Thursday, May 28, 2009

Gödel's modal ontological argument

Previously, I've explained a simple ontological argument, and also a modal ontological argument. However, there's another modal ontological argument which interests me: That which was proposed by Kurt Gödel. It's Gödel's ontological argument!

It's all so clear now! There must exist an x such that G(x)! (That is, God exists!)

What's that, you weren't convinced? Indeed, most skeptics would be unimpressed, since it looks so obscurantist. The argument seems to work solely on the principle of "blinded by Science!", or in this case, "blinded by Symbolic Logic!" Obviously, the proof doesn't really prove what it purports to prove, or the argument would be backed by philosophers everywhere. I certainly don't expect everyone to have the knowledge to be able to read the proof, much less pinpoint the precise location where it might go wrong.

However, as it happens, I had already taught myself a bit of modal logic, so I thought, why not try reading the proof, you know, for fun? I'm not going to go through it step by step (unless my readers really want me to), but instead, I'll outline the basic idea, and explain where I think a flaw is. [Update: By request, I went through the proof, step by step.]

The basic idea of Gödel's argument is based on the previous modal ontological argument. According to the modal ontological argument, God, by definition, necessarily exists. Therefore, if it's even possible that God exists, then God must exist in all possible worlds, including the one which we happen to live in. The problem is that we still have yet to show that God is possible.

And this is where Gödel comes in. Gödel supplies a proof of the fact that God is possible.

But first, we must define a "property". A property is an attribute of an object. For example, we might let "x" denote a particular apple, and "R" denote the property of being red. Thus R(x) is the statement "This apple is red." Gödel's central insight was to categorize some properties as "positive" and other properties as "not positive". We use P(R) to denote the statement "The property of being red is a positive property."

Gödel never tells us exactly what he means by "positive". That's because it doesn't really matter what exactly it means. The proof works, regardless. All that matters is that the positive properties obey a few axioms.
  • Axiom 1: If property A is positive, and if property A entails property B, then B is positive.

  • Axiom 2: If property A is positive, then the property not-A is not positive.

  • Axiom 3: The property G is a positive property. (G is the property of being "God-like"; an object with property G has all positive properties)

  • Axiom 4: If a property is positive, then it is positive in all possible worlds.

  • Axiom 5: Necessary existence is a positive property.
The question is, is it even possible to construct a concept of "positive" such that it obeys all these axioms? Maybe, maybe not. The power in these axioms is that they seem so intuitive. If they were not intuitive, then I would simply disagree with the axioms, and be done with it. But since they are intuitive, I must pinpoint exactly why our intuition goes wrong. And then I can disagree with the axioms, and be done with it.

Let's focus on axiom 1. What does it mean for property A to entail property B? That means that for every possible object in every possible world, if it has property A, then it also has property B. Of course, if there is not a single object in any of the possible worlds with property A, then A automatically entails B.

For example, let's consider the property of being an invisible pink unicorn, which we will call U. Let us presume that U is an impossible property. There is not a single object in any possible world which has the property U. Therefore, U entails O, which is the property of being omnipotent. I mean, have you met any invisible pink unicorns which are not omnipotent? I doubt it. Similarly, U entails every property. All of them, every single one. U entails the property of being solid dark blue, the property of being a flying spaghetti monster, and the property of not being a flying spaghetti monster.

So if U were a positive property, then every property would be positive. Obviously, this is not the case (see axiom 2). Therefore, U is not a positive property.

But this contradicts our intuition of what makes something positive (and recall that these axioms were based on intuition in the first place). Surely, there are some things which are positive, but sadly impossible. For instance, the invisible pink unicorn may be impossible, but if she existed, she would surely be a force for good. Wouldn't I be justified in saying that U is a positive property? No, because if I assume U is "positive", then I am essentially assuming that U is possible.

You might notice that in axiom 3, we assume G, the property of being God-like, is a positive property. Here, we're basically assuming that G is possible. Oops!

[An aside: In discussions of G
ödel's modal ontological argument, people will inevitably start talking about Gödel himself. I haven't really seen any evidence that Gödel necessarily saw this argument as anything more than an interesting exercise in modal logic. Which is, of course, how I see it.]

42 comments:

The Barefoot Bum said...

I'm not going to go through it step by step (unless my readers really want me to) -

Yes, please. :-)

miller said...

Gee, it'll take some work on my part. Anyone else want to second the Barefoot Bum's request?

Logical Positivist said...

Sceptic, you go wrong when you start talking about U as an impossible property. For you to attack Godel properly you must find a contradiction i.e. you must articulate why an object possessing all positive properties results in a philosophical contradiction (P ^ ~P). Furthermore, you conflate the assumption of a positive property with the property being logically possible. The obvious counter example to this is to define positive properties as anything that results in a philosophical contradiction -- call it U. I assume that U is positive but it is not the case that U is possible, which is further codified by this statement ~⧫∃x(Ux) {It is impossible that there exists an x such that x possesses the property U or it is necessarily the case that there does not exist an x such that x possesses the property U}. Furthermore, thinking and defining something, like say an omnipotent pink unicorn, is enough to establish that its existence is logically possible -- provided that the definitions do not result in a contradiction.

miller said...

I disagree on several points.

First, I do not need to show a contradiction in any of Godel's axioms. Rather, I only intend to show that there is no more reason to think they are true than to think they are false (pending further arguments).

Second, thinking and defining something is certainly not enough to establish logical possibility. After all, according to Godel's second incompleteness theorem, you cannot prove logical consistency without referring to a higher logical system.

Furthermore, at no point does this argument refer to logical possibility. Rather, it refers to some concept of "possibility" as defined by the axioms of modal logic. I'm inclined to think that the axioms of modal logic are more in line with some sort of metaphysical possibility rather than some sort of epistemic possibility.

Third, yes I do conflate "positive" properties (in the sense defined by Godel) with "possible" properties (in the sense defined by modal logic axioms). This conflation is justified, because there is a proven link, right there in Godel's ontological argument. If there were not a link, then Godel's argument would fail from the start.

Please clarify your counterexample. What is it a counterexample against? Are you making sure to use "positive" in the sense defined by Godel's axioms? Are you making sure to use "possible" in the sense defined by the axioms of modal logic?

Scott said...

you conflate the assumption of a positive property with the property being logically possible.

Isn't this exactly what theorem 1 states though? I read Th. 1 as "If a property is positive then in some possible world there exists an object with that property."

Scott said...

The invisible pink unicorn may be impossible, but if she existed, she would surely be a force for good.

Would she though? As you've just argued, her existence would entail all possible properties, including the property of being an ultrasupermassive black hole that will destroy the universe. Sounds fairly evil to me...

Logical Positivist said...

Ok, thinking of and defining something is a sufficient enough condition to establish that its existence is logically possible. Something is impossible (logically) if it results in a true contradiction for e.g the statement P^~P (P and not P). You can easily test the consistency of a statement or arguments using the rules of a logical system. Godel's incompleteness theorem refers to testing the entire logical or arithmetic system for consistency. I'm merely speaking of evaluating statements and arguments not logical/mathematical systems.

Godel theorem states that if there actually exists an object with all positive properties then it is possible that there exists an x such that x possesses all positive properties. The existence of such an x is possible because it is non contradictory since x possesses whatever property that is classified as positive and does not consist of positive and negative properties and it is derived from actuality. What you can say is that Godel has to make sure that if he classifies a property, say being tall, as positive then it's negation, in this case being short, cannot be classified as a positive property.

I repeat, by modal logical standards possibility of existence is not dependent upon having a positive or a negative property. The logical possibility of existence is established in so far as the definition of the object/concept discussed does not result in a contradiction. This is the assumption that Godel's Theorem 1 and modal logic operates under.

In my example my Ux cannot logically exist logically because U, the positive property, is defined as a logical contradiction. Anything that results in a logical contradiction is by definition impossible.

Modal logic deals with what is logically possible as well as various modalities and makes distinctions between actuality (which can be derived from necessity), possibility (which can be derived from actuality, as well as thought providing the conception does not entail a contradiction) and necessity.

By the way, it would help your discussion of axiom 1 to translate the statement properly. Axiom 1 has a necessity quantifier (the box), a for all quantifier (the upside down A) that do mean something. Perhaps it would clear up your confusion if you clarified the axiom as it was meant. It would also help if you took a class in modal logic -- it takes at least one semester for people to learn that stuff and at least 5 weeks to grasp the basics and do proofs.

P.S. I deleted the last two comments due to grammatical errors -- they basically say and mean the same thing.

Logical Positivist said...

"Godel never tells us exactly what he means by "positive". That's because it doesn't really matter what exactly it means." Godel does define what positive properties mean and it is in his writings and on wikipedia: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." In other words, Godel's notion of positivity is absolute. The definition of positivity does matter. If you can derive a contradiction from the definition of a concept then concept is necessarily false or to put it logically -- impossible. When it comes to arguments definitions matter in any logical/mathematical system

miller said...

"Ok, thinking of and defining something is a sufficient enough condition to establish that its existence is logically possible." I disagree. When I say that i've thought up and defined the invisible pink unicorn, it's not as if that means I've constructed it ground up from a consistent set of axioms. I just thought it up.

I mean, how do we know that the human mind is a consistent logical system, and thus only produces logically consistent ideas? It seems highly questionable that we would call it a logical system at all.

If thinking and defining a statement were sufficient to establish its possibility, then I could easily conjure up several apparent paradoxes. 1) I can think of a possible world where a god exists. I can think of a possible world where nothing exists. But these two possible worlds are mutually exclusive (see the modal ontological argument). 2) I can think of a shield so strong that no possible sword could break it. I can think of a sword so strong, that it can break every possible shield. But the possibility of these two objects is mutually exclusive.

But I should reiterate that this is all irrelevant, since the argument is not about logical consistency.

Well, yes, you could say that a property is possible if it does not result in contradiction. But this is a two-way street. If it just so happens that a property does not occur in any possible worlds, then it must entail everything, including contradicting properties.

We cannot determine whether a property entails contradiction just by thinking about the property and saying, "oh, that makes sense, and has no contradictions." We must also determine that the property is possible, otherwise it must entail everything, the same way that a false statement must imply everything.

miller said...

(we're cross-posting a bit here)

"Godel does define what positive properties mean and it is in his writings"If this is the case, then I stand corrected. However, I still think that the precise definition of positive properties is irrelevant to the proof. The proof is valid, given the axioms, regardless of any further definitions.

However, the definition of positive properties is relevant when we are deciding whether we agree with the axioms.

miller said...

"In my example my Ux cannot logically exist logically because U, the positive property, is defined as a logical contradiction."Please explain why this counterexample does not contradict Th. 1 (which Scott had stated for us above). If your counterexample is inconsistent, then it is not much of a counterexample.

I agree that it intuitively seems as if we can have a positive property which is also logically impossible. But this contradicts Godel's proof. I think this only goes to show that Godel's axioms are intuitively a poor model of the concept of "positive" properties. In fact, that was my main point in the original post.

miller said...

"By the way, it would help your discussion of axiom 1 to translate the statement properly."Are you seconding Barefoot Bum's request?

Logical Positivist said...

Sorry I translated theorem one wrong -- my mistake. I said (Godel theorem states that if there actually exists an object with all positive properties then it is possible that there exists an x such that x possesses all positive properties), but what I meant to say is that the Theorem states if there actually exists an object with one positive property, then there exists an x such that x has all positive properties and is positive. To prove attack Godel on this theorem you need to show that an object having all positive properties results in a contradiction. You can do this in two ways, you can show that an object possessing all positive properties possesses a negative one or that an object possessing all positive properties (an object with a conjunction of positive properties) is not positive.

"I disagree. When I say that I've thought up and defined the invisible pink unicorn, it's not as if that means I've constructed it ground up from a consistent set of axioms. I just thought it up." Ok, how does a pink unicorn result in a philosophical contradiction? In other words, what makes it necessarily false like the statement: P^~P (p and not p).

Logical Positivist said...

"If thinking and defining a statement were sufficient to establish its possibility, then I could easily conjure up several apparent paradoxes. 1) I can think of a possible world where a god exists. I can think of a possible world where nothing exists. But these two possible worlds are mutually exclusive" It's ok to think of a world where nothing exists just so long as you're not thinking that there is a possible world that is existent and non-existent at the same time.

"I can think of a shield so strong that no possible sword could break it. I can think of a sword so strong, that it can break every possible shield. But the possibility of these two objects is mutually exclusive."

Your second example is a bit trickier. If you make a shield so strong that no possible sword can break it then you can't conceive of a sword than can break the unbreakable shield. The sword can break every other shield except the unbreakable one.

"Well, yes, you could say that a property is possible if it does not result in contradiction. But this is a two-way street. If it just so happens that a property does not occur in any possible worlds, then it must entail everything, including contradicting properties."

Logical Positivist said...

This is what definitions are for. Definitions point you to necessarily, possibly, and actually existing concepts and what concepts are impossible. You state: "If it just so happens that a property does not occur in any possible worlds, then it must entail everything, including contradicting properties." You're conflating possible existence with actual existence. Modal logic makes a distinction between the two that you're not paying attention to. Something that actually exists occurs in this world as well as possible worlds. However, if something does not actually occur in all possible worlds that does not mean it entails everything including a contradiction. It only entails a contradiction if the property in question cannot under any circumstance possibly occur -- like a square circle for instance. An omnipotent pink unicorn is not in the boat of impossibility like a squared circle or a married bachelor because it doesn't result in a philosophical/logical contradiction.


My counter example does not contradict Theorem I it contradicts your assertion that an object having a positive property guarantees its existence. Let me recapitulate the counter example again.

Let P be the statement: P^~P (P and not P). Let ~P (not P) be the negation of P: ~(P^~P) [ it is not the case that p and not p]
Let P be positive and its negation negative. The statement P is positive, but it is necessarily false and cannot be said to exist in any possible universe whatsoever -- its existence is impossible.

This confusion would not arise had Godel's theorem been clarified. It just merely states that if there actually exists an object with a positive property, then it possible that there exists an x s.t. the x possesses a conjunction of all positive properties and is positive. The positivity is not guaranteeing the existence of the property. If you grant that there actually exists one object with one positive the property, then it logically follows that that it is possible that there exists one object with a conjunction of positive properties. Even if you don't grant that there actually exists an object with a positive property the statement would be true. (A conditional statement is only false if the antecedent is true and the conclusion false). As I've stated before, you must show that an object having all positive properties results in a contradiction i.e. show that it is not positive or identify a negative property in it.

With respect to Axiom 1 you have translated it in such a way that does not capture its meaning. If you're going for his propositions you must translate the symbols in such a manner that captures their meaning.

That is all

Logical Positivist said...

Posting to get follow ups

miller said...

I feel that we are both going to start repeating ourselves very soon. Let's refocus on the point on which we most definitively disagree.

"My counter example does not contradict Theorem I it contradicts your assertion that an object having a positive property guarantees its existence."You have misstated my assertion. I intended to say that if Godel's axioms are true, and if a property Q is positive, then this guarantees the possibility that there exists an object with property Q. This is Th. 1, as shown on the third line of the proof.

Let's consider your counterexample. Assume that property P is positive, and it entails both Q and ~Q. By axiom 1, Q and ~Q must both be positive. By axiom 2, Q is both positive and not positive. This is a contradiction, ergo your counterexample is inconsistent.

If you disagree with my argument, I'm afraid it isn't truly me you're disagreeing with. You are in fact disagreeing with a vital step of Godel's proof. Either you disagree with the axioms, or with the reasoning. Either way, Godel's proof fails, and I have nothing more to show.

Logical Positivist said...

It's clear that you don't understand my counter example -- it's not in reference to Godel's Theorem 1 it's in reference to your interpretation of it which states:"You have misstated my assertion. I intended to say that if Godel's axioms are true, and if a property Q is positive, then this guarantees the possibility that there exists an object with property Q. This is Th. 1, as shown on the third line of the proof." You've also stated theorem one wrongly, but let us give you that it is what you interpret it to be. Even so, the inference rule that you claim Godel uses is not the inference rule he is using. What I am stating is that Godel's inference has nothing to do with truth value and positivity of the properties. Something that actually exists guarantees its possible existence in modal logic -- not its actual truth value and positivity -- this is the inference rule Godel is using in Theorem 1. But what if it is the case that the antecedent is false. The entire statement will be true and it will be possible that there exists an x s.t. x possesses a positive property. Take this statement for example: it is not the case that there actually exists a man named Richard Dawkins who wrote God is not Great: How Religion Poisons Everything. This statement is true in what we know to be the actual world. However, its negation [It is possible that there exists a man called Richard Dawkins who wrote God is not Great: How Religion Poisons Everything] is not impossible in our actual world.

Again, I'm not sure how to place this in a more succinct manner. The inference rule that Godel is using is that actual existence of an object is a sufficient condition for the object's possible existence. Even if you don't grant that the object with that property actually exists, it doesn't follow that the object with that property cannot possibly exist.

Scott said...

Without getting into all of this -- which seems very complicated and confusing -- I will simply point out that, Logical Positivist, you seem to be in error here:

But what if it is the case that the antecedent is false. The entire statement will be true and it will be possible that there exists an x s.t. x possesses a positive property.

For precision, I'll be explicit and restate the theorem in symbols and in words. Note that []=neccesarily and <>=possibly:

P(rho) -> <>Ex[rho(x)]

If property rho is positive, then there possibly exists some x such that x is rho.

The statement "the antecedent of Th. 1 is false," is equivalent to the statement "property rho is not positive." If rho is not positive, the conclusion says nothing about whether x has or could have a positive property. It simply states that some possible x is rho.

Furthermore, that proves nothing at all, since the conclusion of an inference in which the antecedent is false may be true or false without invalidating the inference. If P(rho) is false, then rho(x) may be true or false, possible or impossible, without being a counterexample to Th. 1.

On the other hand, if rho is positive, then some possible x must be rho. If one could find some rho such that P(rho) ^ []Ax[~rho(x)], then we could conclude that Th. 1 is false, and that therefore either Ax. 1 or Ax. 2 is false.

Miller claims (in effect) to have done so with his invisible pink unicorn, by asserting that being an invisible pink unicorn ought to be considered a good thing, but that there exists in no possible world an invisible pink unicorn. Miller is, in this respect, more of an "idealist" (in the political sense) than me; I am a pragmatist and I think that properties that couldn't possibly exist in any world can't possibly be good ones. (This not to say that I have anything against any invisible pink unicorns who might be reading this!)

I hope I've summed things up correctly.

Logical Positivist said...

Scott your criticism is m00t, Theorem 1 is a conditional statement. Conditional statements are true under the three following conditions: when the antecedent and consequent are both true, when the antecedent and consequent are both false, and when the antecedent is false and the consequent is true.

The only instant in which Theorem 1 would be false is when the antecedent is true and the consequent is false -- making your critique nonsense

Hence, the reason I told miller -- long ago -- that he must state that a property with all positive properties results in a contradiction i.e. a property with all positive properties has a negative property or is not positive.

I suggest you go back to basics and look at what makes a logical statement true/false

miller said...

Logical Positivist,
I feel like we are reading completely different proofs here. Thus, you are getting the impression that I am not very familiar with modal logic, and likewise, I am getting the impression that you are not familiar with very symbolic logic.

Please state what you think Th. 1 is in a single sentence. You do not need to explain the concepts of necessary, actual and possible existence, nor the meaning of implication. I am already familiar with those concepts, and have written expositions on the subject.

Logical Positivist said...

I have a few problems with translating logical statements, but I can follow a proof. Perhaps it will help if I actually try to prove Theorem 1 What Thereom 1 is basically stating is that if you have a positive property, then it is possible that there exists some object call it x that has that positive property.

Let us grant that we have an actual positive property here in our hands. By definition that property must be assigned or belong to an object. Since, the positive property belongs to an object, we have an actual object with an actual positive property here in our hands. By one of the modal inference rules (I have to look this up to tell you exactly which one) since we have an actual object with an actual positive property, it is possible that there exists an object x, s.t. x has the positive property.

This is theorem 1 in its essence. Now as I've stated to Scott, since theorem 1 is a conditional statement (if/then), it does not matter if the antecedent is false, because a conditional statement is false only when the antecedent is true and the consequent is false. The statement is true at all other times. I hope this helps

miller said...

I stated theorem 1 as "if a property Q is positive, then this guarantees the possibility that there exists an object with property Q." You stated theorem 1 as "if you have a positive property, then it is possible that there exists some object call it x that has that positive property." These statements are nearly the same, so I'm not sure what the source of our misunderstanding is.

One difference between our statements is that you say "if you have a positive property". What do you mean by "having" a positive property? Do you mean to say "P(Q) ^ Ex s.t. Q(x)" (translation: "Property Q is positive, and there exists an object such that it has property Q")? That's not what the antecedent states. The antecedent only states that Q is a positive property. The antecedent does not require that we actually have any object with property Q.

Logical Positivist said...

My fault I should be more clear. What I mean to say when I say if you have a positive property is that there actually exists a positive property.

If there is actually a positive property then by definition of property it must belong to some object. You can't have a property without an object because an object is ontologically prior to its properties. If we are assuming the antecedent for a conditional proof then we need to have some way of deriving the consequent, thus we must rely upon the definition of property and introduce the object to arrive at the consequent or in this case the conclusion.

Cheers

miller said...

The inference rule you were referring to is "s => <>s". This is the contrapositive of axiom T, as applied to ~s. This is not the only inference rule needed to prove Th. 1. Th. 1 requires a multi-step proof, and relies on the truth of Ax. 1 and Ax. 2.

miller said...

(we may be cross-posting again)

Existence is not a modifier that applies to properties. Existence applies to objects. There is nothing in the logical definition of "property" that requires it to apply to some possible object. There is no axiom that states "Q => Ex s.t. Q(x)" (Given property Q, there exists an object with property Q). Even if there were, Godel's proof does not require it.

Would you like me to go through a step by step proof of Th. 1?

Logical Positivist said...

I don't think you're carefully reading what I have to say. With that said I'm going to give my closing arguments. Existence does not only apply to objects it is a quantifier that ranges over objects and their properties.

My problem with your proof is two fold. You're not disproving it on its own terms i.e. logical terms. As I stated before, you must prove that the argument leads to a contradiction. One way to go about this is to prove that an object that has a conjunction of all positive properties is not positive.

If you're going to rely on extra-logical principles, say empirical evidence, you must provide a justification as to why logic is beholden to that first principle (empiricism/science/philosophy/art whatever it may be).

Sidenote: As to the modal logical inference rules they are not if/then (conditional statements). For example M1 states P is necessarily true, so it follows that P is actually true. Modal logical rules tell you what statements follow from what.

My second problem lies within your translation of the first axiom. You completely ignored the for all and the necessity quantifier in the translation. Neglecting them is costing you because you haven't captured the semantics behind the symbols.

What is more problematic is that you state that Godel's theorem implies something that it does not -- mainly that the antecedent must be positive and true for the entire theorem to be true, which is not the case. Since the theorem is a conditional statement it is only false under one condition when the antecedent is true and the consequent false. To see what I'm talking about read this http://en.wikipedia.org/wiki/Logical_conditional#Truth_table

This is why I say contradictions are important if you must beat Godel.

Scott said...

The only instant in which Theorem 1 would be false is when the antecedent is true and the consequent is false -- making your critique nonsense.

How does that make my "critique" nonsense? That was the foundation of my entire post, and I stated it explicitly twice!

My argument was simple: if the antecedent of Th. 1 is false, then Th. 1 tells us nothing about the consequent -- neither whether it is true or false, nor whether it is possible or impossible.

I will clarify that in my above post, I made the error of using the term "conclusion" in place of the term "consequent." Just read "consequent" when you see "conclusion," and it will parse correctly.

miller said...

Woah, when did I ever state that Godel's proof is invalid? It is valid, as I've stated already. The problem is that we have no reason to think the axioms are true. Thus, even though the proof is valid, we have no reason to think the consequent is true. (This is the same point Scott has been making.)

Are you trying to claim that the proof is valid, or are you trying to claim that the conclusion is true?

"You completely ignored the for all and the necessity quantifier in the translation." This is demonstrably false. I stated Axiom 1 as "If property A is positive, and if property A entails property B, then B is positive." At a later point, I stated that A entails B iff "for every possible object in every possible world, if it has property A, then it also has property B." Do you see the words "for every"? That's the quantifier right there.

miller said...

You may be getting the impression that I am not reading you carefully, but I gotta say I'm getting the same impression of you. You may have taken a class in modal logic, but all I seem to be getting is repeated restatements of basic modal axioms and repeated statements of the definition of conditional statements. And you won't even state anything symbolically, which I would have thought much easier for the both of us.

Look, I already know that p => <>p. I know that []p => p. I know that (p => q) <=> (~p v q). I hope you don't think you are stating these things for my benefit.

Logical Positivist said...

What I am trying to claim to Scotty is that Theorem 1 is true in its entirety when the antecedent is false regardless of the truth value of the consequent. I grant what he is trying to say with the consequent, but if the antecedent is false the truth value of the consequent does not matter because the entire Theorem is true under that circumstance.

"I stated Axiom 1 as "If property A is positive, and if property A entails property B, then B is positive." That is not what axiom 1 states at all. It states: "If property A is positive and it is necessarily the case that for all x if x is..." This is what I'm trying to tell you. Axiom 1 is dealing with the issue of conjoining all positive properties to a single object and that object with the conjunction of positive properties is positive itself. This is why I was telling you to go for the contradiction.

Logical Positivist said...

But what I'm actually here for is to find some middle ground between your empirical scepticism and logic.

In quantificational logic there is a rule called the existential instantiation. What it does is to remove the existence quantifier and assign a specific subject that is not a variable. There are two restrictions on an existential instantiation (1). You cannot instantiate to a variable (x,y,z) (2). You cannot instantiate to a letter in the preceding line of the proof. Let us take Godel's conclusion

1.It is necessarily the case that ∃xG(x)
2.∃xG(x) 1, M1 {Modal rule 1 from Hurley's Many World's of Logic}
3.G(m) 2, EI
4.G(a) 2, EI

The problem here is that I can instantiate up to 23 specific Gods. As you can see in line 3 I've instantiated the god like object to be maths (m) and in line 4 I've instantiated the God like object to be Allah (a). The problem here is that we do not know what/who the god like object is and we can instantiate to any consonant we like so long as we do not repeat it. Likewise, the same can be done by anyone in any possible world. Hope that helps

miller said...

Logical Positivist,
You can see that I'm stating Axiom 1 the exact same way it's stated on Wikipedia (though on Wikipedia it's named axiom 2). All the details are contained in the concept of "entailment".

miller said...

Actually, I also disagree with your conclusion that Godel's theorem implies many gods. If we accept Godel's conclusion (that there necessarily exists an object x such that G(x)), then we must conclude that there exists only one object (or multiple identical objects) such that G(x).

Let's say that we have two distinct objects m and n, such that G(m) and G(n). Because they are distinct, there must be some property D such that D(m) and ~D(n). But if D is positive, then all god-like objects must have D. If D is not positive, then ~D is positive, and all god-like objects must have ~D. Either way, there is a contradiction. Objects m and n cannot have any differing properties.

Symbolically:
1. Suppose []∃xG(x)
2. ∃xG(x)
3. G(m), G(n)
4. Suppose ∃D D(m)^~D(n)
5. P(D) => D(n) (contradiction)
6. P(~D) => ~D(m) (contradiction)
7. By contradiction, ~∃D D(m)^~D(n)

This reasoning is captured in Th. 3 of Godel's proof, which says that if an object has property G, then G is the "essence" of that object.

Scott said...

LP: You seem to think that I don't understand implication. But I do. I understand it perfectly. Tell me what I have said that makes you think I don't understand implication, and I will clarify, or admit my error if I have made one.

If you will not do that, I will surmise from your rude demeanor that you are a troll and say no more.

DeralterChemiker said...

Perhaps I am being a purist, but you should spell the name as Gödel. The umlaut changes the pronunciation.

miller said...

You are totally correct, DeralterChemiker. I've changed it. But unfortunately, I can't put special characters in the title.

The Barefoot Bum said...

Ok, thinking of and defining something is a sufficient enough condition to establish that its existence is logically possible.

The set of all sets that do not contain themselves.

The Barefoot Bum said...

Doesn't axiom 5 trivially contradict Kant's position that existence (necessary or possible) is not a property? If existence were a property, how would we make sense of statement (A)?

A: There exist objects (such as unicorns) that have the property of not existing.

Well, do they exist (to have the property of not existing) or do they not exist (i.e. having the property of not existing)? There's a reason the existential qualifiers don't use predicate or property semantics.

Godel of all people should be wary of mixing meta-levels, and he does this twice: not just calling existence a property, but by talking about properties of properties (i.e. a property, not an object, can have the property of being positive).

miller said...

Yes, yes it does contradict Kant. However, I would disagree with Kant.

If we were using Kant's system, we would have to find some set of rules on how to construct properties. I believe we could find a consistent set of such rules, but it's not immediately obvious to me what they would look like. I also believe it is possible, and perhaps more intuitive, to build a consistent system which does include existence as a predicate.

I would say, under this system, that statement A is simply false. There cannot exist an object which does not exist. If there exists no such object, we cannot apply any properties to it. However, we could apply properties to our conception of that object (provided that our conception exists).

Anonymous said...

Good blog, you have it spot on. I see it like this:

Axiom 1 automatically means that for any predicate R that has the property P, there must be at least one entity that satisfies that predicate R. In other words, for any predicate R that does not hold for any entity then that predicate does not have the property P.

So that when you get to Axiom 3, which is that the predicate G has the property P, that means that there must be at least one entity that satisfies the predicate G. And since G is interpreted as being `godlike', then at that point you have assumed that there is at least one `godlike' entity.

So what does it all prove? Nothing.

miller said...

The first premise you must accept is that Godel was almost certainly much, much, smarter than you.