For purposes of the ontological proof, G is defined as G and I’ll say two things about that.
One, a couple of people have protested that by defining G this way, it is defined into existence since necessary G implies actual G according to the Modal Axiom. However, as I’ve explained each time the protest is raised, proof of necessary existence in S5 is not the same as proof of G in actuality. (Note: the second link is to an alternate version of the proof, and is the simplest version I’ve found.) A definition merely establishes what will be meant by G when the conclusion is reached. It is like defining a triangle as having three sides and angles that sum to 180 degrees — then — proceeding to prove that triangle X is indeed a triangle. We aren’t defining X into existence by defining triangles. Stating that G is defined as G does not prove that G exists. Those who protest have it backwards. G does indeed imply G, but G does not imply G.
Two, the definition of God as G is perfectly natural since the phrase “Supreme Being” is universally understood to mean God. The phrase is a direct transliteration of the modal symbols; i.e., Necessary -> Supreme | Being -> Existence. Thus, the Supreme Being is defined as Necessary Existence. A couple of people protest this as well, but it is hard to imagine a more appropriate definition of God than Supreme Being.
(By the way, a couple of people also have protested that the modal proof does not prove the existence of Jehovah. Of course, no one has ever claimed that it does, but that is not what Suzene asked for. Specifically, she said, “Now prove to me that your God – or any other – exists. Can’t be done.” Emphasis mine.)
I saw Ultrafilter’s post after submitting mine. Ultrafilter is correctly distinguishing between validity and soundness. The proof is prima facie valid, but it is sound if and only if you accept both of these two premises as true:
Whenever someone tells me they are praying for me I open my eyes really wide and start looking around with a real confused look on my face and say “Uhmm… OK” just as you would respond to a crazy person who just spouted off some totally nonsensical phrase. Usually they confused and wander off but if they continue talking to me I play dumb about the whole idea of religion. If they go into the whole religous sales pitch mode I listen, ask stupid questions and then tell them that it sounds fascinating but I think I’ll give it a miss.
If I’m feeling snarky I’ll tell them “No thanks, Satan’s got me covered”
If they really manage to piss me off I like to offer them a "Tall Frosty Glass of Shut the Fuck Up. "
Mostly though I just go with the first option as it’s more fun and less offensive.
You say you define G as G. Do you mean that literally, which sounds a bit circular, or that G is defined in such a way as to make G equivalent to G? (Or am I confused and these are the same thing?)
For the moment I’ll accept that definition, and that it makes the proof work.
Could you elaborate on necessary->supreme?
You have proved that G is necessarily true, right? In what way is that ‘supreme’? To me supreme being suggests most powerful, most good, or something like that. Is G this?
I assume that if it is this must have something to do with the definition of G, else everything would be necessary and supreme :eek:
Come back again to the triangle definition, which might be more along the lines of what you’re used to. Suppose you saw an ontological proof that began by defining a triangle this way…
T = T[sub]3[/sub]A[sub]180[/sub]
…meaning that a triangle is defined as a shape with three sides and angles that sum to 180 degrees.
Notice that at this point, you have not proved that any triangle exists. You’ve merely said, “When I speak of a triangle, I mean a shape with three sides and angles that sum to 180 degrees.” You have not yet established that any shape is a triangle.
By the same token, G = G is not saying that any God exists, but merely, “When I speak of God, I mean a being whose existence is necessary.” You have not yet established that any being is God.
Definitions are not themselves proofs, but are merely contextual references that give meaning to symbols in the proof they precede.
You can also state the definition as G = B if that makes things clearer to you, but then extra steps are necessary in the proof to show that necessity obtains across all possible worlds. It’s just a matter of deciding to begin with X = 2 rather than X = 2y / y.
Sure. In his original proof (his second one, the one that has been converted to modal logic), Anselm defined God as “that than which nothing greater can be conceived.” In modal logic, necessity is a state that obtains in every possible world. It is easy to understand that there can be no greater state, since possilbe worlds are the only worlds that exist. Therefore, the greatest (or supreme) existence is necessary existence because any other state of existence is less.
An important question, and possibly a key to understanding the proof from a lay perspective. The answer is that you proved that in proposition 10, just before the conclusion is reached. The conclusion is not that G is necessarily true, but that G is true in actuality. Without the Modal Axiom (proposition 4), you couldn’t have reached your conclusion. And the Modal Axiom was not presented as a premise because until proposition 3, it was not established that your proof was in the context of a Euclidean frame, where the accessibility relation among possible frames obtains necessarily.
The condition on our frame is expressed this way…
(wRv AND wRu) -> vRu
…where w, v, and u are possilbe worlds that range over W (all worlds).
This condition implies that <>A -> <>A, which in turn identifies the frame as S5.
G is that which exists in every possible world. It follows then that G is the most powerful, the most good, and so on, since there are no worlds in which G is not G. But G can also be the most impotent, and the most evil. etc., so long as that is what G is in every possible world. Whether G is good or evil is outside the bounds of ontology, and inside the bounds of ethics. The goodness or badness of G depends on what aesthetic G values. If G values goodness, then G is perfectly good. If G values badness, then G is perfectly bad.
A single counter-example can prove that NOT everything else is necessary and supreme. A real number, R, such that the square root of R is -2, for example, does not exist in every possible world (including the actual world), and so therefore does not necessarily obtain.
I misstated the real number example. I meant to say that there exists no real number R in every possible world that is the square root of -2. But I bet you caught that!
Yes, could you state the definition a bit more explicitly?
G is going to say something like “God exists” where God is not yet defined, right? And then we colloqially refer to G as God… Are we defining G so G->G is true? Or just making G assert the existance of something, and then making G->G as axiom to define a necessary being.
Also, if we’re defining God as a Necessary being, then I have to question axiom 2. I accept the possibility of a perfectly Good, perfectly powerful, type being, the colloquial God, so accepted axiom 2 on first reading. But why is it reasonable to assume the possibility of a necessary being?
Whew! That post took longer than I expected to write.
I’ll respond to the necessary/supreme bits in a second post.
Assuming we have prooved the existance of a necessary being and hence his existance in every world, (and a property called goodness defined in some way )…
So have you asserted the existance of a being with the extreme of each combination of properties. Eg: the most impotent and most good, or the largest and most evil? Why not being the largest integer?
Also, I’m still not quite clear on what you’re saying. In this world, for instance, is God the most good being in this world, or the most good being possible?
Anslem’s quote seems to say the second, but then can there be any possible but not necessary beings? I exist, so can I define a property me-ness, and deduce than a being with maximum me-ness exists in all worlds, ie. I do?
If the first, then aren’t you merely asserting the existance of a most-good being, which isn’t surprising given a finite number of people and an objective measure of goodness?
If you intend to spew obscenities and vulgarities in public, either learn to speak more discreetly or expect perfect strangers to be react negatively to your boorish behavior.
I mean, you know you are offending people, yet you gleefully do nothing to alter your behavior. That pretty much makes you an asshole.
So instead of doing less to provoke perfect strangers, make sure you have a snappy comeback for the next person who you’ll offend. Spread the good will, dude. Spread the good will.
Actually, no. The definition as stated is the most explicit possible. What I can do, however, is state it more ambiguously. Sometimes, a thing is so simple that we get lost in attempting to make more of it than it is, and a less simple definition might help clear things up.
To say that God is the Supreme Being is to say that, in every world where existence is possible, He exists. It is also to say that in every possible world, He is capable of whatever is possible in that world. In our world, for example, God is not capable of creating a square circle because in our world, that would be a contradiction. In Flatland, God’s height is zero. But in FourSpatialDimensionsland, God’s kata is without bounds. In our world, God has no kata dimension, just as in Flatland, He has no height.
These are metaphorical references, of course. We do not mean to say that God is this or that height, at least not in the context of ontology. We aren’t talking about attributes of God qua Being, but attributes of God qua Supreme. Nor is it sound to interpret the God of the proof as pantheistic, although it is sound to interpret the God of the proof as panentheistic. And that is exactly how most materialist logicians interpret G. You can think of these boundings much in the same way that cosmologists speak of a finite universe with infinite space.
Another way to define God more vaguely is to say that, what with being necessary and all, were He NOT to exist, then neither would you or anything else. Necessity is the only assurance of possibility, and actuality is an assurance of necessity. This is the core reasoning behind the proof of NE in S5 (to which I linked you earlier): because there is at least one actual world, there is at least one necessary world. (That might also help you to see the difference between proof of NE in S5 and proof of G in actuality.)
I could go on and on, but I’ll give you one more vague definition. Essentially, God is that Being Which is the source of all existence. His existence is necessarily eternal because He must exist in nontemporal worlds. That is why it is logical to deduce His existence as a spritual being as opposed to a material one.
Let me split those up:
G is going to say exactly “G exists in the actual world”. To make certain that the interpreter knows what G is, G is defined. The definition is not necessary to the proof, but it is helpful. Notice, for example, that Peanno never defined “successor” in his proof that 1 + 1 = 2. Had he done so, the Induction Axiom might have been more intuitive.
No. It is a mistake to consider any of the definitions that precede a proof to be a part of the proof in se. Definitions prove nothing. They merely give meaningful context to the symbols in the proof.
G -> G is obviously true if G is necessary. In other words, anything that exists both in necessity and in actuality must exist necessarily. If you examine the converse of that axiom, you’ll see why. G -> G is a restatement of the Modal Axiom and is intuitively true, but it does not imply its own converse. Stating that G -> G -->> G -> G would be a logical fallacy. Thus, G -> G is offered axiomatically, and not as the result of some other implication.
And that’s where the definition is helpful. At that stage of the proof, the interpreter thinks something like this: “Ah, since G = G, then by substitution, G -> G -->> G -> G, which obviously is true.” That is, the interpreter has now recognized the Derived Rule of Necessitation (called NA), which is the underpinning metatheorem of modal logic, and is the basis for formulation of Becker’s Postulate (cited in the proof). Necessity obtains.
No, not that either. Axioms don’t define anything. They are implications that merely conditionalize inferences.
So, G -> G may be read this way: IF [sub]God exists in actuality,[/sub] THEN [sub]He exists necessarily[/sub]. Often overlooked in logical implications is the fact that they are conditional. If… then… G -> G does not assert the existence of anything. It asserts a condition. The only statement in the proof that asserts the actual existence of something is the conclusion.
From the context, I think you meant axiom 1 here (~~G). Correct me if I’m wrong, but I’m going on that assumption.
It is reasonable to assume the possibility of God’s existence because denial of the possibility would be a logical fallacy, namely, a substantive denial of a positive ontological proposition, and is similar in nature to the famous assertion that you cannot prove a negative. That assertion condemns itself: if you cannot prove a negative, then you cannot prove that you cannot prove a negative. In fact, you cannot prove that you cannot prove that you cannot prove a negative, and so on. That is the logical nature of contradiction. Whereas tautologies are proved by everything, contradictions prove everything. If A -> ~A, then everything is true!
Consider the statement, “It is not possible that God exists.” Modally, it is rendered as ~<>G. In that modality are two elements: “~” and “<>G”. Because the second element is a positive ontological proposition, the first element is a substantive denial of the second. Now, here’s where your mind must be flexible or else this will fly over you, but stop to think it over a minute, and you’ll get it. If there were nothing in the second element that were possible, then the first element would have nothing to deny.
Let me repeat that: if there were nothing in the second element that were possible, then the first element would have nothing to deny. In other words, the entire statement “~<>G” is contradictory. G must be possible, else there would be nothing to negate. Now, notice that this is not the same as <>~G. ~<>~G is the opposite of <>~G and is the same as G. In fact, means ~<>~.
One thing that all this means in real world terms is that so-called “hard atheism” is a logically untenable position, whereas the position of so-called “soft atheism” is logically sound. The hard atheist says that God’s existence is not possible, but the soft atheist merely says that God’s nonexistence is possible.
No. We have not asserted anything about properties of the being, but only about properties of His supremeness. Ontology is concerned with the nature of existence itself, and not with the nature of what exists.
An excellent question.
In this world, God is the most good being that is possible in this world. You can think of necessary existence as a sort of meta-existence, and to stretch our metaphor, you may consider God to be a critter with as many legs as there are possible worlds, and with one of His legs in each world. Although they all are His legs, the perceptions of His legs are contingent upon the rules of the worlds. NB: his legs are NOT contingent — necessary existence is contingent on nothing by definition — but the perceptions of His legs are contingent on the rules of the worlds.
So… In Flatland, His leg would seem to have two dimensions only, but in our world, the leg we see would have width, breadth, and height. And in RoundSquare world, it is hard to imaging how His leg might be perceived, although given the nature of contradiction, it is reasonable to assume that His leg would be perceived as being the whole world.
And if you think about it, that is the nature of God’s existence generally in the actual world. Those of us who love Him perceive Him as perfectly good and loving. Those who hate Him preceive Him as the most horrible possible monster. And those who merely doubt His existence but don’t otherwise think about Him much, perceive Him as something vague that might be “out there”. And might not.
A circle whose diameter is a ratio of pi to its circumference is one example of a possible thing that is not necessary, and in fact does not exist in the actual world, since the space in the actual world is curved. But proof of the possible existence of such a circle is well documented, and is understood even by school children.
With regards to your me-ness, you cannot prove your own existence logically. That is because before you could begin your proof, your existence would have to be axiomatic. (If you’re writing a proof, then you can’t be nonexistent.) Eventually, your conclusion will be that you exist. And now you have a problem: your conclusion — that you exist — is a restatement of your axiom — that you exist. Therefore, your argument is invalid. It is a petitio principii, and you have “begged the question”.
I think you see by now that the first case is not what Anselm was saying.
Each time I see these threads I find something new in a familiar setting, like a favourite old film seen again on DVD.
Though you may have answered similar queries before, Lib, I pray you indulge me one more time:
Why should the thing with necessary existence be a being?
You see, it is not the supremacy I have trouble with, nor the distinction between S5 (or any other) logic and “reality”. It is this step : Necessary existence is true. Existence -> Being.
The argument, to me, still goes: If the Supreme Pencil-Sharpener existed, it would exist in every possible world. After all, hardly “supreme” would the Supreme Pencil-Sharpener be if it didn’t actually exist! …Therefore, the Supreme Pencil-Sharpener exists in actuality.
We all have a tendency to anthropomorphize, and I often have to remind myself when I’m doing it that it’s nothing more than a metaphor. Scientists often anthropomorphize nature, sociologists anthropomorphize society, and people everywhere anthropomorphize God.
When you think of a being, it is natural that you think of a human being or a being from another planet, or some other such biological organism. Seldom do you think of a rock or a piece of paper or a piano bench when you hear “being”.
Being as it is used ontologically is simply the ordinary definition, which I give here from Merriam-Webster: “To exist in actuality; have life or reality”. (Emphasis mine). The disjunction is not trivial. It is simply a matter that that which is is being. Any entity that exists is a being. To say that God is a being is merely to say that God is real, or exists in reality.
A supreme coffee cup, if it existed, would indeed exist in every possible world. But if you try to prove that it exists in actuality, you run into problems right away due to the fact that you’ve decided to assign attributes to the existence — i.e., it exists as a coffee cup. You’ve collapsed the wave, as it were, and no longer are speaking of something necessary, but only of something either actual or possible, and therefore contingent. To exist as a coffee cup, for example, is to be contingent on the existence of both coffee and cups. To exist as a pencil sharpener is to be contingent on pencils. By its very definition, necessary existence must not be contingent on anything.
The ontological proof of God assigns no attributes to His existence. It does not declare that He is a coffee cup, a pencil sharpener, a man, or a creature from outer space. It declares merely that all of existence owes its state of being to the Supreme Being. Nothing exists that did not come into existence through Him. (See? I’m anthropomorphizing, but I realize that I am.)
Going off-subject just a bit, one of the things that so attracted me to the teachings of Jesus was His understanding of the nature of God. He teaches that God is spirit, and is eternal, and is the very basis of all existence. He used anthropomorphologies to describe God, but that’s all they are — descriptions, not of His attributes, but of His nature. When He speaks of life, for example, He does not mean biological DNA replication. He means God.
Anyway, I could go on about Jesus forever, but you get the point.
I don’t think I made my argument explicit enough. Of course there are are possible non-necessary things (except in the pathological case of only one possible world, or equivalently all possible worlds being the same). I won’t debate the correctness of your example, though shouldn’t you have specified in what set of possible worlds? I thought that your proof showed that all possible things were necessary, which would contradict this.
The me-ness thing was just an example. The argument I was trying to make is as follows.
It was intended to be show that if Lib’s proof shows that God is as good as possible, ie. as good as any object in any world, then this leads to ridiculous conclusions. I now don’t think that’s what he meant, so the argument may not be relevent.
Suppose A is possible.
Let a(X) be a measure of how similar X is to A (eg. if all properties can be expressed numerically, the sum of the differences)
Then by analogy with the argument that there must be something most-good in world W, there must be something most a(X).
Call this A’ and then A’=A.
So A is in W, and similarly each possible object in each world, so all worlds are the same.
Well, a circle whose circumference is a ratio of pi to its diameter would exist in a world of flat space.
I’m not sure where that came from. […scratching head…] The proof shows that a particular being, Who exists necessarily if He exists at all, indeed exists necessarily and therefore exists in actuality.
<>A does not imply A.
Okay, fair enough.
Right, but that’s a reflexive frame where the condition on the accesibility relation is wRw. That is the frame that implies the Modal Axiom, and is not itself sufficient to prove the existence of a necessary entity. A reflexive frame can only prove the existence of an actual entity when it is already known that it is necessary.
Possible means contingently true, and this world means the actual world, so it means true in the actual world (the rules of the actual world are the contingency). That means that God (if He is good) is perfectly good, but we might not be able to perceive His perfect goodness because the rules of our world allow for evil. His goodness doubtless is beyond our language rules for describing goodness.
To say that something is possible means that it is contingently true. In a Euclidean frame like S5, contingency is accessible across all possible worlds (i.e., all worlds that contain at least one truth).
It isn’t that God Himself is different in all those worlds, but that those who perceive or describe Him depend upon the rules of their world for their perceptions and descriptions. God is NOT different in each world; it is each world that is different.
It isn’t that God cannot make a square circle definitively, but merely that in our world, contradictions do not exist. It is our perception, and not His existence, that is contingent — due to the rules of our world. In other words, if He were to make a square circle and show it to us, we could not see it as what it is.
It is an adaptation of Tisthammer’s proof, which itself is an adaptation of Suber’s proof.
Modal logic was developed in the twentieth century mainly for its applications in computer science. States of necessity and possibility are amenable to computer states. It wasn’t until the late twentieth century that it began to dawn on philosophers and locigians that Anselm’s second proof could be expressed modally. The resurrected ontological argument has been quite the buzz in philosophy circles for the past twenty-five years or so.
I’m not sure I understand your question about the axiom. A proof without any initial premise is invalid right out of the gate.
Definitions aren’t used in proofs. They merely tell the interpreter of the proof what the proof is talking about. For example, if I prove that X is a triangle but have not defined what a triangle is, then if you don’t know, then you don’t know what the proof means.
The definition isn’t used by the proof, but by the interpreter. As I said before, one example is when he sees G -> G. He knows that this is an acceptable axiom because when he substitues G for G, he gets G -> G. That is a transitive frame where the accessibility relation is expressed as (wRv AND vRu) -> wRu. And that is S4, which is one frame removed from S5.
So the definition merely helps the proof’s interpreter understand what the proof is talking about. That’s all any definition does in any circumstance. It helps to clarify a word or symbol.
Aside: there are some systems of logic where a rule of inference allows you to construct true sentances without any axioms.
Anyway, what I meant was that you seemed to be able to show <>G was true. If so, you could formalise that argument and append it to the start of the proof, and not need <>G as an axiom.
