Jul 262006

This page is an attempt at simplification of the material on Christopher Small’s website, at http://www.stats.uwaterloo.ca/~cgsmall/ontology2.html If you want to go through the complete agonizing detail, I recommend you check it out there. Ok, to business.

What does it mean to say “God is great” or “greatest”? It is to judge between some attributes, and call them positive.

Kurt GodelKurt GodelGodel invented a notation for this as follows:

Pos(x) means that “x” is a positive attribute. What does this mean? Well we could judge this on some sort of aesthetic basis, looking for beauty or moral goodness. Godel choose for his ontological argument to keep it in the context of “pure attribution”. By this he seems to mean things that ADD TO BEING, rather than being a lessening of being.

Pure positive attributes are also absolute- and don’t depend on the changing nature or accidental structure of the world. In modal language:

Pos(F) >> #Pos(F) (for an attribute to be positive implies that it is NECESSARILY positive)

Some more details about positive attributes:

Axiom G2: Pos(F) >> ~Pos(~F) (if F is positive, it’s opposite is negative)

This contradicts Brian, who insisted that, for example, God would have to be both absolutely good and absolutely evil to be the “greatest”. In Godel’s proof, only attributes with add to being or indicate MORE BEING are positive, and their opposites are absolutely negative.

Godel next proved, through a process I won’t repeat here, but which you are welcome to pursue on the page I listed, that positive properties had to be “consistent” – and by that, he means “possible”. It makes no sense to claim that an impossible property is positive, and Godel proved that it would involve a logical contradiction. He wrote this as follows:

Axiom G1: Pos(F) >> * (Ex) Fx “For every positive property, it is possible that there exists an x which has that property)

Definitions: with this in mind, Godel defines formally what he means by “God”. A Godlike individual, says Godel, would be one for whom every essential property is positive, and every positive property is essential. Christopher Small translates this as:

Definition G1 – Gx = df(F) [#Fx == Pos(F)]

“In a Godlike individual x, for every property F which is positive, it is necessary that x have that property”

As a note, this does not say that God can’t have accidental properties that are not positive.

I will skip over some of the intermediate work of definitions and cut to the chase, since we’ve lost most people by now.

Axiom G3: Pos(G) “The quality of being God-like is positive”. There is considerable verbiage devoted to this axiom on the website, but I think most of us can simply nod.

Corrolary G1: * (Ex) Gx “It is possible that a Godlike individual exists” (Proof: Theorem G1 and Axiom G3)

Axiom G4: Pos(NE) “Necessary existence is a positive property” (Proof: already discussed previously. Necessary being would be greater being)

Theorem G2: Gx >> G Ess x “x being a Godlike individual implies that Godhood is the essence of x” (I skipped the disscussion of essence. Read it if you like. Basically it implies a property that isn’t accidental or contingent – but essential. The proof involves theorems I skipped. Again, I think this one’s pretty obvious. God isn’t just simply God by a strange quirk of chance 😉

Theorem G3: #(Ex) Gx “It is necessary that there exists an individual who is a godlike individual”

This is basically the QED – it is necessary that God exists, so I’ll try to show the proof.

We stated in definition G1 that being Godlike involved having every positive property necessarily. We stated in G4 that necessary existence is a positive property (and hence the godlike being would have it necessarily) We stated in theorem G2 that if there is a godlike individual, godhood is his essence. From the definitions of necessity (which we skipped) we conclude that if any individual x is God-like, then the property of being God-like is necessarily exemplified. This can be written symbolically as:

(Ex)Gx >> # (Ex)Gx “if there exists an x, and x is godlike, it is NECESSARY that there exists an x who is godlike”

Now it’s just cranking the logical equations…

  1. [(Ex)Gx >> # (Ex)Gx] (necessitation axiom of modal logic) “it is necessary that if there exists an x, and x is godlike, it is NECESSARY that there exists an x who is godlike”

Brief pause to introduce a new theorem of modal logic:

  1. [p>>q] >> (*p >> *q) (let’s call it the possibility theorem) “if it is necessary that p implies q, then the possibility of p implies the possibility of q”

Applied to the above, this gives us:

  • (Ex)Gx >> *#(Ex)Gx “the possibility that their exists a godlike individual implies the possibility that the existence of a goodlike individual is necessary”

But Corollary 1 told us * (Ex) Gx “It is possible that a Godlike individual exists”… therefore:

  • #(Ex)Gx “it is possible that it is necessary that a godlike individual exists”

but, according to theorem S5 of modal logic…

  • #p >> #p “if it is possible that it p is necessary, then p is necessary”

Applied to the above,

  1. Gx – it is necessary that God exists.
Jul 262006

Next in my tour of arguments for the existence of God, a perennial favorite, the Ontological Argument for the Existence of God.

St. AnselmSt. AnselmThe argument is originally credited to St. Anselm, who probably developed the ideas as a meditative technique, rather than an actual argument. Since his time, a number of brilliant minds have both defended and attacked the argument. It’s proponents have included Leibniz, Godel and Plantigna. At the current time, my assessment is that the proponents of this argument have the upper hand over it’s detractors. The version I will give here is one that uses a form of logic called modal logic. There have been some attacks on modal logic, and there are forms of the argument that do NOT use it, but they are a bit complex, and the attacks on modal logic also seem to have petered out. If anyone is a real critic of modal logic, I can post the other version of the argument or links to it. First, I’ll explain the symbolism I’ll use for the logic and the basic axioms of modal logic.

Symbols of modal logic:

1. “>>” Material implication. Example x>>y is “x materially implies y” (usually written like a side-ways horse shoe)

2. “~” Negation. Example ~x is “Not x”

3. “*” Possibility. Example *x is “It is possible that x” (Usually written as a diamond)

4. “#” Necessity. Example #x is “It is necessary that x” (Usually written as a square)

5. “V” the “or” operator. Example x V y is “Either x OR y”

Basic axioms and postulates of modal logic:

1. #x >> x “If it is necessary that ‘x’ is true, then ‘x’ is true”

2. x >> *x (contrapositive) “If x IS true, then it is possible for x to be true”

3. #(x>>y) >> (#x >> #y) (called ‘modal modus ponens’) “If it is necessary that x being true implies y is true, and x is necessarily true it implies y is also necessarily true.”

4. #x is true if x is a proven axiom (the principle of necessitation) “If we can prove that x MUST always be true, then x is necessarily true”

5. #x V *~x (law of excluded middle) “Either it is necessary that x is true, or it is possible that x is NOT true”

6. #x >>##x (Beckers first postulate) “If x is necessarily true, then it is NECESSARY that x is necessarily true”

7. *x >> #*x (Becker’s second postulate) “If it is possible that x is true, then it is NECESSARY that x is possibly true”

The Modal Ontological Argument (g = God exists)

Axiom 1: *g (“it is possible that God exists”)

Axiom 2: *g >> #g (“if God exists, he exists necessarily (he is not a contingent being)”)

The proof

1. *g >>#g (axiom 2) “if God exists, he exists necessarily”

2. *~g >>#*~g (Becker’s second postulate) “if it is possible God doesn’t exist, it is necessary that it is possible God does not exist.

3. #g V *~g (excluded middle) “Either God necessarily exists or it is possible he does not exist”

4. #g V #*~g (substitute 2 into 3) “Either God necessarily exists or it is necessarily possible he does not exist”

5. *~g >> ~g (contrapositive of axiom 2) “If it is possible God doesn’t exist, God doesn’t exist”

6. #(*~g >>~g) (necessitation postulate on 5) “It is necessary that if it is possible God does not exist, God does not exist”

7. #*~g >> #~g (modus ponens on 6) “If it is necessary that it is possible that God does not exist, then it is necessary that God does not exist”

8. #g V #~g (substitution of 4 into 7) “Either it is necessary that God exists or it is necessary that he does not exist”

9. ~#~g (axiom 1) “It is not necessary that God does not exist”

10. #g (8 and 9) “It is necessary that God exists”

A Possible Worlds Version:

Assuming that the modal proof above was a tad hard to follow for some, let me give Plantigna’s “possible worlds” version, which I have used here before.


Maximal excellence: To have omnipotence and omniscience in some world

Maximal greatness: To have maximal excellence in every possible world.

Why is it maximally great to have maximal excellence in every possible world? Because this indicates that God’s greatness doesn’t depend on this or that particular circumstance. God MUST have maximal excellence, regardless of the possible world in which he is present. Nothing can prevent him from having maximal excellence.

1. There is a possible world (W) in which there is a being (X) with maximal greatness.

i.e. It is possible that God exists.

2. But X is maximally great only if X has maximal excellence in every possible world.

3. Therefore, X is maximally great only if X has omnipotence, omniscience and moral perfection in every possible world.

4. In possible world W, the proposition “There is no omnipotent, omniscient being” would be impossible – that is, necessarily false.

5. But what is impossible does not vary from world to world.

6. Therefore, the proposition, “There is no omnipotent, omniscient being” is necessarily false in this actual world too.

7. Therefore, there actually exists in this world, and must exist in every possible world, an omnipotent, omniscient being.

The last time we tried this, we got hung up a lot on point 5, with some suggesting that there are possible worlds where what is impossible DOES vary. This is a misunderstanding of what is meant by “possible worlds”. A “possible world” is a theoretical construction of a world that differs from ours only in it’s contingent details, NOT in the laws of logic. A world where the impossible is possible is not a “possible” world, but an “impossible” one. Without resolving the question of whether the laws of logic can in fact change, let’s just agree that for this example, we are considering only that set of worlds that obey the laws of logic – where that which is logically impossible does not vary.

What do these arguments tell us?

There are several trivial attacks against the ontological argument, and a few serious ones. One of the most significant attacks is to point out that the logic of these proofs can be reversed. You can prove with the same analysis that if it is POSSIBLE that God does not exist, then it is absolutely impossible for him to exist.

What these proofs REALLY tell us is that, contrary to what we might assume, God is not a mere possibility – something which may or may not exist. He is either absolutely necessary, or he is logically impossible. Any middle ground is an illusion based on not understanding the concepts involved. Which proposition, then, can marshall more support?

1. It is possible that God exists (and hence he is logically necessary)


2. It is possible that God does NOT exist. (and hence he is logically impossible)

I think the best support can be gathered for #1. It is easy to see that if God exists, he would be the fundamental creator and/or sustainer of every atom and every photon – an absolute necessity. In fact, as we saw in the cosmological proof – if our notions of cause and effect, and the principle of sufficient reason, have any application on the cosmic scale, God WOULD be absolutely necessary.

On the other hand, it is difficult to see why the existence of God should be a logical impossibility. For one thing, we can coherently form a conception of God – something we really can’t do of logical impossibilities. We can form coherent concepts of CONTINGENT impossibilities (like pink unicorns) but NOT of logical impossibilities (like square circles). I can get my mind around the concept “If God exists, he exists necessarily”. The contrary, that if God does NOT exist, his non-existence is a logical necessity – just doesn’t seem as convincing. And it it much harder to summon up the idea “There is a possible world W where God is logically impossible” than the contrary.

Secondly, millions of people claim to have had some kind of contact with God in mystical or religious experience. This should give the benefit to at least the possibility of God’s existence – and, as we have seen from the proof, if God is at least possible, then he necessarily exists.

Godel also formulated an argument for God’s possibility based on some very interesting principles, which I can introduce later if anyone’s interested.

Jul 262006

There are a numberof versions of cosmological arguments for God floating around out there, and a number of serious refutations to the classical version of this proof. I want to introduce to the readership here a “new and improved” version of the cosmological proof developed by Mortimer J AdlerMortimer J AdlerMortimer Adler, professor of philosophy at the University of Chicago (chairman of the “Great Books” program, founder of the Aspen Institute, etc. etc..)

I’ll try to give a bit of a simpler (and hence slightly less precise) version than Adler uses.

A Cosmological Argument for the Existence of God.

First of all, many critics of cosmological arguments have reasoned that we don’t need a first cause. There can simply be an infinite series of causes, each explaining the other for eternity. Let’s grant this possibility. Perhaps existence can be passed on through some sort of “inertia of being” from one link in an infinite chain to the next. Let’s assume, then, for the purposes of our improved cosmological argument, that the cosmos is eternal. On to the simplified argument:

1. If something exists, and requires a cause, then the cause also exists. 2. The cosmos as a whole exists. 3. The cosmos as a whole is contingent (ie only possible) rather than logically necessary, it therefore requires an ongoing cause outside itself preventing its non-existence. 4. Any cause outside the contingent cosmos is both supernatural (since the cosmos includes all of nature) and non-contingent, this supernatural, necessary cause is God.

The controversial premise is clearly #3. Let’s elaborate on it a bit.

One of the points critics try to make is that perhaps the cosmos as a whole is logically necessary. I would argue that it is much more likely to be contingent. Here’s why.

The UniverseThe UniverseWe can imagine alternatives to the cosmos. We can imagine a cosmos with different natural laws, different structure, etc. We don’t see any compelling reason why these alternate cosmoses are logically impossible. Ordinarily, it is impossible to coherently imagine something which is logically impossible, such as a square circle. The fact that we can coherently imagine an alternate cosmos strongly suggests that our cosmos is not logically necessary, but merely possible, something which might have been different.

Anything which MIGHT have been different… might also not have existed at all. It’s non-existence is a real possibility.

BUT – given infinite time, all real possibilities would have been realized an infinite number of times. Hence at some point, the cosmos would have ceased to exist. And once replaced by absolute nothingness, the cosmos can no longer pass on its existence.

Hence, the only way this merely possible cosmos can continue to exist – is if a Necessary Being maintains it in existence.

The only alternative is that, despite all appearances, the cosmos as a whole IS logically necessary, even though we have absolutely no proof of that.

It is therefore reasonable to believe, in fact likely, that a supernatural being maintains the cosmos in existence.

Several articles by Dr. Adler discussing these ideas can be found at : http://www.radicalacademy.com/adlertheology1.htm

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