Suppose the possible worlds were {<10 ; 0,1,2,3} and {<10 ; 1,2,…,10} and {<15 ; 1,2…15} where <10 or <15 is a ‘natural law’ of the the world, and the numbers are the things in that world. ‘Good’ is interpreted as ‘large.’
Is this ok? Can we apply your argument to this simpler set of possible worlds?
If so, it shows G is necessary, ie. exists in all worlds. You would say this means G must be the most good, or large? But why not 1? That’s what exists in each world. Why does G being in each world mean G is the largest
Have you got a link that explains contingency? In world 1, you would say the ‘largest possible in that world’, hence G, is 10, being the greatest consistent with its natural law, right? Why is the natural law what it’s contingent on? And if so, then G doesn’t exist in world1. Wouldn’t this contradict the proof?
Once again, consider carefully the difference between ~<>G and <>~G.
“It is not possible that G exists” is not the same as “it is possible that G does not exist”. We have not shown <>G to be true. Remember that we’re talking about modality here — i.e., states that obtain. <>G does not imply that G is true because both <>G and <>~G are states. It is simply that ~<>G is not a state. We therefore present <>G as an axiom.
It is not ridiculous to say that everything is possible. What is ridiculous is to say that anything is not possible. A seasoned computer programmer is well aware of this subtle but critical difference, and leaves no contingency to chance when he writes his error handlers.
As an aside, I recall working on a project with a gentleman who was one of those self-taught, seat-of-the-pants programmers. (I too am self-taught, but not so seat-of-the-pants.) He was populating a data grid, and I noticed that there was one possible error condition he had not addressed. “Bah,” he waved me off, “that really can’t happen because the data is already filtered by the SQL call.” I was responsible for the project, and so I insisted. He heaved a heavy sigh and began inserting the code as I walked away.
A couple of weeks later, our customer called and told me she’d gotten a strange message when she tried to do her clock-out summary that morning.
“What was the message?” I asked.
She replied, “Well, it says, ‘Here’s your stupid error handler which you’ll never see.’”
As far as I know, any system of logic will allow you to construct true sentences without any axioms. But if you’re going to put sentences together to form a proof, then you have to start with clearly stated premises. Failure to do so is bad practice. While not a fallacy per se, it certainly renders the proof suspect. You can Google the phrase Audiatur et Altera Pars for more detail.
Once again, ontology deals with the nature of existence and its properties, not with the nature and properties of things that exist. This particular modal ontological proof can be applied to any world-set so long as one of the worlds exists in actuality. If you prefer the symbol “1” over the symbol “G” to represent that existence which is necessary, that’s fine. (Although you would raise a few eyebrows and have to explain yourself repeatedly for each new interpreter.) Also, if you prefer to call “goodness” “largeness”, you may, so long as you make it clear to all interpreters that you are redefining terms. Failure to do so is called equivocation (and in some special cases, amphiboly.)
You would want to begin your proof with 1 = 1 so interpreters would know what you are talking about. And of course, as I’ve explained already, “largeness” (which you use to mean goodness) is not relevant.
Contingency per se is an awfully big topic. I’ve given you the essential, broad meaning, but I applaud your initiative to explore further. I recommend that, if you are not firmly grounded in the philosophy of logic, you start here. Dr. Sion is an excellent logician with impeccable credentials and is a skilled expositer. Read up on the history of modality theory beginning with Aristotle, and then come forward to today. You can learn how the various schools of truth view the modal states.
Well, no. The “largest” is 1 because it exists necessarily. It’s just that the rules of world 1 cause those who are bound by its rules to perceive 1 as “smaller” than 10. It is important to view the problem, not from the point of view of the contingent beings, but from the point of view of the necessary being.
(In fact, what you are doing even in discussing your worlds is imposing rules from the actual world upon them, specifically rules of ordinality and cardinality. They are not rules that are stated for your worlds, and therefore do not apply. And if you apply them, then you are merely modeling actuality.)
It isn’t. It isn’t contingent on the natural law. It obviously isn’t contingent on anything at all if it exists in all your possible worlds.
If you remember nothing else from this post, remember this: The rules of your worlds do not restrict 1, they restrict merely the interpretations of 1 in those worlds — there is a radically different view of reality from the reference frame of 1, and the reference frames of the contingent beings.
Clearly, 1 is not restricted at all. It is everywhere that existence is possible. Just as the number 1 is not the same as the symbol that represents it, so is it true that the being 1 is not the same as the perceptions that interpret it. It is the latter, and not the former, that is bound by rules.
I just happened to recall something I’d read that might help you better understand the nature of substantive denial errors.
Consider this sentence:
“It is not possible that G exists”. Note that, if your sentence were true, then everything following the that in your sentence is nonsense! Therefore, what exactly is it that the sentence is denying?
So you’re saying everything is possible (or at least can be)? ie. <>X is true whatever X is? How about X being | (the ‘false’ symbol from propositional logic)? How about X being ‘~G’?
How about, "Why don’t you cut out the middle man and blow me for Pete’s sake?
Or, “You’d really do that for me? That’s impressive, considering that I just prayed for you to get brain cancer.”
To me these statements seem contradictory, and intuitively negations of each other, though I haven’t put it into logic so I may well be wrong. Anyway, I can’t go much further until we agree on this aspect of modal logic, so if you can clear up my confusion in the last post first, I’ll be grateful.
It looks to me like it says that not stating assumptions you make implicitly is bad practice, with which I certainly agree. I don’t think it means you need to state some asumptions when you don’t have any, even implicitely, as unlikely as that may be. (Eg. replace assumptions in your logical system with rules of inference that allow you to deduce them from nothing.)
Anyway, this is just a hijack into semantics of a phrase. Sorry, I’ll try and leave it alone.
However, one other hit, while not relevant, I couldn’t resist sharing:
Thanks, but I still don’t get it. Why can’t “that G exists” be well-formed, but false in each world? cf. my previous example of |.
Or are you trying to say that everything that is conceivable, or stateable, (not just every sentace expressible in modal logic) has to be true in some possible world? In that case, I seriously need to read more about modal logic.
I am using largest in the normal sense. I thought we decided ‘good’ was arbitrary, so could be used to mean ‘large’ for the purposes of this example.
I have proposed here a set of possible worlds that you appear to accept. I thought that your proof showed that there must be a necessary being, (called, for the sake of argument G) which is equal to one of the things in each world. I now think I have misunderstood you. If so, can you please explain how the proof does in fact apply in this case, and ignore the rest of this post. Thanks.
But… then what point is the argument? I have a small orange here. Suppose that existed in each world. Then it is necessary, and hence, according to you, most good, most powerful, etc. Yet it just sits there going rotton. You can say ‘it’s most powerful, really!’ but I don’t think this is what anyone else means by ‘having shown the existance of a most powerful being’.
Sorry if that sounds facetious, it’s late, and I’m struggling with this.
In fact, can we run through the proof again and see where I think I need more clarification?
First, what is “G”. Is it a statement of modal logic? A symbol representing an entity?
<>G Discussed in the post above
G->G We discussed this before; I need to go back to that
At this point in the argument, what is G? Is G arbitrary, just allowing us to state the axioms? Is G something like “a most good being”? Or a description of God?
Then a valid proof in modal logic, finishing:
G G is true in our world. Then you use the fact that G is necessary to show that G is most good or something
Sorry, I know most of these guesses are wrong. I’m just realising that my attempts to clarify just lead to more confusion. Where I’m wrong here, please don’t waste time explaining why - it was just a stab - explain what you do mean at that stage.
PS. I need to sleep now; hopefully I’ll post more tomorrow. And thanks for continuing to explain, I do find this very interesting, even if I don’t get your proof yet.
Well, the last page of this thread (which is all I’ve read thus far) is an interesting delve into dei-physics, but getting back to the OP, sort of, I just say thanks.
It’s not a threat, it’s not an insult, it’s just people being kind or stupid, and in either case, they are are of no consequence to a person who knows where they stand.
Just say thanks and be silent. Usually does the trick.
Possibility is just a state. If everything is E, then <>E is not absurd. But neither is <>~E. What is absurd is ~<>E.
<>E — not absurd
<>~E — not absurd
~<>E — nonsense
Regarding your specific examples, there is an interesting paper by Bernard Linsky, of the Department of Philosophy at the University of Alberta, and Ed Zalta, of the Department of Philosophy at Stanford University, two well respected logicians, called Defense of the Simplest Quantified Modal Logic, that combines classic quantified systems with K logic (a stronger form of M — all modal logics are formed by adding axioms to K).
On page 9, they begin dealing with the objections of actualism to QML. Note that they make the same point that I’ve been making here: if you draw no distinction between “there is” and “there exists”, then your statements are existentially loaded. That’s what causes you to interpet the statement “<>E” as “everything exists” when all it means is “everything is possible”. To actualists, only that which exists in actuality is possible.
For obvious reasons, actualism is not applied to computer programming. How would you deal with division by zero, for example — a state which does not obtain in actuality? And yet if your error handling doesn’t deal with a division by zero state (an impossible state to actualists), then your program will crash.
Thus, from your examples:
<>(X -> |) — not absurd
<>~(X -> |) — not absurd
~<>(X -> |) — ridiculous
and
<>(X -> ~G) — okay
<>~(X -> ~G) — okay
~<>(X -> ~G) — meaningless
As you likely know from the elementary truth tables, X -> ~G is false no matter what the truth value of X is; therefore, the state <>(X -> ~G) obtains and the state <>~(X -> ~G) does not obtain.
I hope that I did so, but I need to point out that this is not a matter of modality but of ontology. The error is in the substantive denial of the positive ontological proposition. It is when this error is rendered modally that its absurdity becomes most clear.
Whatever begins with ~<>, the rest is gibberish because the ~<> has rendered it all as an impossible state which cannot obtain.
I think you might be mixing up two things: (1) the truth of a proposition, and (2) the state of a proposition. First order logic is sufficient to deal with (1) but K logic and higher is necessary to deal with (2).
~G is simply a false statement (which we have proved).
~<>G is a state that cannot obtain (and so is absurd).
<>~G is a state that can obtain (suppose, for example, God decides to eliminate His own existence).
I thought I already did. In this case, G is true if one of your worlds is actual, simply because G (or 1) is in all your worlds.
Okay.
G is a symbol representing a supreme being.
This is derived from ~~G (“it is not necessary that God does not exist”). ~~G is a reasonable axiom since we know that we do not want to deny substantively a positive ontological proposition.
Remember that G is G. The Rule of Necessitation states:
[symbol]"[/symbol]x[symbol]$[/symbol]y such that y=x
Since G = G, G -> G means G -> G, which in modal logic is called the “4 Axiom”. Look at this map of the relations among modal logic systems. Find the system with the transitive frame. There you will see A -> A.
G is still a supreme being.
Right.
Well, but I specifically said that in doing so, we were stepping out of the ontological argument and into a discussion of ethics. The goodness of G (and any other property of G as a thing that exists) is not relevant to the proof. Ontology deals with properties of existence, not properties of things that exist.
I’m enjoying the discussion with you. Your questions are good ones and indicate your willingness to open your mind to new information. At the very least, I hope that by now you realize that Suzene’s summary dismissal of an ability to prove the existence of “any God” was premature.
AHA! I get it. I hadn’t realised you could have a possible world where G and not G are true. OK, I accept it’s reasonable to have <>X for any X. Thanks for being patient.
Nope, I’m still not clear. Do you mean represents a being who is most powerful, etc? Or that we later show, due to the fact that G has been proved to be necessary, that G is most powerful?
Do y’all maybe wanna take this hijack to a “new” thread? (“New” only in the sense of its having its own OP, not in the sense of its covering any new ground)
To this OP, I think honesty and shame are appropriate responses. As in, turn to the lady, cock your head, smile sweetly, and say, “Why don’t you pray instead that God teaches you some manners and humility, you sanctimonious bitch?”
<>X and X are not the same state. Your primary source of confusion seems to be between tables of truth and states of existence.
Actually, I mean what I say. G is a being Who exists necessarily if He exists in actuality.
That’s not quite the right way to state the RN. If A is a theorem of K, then so is necessary A. Or A |- K -> A |- K.
Well, things for which the state of necessity obtains are necessary. That is to say, things that are true in every possible world are necessary.
There is nothing “special” about G that makes it supreme. It’s supremeness is deduced. We applied a disjunctive syllogism on propositions 9 and 4, such that G -> G and G V ~G -> G. (See proposition 10 of the proof.)
I don’t know what you mean about | -> | being necessary. Why is it necessary?
In this case, G is true if one of your worlds is actual, simply because G (or 1) is in all your worlds.