s proof deals with one world only: the world of God, which bridges all other worlds (which is why I like “convergence”) owing to the nature of His existence and of necessary existence generally. The proof’s underlying presumption is that there exists, for every possible world, a greatest possible existence, namely, necessary existence, and this can be proved:
Call necessary existence [symbol][/symbol]. Select any arbitrary interpretation, I, and show that the open modal formula, [][symbol][/symbol][sub]y[/sub](y = x) is T[sub]I[/sub] at W[sub]0[/sub], where y and x are things that exist, T is truth (and so, true at I), and W is an actual world. If the proposition represented by the formula is true, then it will have to be true for every assignment function, F, meaning that [symbol]"[/symbol]x[symbol][/symbol][sub]y[/sub](y = x) will be true. That requires showing that [][symbol][/symbol][sub]y[/sub](y = x) is T[sub]I[/sub] for every possible world.
Select an arbitrary world, say W[sub]a[/sub]. Now, show that [symbol][/symbol][sub]y[/sub](y = x) is T[sub]I[/sub] at W[sub]a[/sub] such that any and all assignments of variables to F, in T[sub]I,F[/sub], are true in W[sub]a[/sub] for [][symbol][/symbol][sub]y[/sub](y = x). Select an arbitrary assignment for F, say F[sub]a[/sub]. If the proposition is true, then [symbol]$[/symbol][sub]y[/sub](y = x) is true for T[sub]I,F[sub]a[/sub][/sub] at W[sub]a[/sub].
Call the individual that F[sub]a[/sub] assigns to X, D[sub]a[/sub], representing a nonempty domain. Now, show that there is some assignment function, F’, that differs from F[sub]a[/sub] at most in what it assigns to y, and such that y = x is true (specifically T[sub]I,F’[/sub]) at W[sub]a[/sub]. This will show that [symbol]$[/symbol][sub]y[/sub](y = x) is T[sub]I,F[sub]a[/sub][/sub] at W[sub]a[/sub]. In other words, it is true of any arbitrary assignment for any arbitrary world.
Now call upon the definition of an assignment function, and consider the assignment F’[sub]a[/sub]. It is just like F’, except that it assigns D[sub]a[/sub] to y. Since F’[sub]a/sub = F’[sub]a/sub (implying that d[sub]D,F1[sub]a[/sub][/sub] at x = d[sub]D,F1[sub]a[/sub][/sub] at y), y = x is T[sub]I,F[sub]a[/sub][/sub] at W[sub]a[/sub].
QED.
(Let me know if I’ve mistyped something.)
It is therefore acceptable to define God as Tisthammer has defined Him because we know that necessary existence exists in every possible world. If you won’t accept the lay term “convergence of existence” (which I believe is more intuitively descriptive of the matter at hand), then you may use the tecnical term “necessary existence”.
There is no synthetic property assignment to this existence other than one that you might pull out of a hat. Searching for similarities among properties is a waste of time and effort that does not have anything to do with this proof, and results in irrelevant quantifications. Your function, f[sub]i i+1[/sub] is one such quantification that is not justified because it assigns itself to G[sub]n[/sub]. The net effect is to split God out into “this necessary existence” versus “that necessary existence” as though one were different from the other when it isn’t, as proved above.
Although some manifestion of the existence might differ from one world to the next (symmetric or not), the existence itself is the same. It is like pulling a chainsaw down your midsection in order to make two Erises. Saying that you consist of your right hand, plus your left hand, plus your other parts, and mapping each as P[sub]n[/sub]({a[sub]n[/sub], b[sub]n[/sub], c[sub]n[/sub]…n[sub]n[/sub]}) in order to combine them into a superset competely misses the point of, not only who you are, but of who people are generally. Is a blind man somehow less of a person than you? For every way in which you can describe how his sightlessness restricts him, a compensatory advantage can be offered.
But all of that is completely off the mark with respect to God as defined by Tisthammer. God is not blind; He can see when there is something to see. You are dwelling on the somethings, and God is not found there. If you attempt to apply contingency to an apriori analytic being, you will always fail.
OK, I think we are agreeing, Lib, on some things, but perhaps one of us isn’t stating his objections clearly.
I’m not searching for them. I don’t care what they are! I’m just saying that they have to be there for us to be talking about the same “kind” of being! If the actual properties of god differ from possible world to possible world, we must have a generalized definition availible to us that makes sense in all worlds. I don’t think either of us are wrong on this. I just want to make it clear that the possibility could exist that different worlds might make god so different as to not have any method of proving it similar. I surely won’t attempt to create such a world. But it is a possibility. It shouldn’t be rejeted out of hand.
You say the qualities of god are unimportant (with respect to the proof). I say that they are indicative of what is important: that we are talking about the same thing the whole time.
I believe your formulas are fine, too. But they don’t address the issue I had with the definition: that the definition includes “possible”.
Yes, we have to do this to ensure we are speaking of the same god the whole time. Necessary existence doesn’t change; I’ve never said it did. But the being that necessarily exists does change. In order to ensure that we ar talking about the same being in all posible worlds, something must show that.
Otherwise, perhaps all we’ve shown is that existence itself is necessary in all worlds.
Of course not, but he is different from me. So something must be done to demonstrate that the blind person in World[sub]n[/sub] is me.
I am doing this because I will not accept the definition of god as “the greatest possible being”, but rather, “Given all properties World[sub]n[/sub] has to offer which have inherent maximums, god is the being which has all these properties at a maximum.”
That could certainly be the case, and it’s probably I. Exposition is not my strong point. I too often seem to say more than is necessary to make the point, even as I am doing now. […sigh…]
Okay. And I think that that’s exactly what Tisthammer has done with his first axiom. By defining God as necessary existence (i.e., the greatest possible existence), he covers those bases. Whatever exists necessarily exists possibly, since necessary existence exists in all possible worlds and possible existence need exist in only one or more of them. The definition, together with Axiom 1, make Axiom 2 self-evident.
Now that he has moved his Becker’s Postulate statement with its excluded middle to the set of axioms (which I think was an unnecessary adjustment and, if anything, makes the proof more controversial) the only possible point of contention that remains is whether the modal status of the excluded middle’s proposition is always necessary.
In plain language, the entire implication means that if it is not necessary that God exists, then it is necessarily not necessary that He exists. And that’s why I think Becker’s Postulate is not controversial (justly controversial) here. If the greatest possible existence does not exist, then that status must of necessity be necessary because it contradicts what otherwise is known to be true, namely, that necessary existence exists in every possible world.
I can’t recall now exactly specific objections to this (it’s hard for us to remember things that don’t make sense to us), and I don’t want to hunt it down. But I think it had something to do with wanting to apply the same postulate to the inverse of Axiom 1 (although I believe that it was stated that it was not derived as an inverse). I allowed then that it may be applied to the contrapositive, but not the inverse (whether derived intentionally as such or not).
No. Please, hear me out.
It isn’t God that is different; it is merely the manifestation of God that is different. I hesitate to present an analogy, but I hope that in this case it will help, and there’s nothing left to do but put it a different way. If I pour blue paint into a red world, purple will happen. If I pour blue paint into a yellow world, green will happen. The effect from the blue paint is different in both worlds, but the paint that poured was the identical paint in both instances.
Similarly, if God does something in World A such that the effect is different than would be the exact same action in World B, it is a mistake to conclude that God did different things in the two worlds. He did identically the same thing. NB: It isn’t God Who is different; it’s the worlds.
I don’t believe that, and I’d like to know why you do.
First of all, I’m quite amazed at erl’s insistent that logic is merely a manipulation of symbols according to some arbitrarily ordained rules. When we question G -> G, we are questioning whether it as we can think of it follows from the definition of God that, if he exists, he exists necessarily. The same goes for transitivity. A-> B->C => A -> C isn’t derived from some desire to have some fun. We conclude it to be true because it makes sense to us. Words in the English language are no different than the variables A or B. The English word “smoke” , in some dialects and contexts, will mean “cigarette”. In others it will refer to, “cloud like substance”. If I say “a hat is a woman and women have menstrual cycles”, then I can say “hats have menstrual cycles”. This is the same as saying A->B and B->C, hence A-> C. I have, in effect, redirected the intentionality of the word “hat”.
Various forms of logic have arisen because we applied our gray matter to the nature of thought. We deem prepositional logic to be limited because it seems to permit some odd meaningless statements like “John is President. John is allergic to peanuts. Abraham is President, so Abraham is allergic to peanuts”. Non-sequitur, you cry out! Logic is nothing else than a (formal) language. If the (formal) language produces nonsense, we object. In this case we object because it’s not John’s presidency that makes him have peanut allergies. Nor are John(Jo), President(Pr) and Abraham(Ab) the same person, therefore the identity symmetry Jo = Pr = Ab is incorrect. How do we know? Because we do! All logical statements are ultimately judged by whether they intuitively make sense. I don’t dismiss that there are sometimes difficulties involved in applying formal logic to the real world in which we live. That is, going from the abstract to the concrete is sometimes problematic. But the fact that a perfect Euclidean circle is hard to find in the corporeal world makes it no less real or meaningful.
For example, to demonstrate my point, take:
[>={{
{{>=}}
}}=>A+D
AD+L
L>=[
Do you agree with me? How could you. You have no idea what it means!!! The criteria placed on any logical system is that it be both *sound[i/] and complete. It’s sound if all arguments proved using its rules are valid. And what does valid mean? It means that they are meaningfully true in the context of our life world. The logical system is complete if all its valid arguments can be proven using its rules. Every step in a logical system must intuitively follow from the previous step, otherwise the system is flawed and should be improved or dismissed entirely.
Furthermore, as for all our discussions of various logical systems and semantics, IMHO we need to reign in our horses a bit. Personally, I’m only interested in such systems that intuitively form a complete epistemic system. If a system fails to incorporate a priori knowledge that I have, it’s too weak for any discussion about God being a necessary conclusion of human reasoning. Therefore, modal systems like K* which have no means of reducing modal operators are weak and epistemically incomplete. Also transitivity and symmetry to me seem so basic that I would claim that systems without them are like houses without walls. I say this because one of Newton’s objections to an argument of mine was it didn’t apply to systems without transitivity and symmetry.
The accessibility relation in our discussion is the possibility of some given GREATEST POSSIBILITY for any positive property. Saying that perfection may mean different things in different worlds is moot because any world where perfection means something other than what we understand under “greatest possible being” isn’t accessible to us. We are not talking about the word but a specific metaphysical attribute of GREATEST POSSIBILITY which is identical in all accessible worlds (the letter combination “perfect” could mean “dog” in some given worlds for all we know, but that’s irrelevant). True , all possible worlds would include the worlds which we can’t even conceive. But please explain to me how we are to discuss (or access) these worlds, erl Yes, I can think of metaphysical properties that are common to all accessible worlds:
P = P
A -> B -> C => A-> C
And this is where I fail to see how the Wade’s definition brings any enlightenment. I can substitute the GREATEST POSSIBILITY with NECESSARY. Therefore, the “being” we are talking about is the set of those world properties (or metaphysical properties) that must necessarily be imposed on a world in order for us to access them (or extend our mind into them). “Shoe lace tying ability” or, for that matter, “love” are not metaphysical properties that make them NECESSARY beings in all accessible worlds. They are mere possible properties. Therefore, saying that these positive statements would lead us to the axiom P -> P is invalid. With other words, GREATEST POSSIBILITY is not a valid meta property of all properties. It’s only a valid meta property for such properties that are necessary (i.e. all properties that apply to all accessible worlds). Which, in fact, are all meta properties other than NECESSITY!
God is then the set of all meta properties, which makes God more equivalent to a Platonic eternal state than a maximally compassionate and maximally loving being. We have in fact stripped the term God of its intuitive meaning.
K doesn’t include the axioms A -> A and <>A -> <>A. Not having such axioms means you can’t simplify redundant iterations of modal operators. And no I don’t know how to get the dagger in a post!
I don’t want to be the layman who hasn’t studied logic coming in to ruin the discussion, but there’s still something I’d like explained. I still haven’t seen a good non-subjective definition of perfect, other than Lib’s mathematical approach that a being that exists in x universes is more perfect than one who exists in x - 1 universes. (and this can’t of course be the only definition, because then your statement would be nothing more than “the being that exists in the most universes exists in all universes”)
Anyway, I can’t think of any other actual ways to define ‘perfect’. Of course you can list lots of attributes – the most knowledgeable, the one with the most physical strength, etc. – but shouldn’t it be possible to think of infinite attributes that are non-boolean and can be more or less perfect? Maybe even attributes that are self-contradictory?
Is the smallest possible being or the biggest possible being the one that is perfect? Or is it just subjective if ‘big’ or ‘small’ is the better one?
You still have not responded to the fact that this is not the reason he threw out <>~G. He threw it out, by his own admission, because it contradicted the definition. Because it is absurd to conclude that “the greatest POSSIBLE being” is not possible.
I agree. And it is clear that the word possible is being used to mean possible in reality. Otherwise it would not have been a contradiction. But this means that a postulate that says it is possible for god to exist is unnecessary, because god is already defined as possible. More importantly, it means that we have not defined god as the greatest being we can imagine, only the greatest being that exists. Therefore there is no explanation for saying that god is necessary. Because how do we know that ANY being is necessary? The greatest being that exists clearly does not imply that that being exists necessarily.
And there is the problem. There is no way we can know that it is possible for any being to exist necessarily. The only way we could get the idea that god exists necessarily is if the definition was using possible to mean the greatest being we could imagine. But I have already shown that the definition must have been using possible to mean possible in reality, because otherwise there would have been no reason to throw out <>~G and pretend it was the problem, and everything else is fine.
“Et tu” says Lib).
You did indeed impose some restrictions…