The Roles of Logic and Science in Questions of Theology

[_marg]

In addition it seems to me that modal logic deals with imaginary possible worlds as well as the world we live in.

Here's a novel idea, instead of just going by your gut instincts on everything, why not learn something?

I've got news for you Gad, the real novel idea here is, this is a discussion and if you disagree with something said then add to the discussion by explaining briefly why you disagree.

The easiest thing to do Gad is to criticize others in a discussion. That takes no brains whatsoever.

[_marg]

Ok Tarski back to the first line

Def'n 1) x is god-like..iff ..........

That sentence is a claim. When one writes a def'n let's say about a rose..i.e. a rose is a type of flower , that is part of a def'n of a rose and it is linked to the world via observations of flowers and classifications. It is also a claim.

X is god-like is a claim as well as a def'n. There is no link to the actual world. Therefore it is an analytic statement. Analytic statement def'n : a statement whose truth can be determined solely through analysis of its meaning. And a proposition whose predicate concept is contained in its subject concept. The predicate is "god-like" X the subject is unknown, but in the argument X is God. So the claim is not saying anything more than what is already in the subject term. And therefore the claim made by the def'n 1 assumes a god.

[_Jersey Girl]

Excuse the intrusion of a know-nothing preschool teacher to this discussion but:

Def'n 1) x is god-like..iff ..........

That sentence is a claim. When one writes a def'n let's say about a rose..I.e. a rose is a type of flower , that is part of a def'n of a rose and it is linked to the world via observations of flowers and classifications. It is also a claim.

X is god-like is a claim as well as a def'n. There is no link to the actual world. Therefore it is an analytic statement. Analytic statement def'n : a statement whose truth can be determined solely through analysis of its meaning. And a proposition whose predicate concept is contained in its subject concept. The predicate is "god-like" X the subject is unknown, but in the argument X is God. So the claim is not saying anything more than what is already in the subject term. And therefore the claim made by the def'n 1 assumes a god.

Tarski has already addressed that the definition is not a claim and already explained why. He does so in a post on the previous page here:

Secondly, definitions are not claims.

Look at the construction below:

Definition: A a smooth function is a function that is continuously differentiable k-times for all k.

This may look like claim to you but it is not at all. It is a definition and makes no claims despite grammatical similarity.
This is the proper grammatical construction for definitions (not claims).

Here are a few more standard definitions from the literature. Note the grammatical pattern:

Definition: A real or complex Hilbert space is a real or complex inner product space that is a complete normed space (Banach space) under the norm defined by the inner product.

Definition: A topological vector space X is a vector space over a topological field K (most often the real or complex numbers with their standard topologies) which is endowed with a topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous functions.

Definition: A differentiable manifold is a topological manifold equipped with an atlas whose transition maps are all differentiable. More generally a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable.

Having read every single post on this thread and followed it as it has taken place, it appears to me, marg, that you either wilfully ignore what he is teaching you or you simply do not understand or do not accept how far out of your depth you really are here. My own intrusion here could be off base, but at least I'm smart enough to know when I don't "know enough".

Jersey Girl

[_Tarski]

Summary of points based on questions by Marg.

I am not even sure that Godel himself used the term "God-like".
The argument is often formulated symbolically without the use of any common language nouns such as "God".

Q. Then why are we discussing the argument in word form?

Ans. It is considered easier to understand in that form as long as one doesn't not read anything more into the words than is given in the definition. JAK is doing that and assumes that the argument pivots on such connotations. It doesn't, at least not in the case of the property defined as
"has as essential properties those and only those properties which are positive". We use "God-like" as a name for that defined property.

Q. Isn't it true that existence is not a property?

Ans. This is the usual criticism of ontological arguments and it is a good one. Why is JAK not satisfied to take that approach?
Godel's argument seems to have a similar problem with "necessary existence".

JAK keeps acting like I am defending the whole argument and its application to theology. I am not.
I only want him to focus on what the real problems are. He says that definition 1 assumes God's existence if I understand him correctly (but it doesn't). That would be begging the question if it did (see the definition I quoted). But then he turns around and say he never said that definition one begs the question. huh? Sounds as if he is confused. If it assumes God's existence then it begs the question of God's existence which is what is supposed to be proved (in Godel's specific sense).

Q. Why should we be convinced that symbols can translate exactly into words with concepts?

Ans. Good question. Indeed "positive" seems unlikely to be an exact match for something real. But also the symbols in a symbolic deductive argument may apply to totally different real situations. The very same symbolic argument might possibly be mapped onto more than one situation which can be described in English. This is one big point about the first order predicate calculus. For example, there is more than one model for axiomatic hyperbolic geometry.

But Godel wants to make a connection with familiar concepts via the notion of "positive" property. He is where one of the main problems lies. Why should we accept that there actually is such a property?
On the other had, God-like is not but a word that he defines in terms "positive property". He is not assuming God-like, he is merely defining it as a word to use to mean what is captured in the long phrase above. For the argument to apply to something real, we must assume that there is a real notion of positive that satisfies his intuitive axioms.


Q. Isn't it true that not all modal logicians accept Godel's argument in word form as being sound. [/quote]

Ans. Right! They don't and neither do I. I am guessing that most don't. But some do consider it to be at least a valid argument (note the distinction).

The analysis in the prestigious Standford Encyclopedia of Philosophy gives the following analysis which is at total variance with JAK's claims about defintion 1 and the issue of inappropriately assuming the existence of what is to be shown to exist.

Q. So what is wrong with the argument? Perhaps it's not so much that a God is assumed but rather Godel's god is..even if Godel's god is simply what can be conceived.

Ans. It seems to me that there are at least two things that need to be assumed for the formal argument to be sound as applied to the actual world. One is that there is a real life workable notion of "positive". This is problematic for sure but it is far more intuitive and accessible than the concept of God. This is why Godel's proof would have some value to anyone who already accepted that such a notion of positive had reality. Second, and related to the first, we seem to need to be committed to some sort of Platonism. His notion of positivity has to be so objective that we probably need the platonic world for it to reside in. But then the platonic world is far from established and seems not to be admissible to empirical analysis.

Q. So what exactly are the properties of Godel's god?

Ans. So far as what he hopes to prove here, Godel's God would be a being that has as essential properties those and only those properties which are positive. He would have shown no more than that if the proof had worked.
But Godel's personal notion of God seems like a sophisticated version of Plato's "The Good". It is entirely unphysical and has the same sort of existence as Plato thought that numbers had.
Not the kind of God that Christians and Muslims seem to accept.



Q. Well what God does the argument prove, what properties does this God have?

Ans. Only the property described by the bolded phrase above and anything else that follows deductive from it. Nothing more.
Not much use to a Christian or Muslim is it?

Given a sufficiently generous conception of properties, and granted the acceptability of the underlying modal logic, the listed theorems do follow from the axioms. (This point was argued in detail by Dana Scott, in lecture notes which circulated for many years and which were transcribed in Sobel 1987 and published in Sobel 2004. It is also made by Sobel, Anderson, and Adams.) So, criticisms of the argument are bound to focus on the axioms, or on the other assumptions which are required in order to construct the proof.

Q, what about the fact that a certain professor says there isn't consensus even among modal logicians with Peter Geach being one who doesn't accept as sound ontological arguments for God?

Ans. Neither do I accept it as sound! Neither does the SEP. They just identify the problems correctly. JAK does not. (Unless somehow I have confused what Marg said with what JAK said)

I claim that definition 1 does not assume the existence of God and that it is a definition and no a claim. I also claim that Godel's arguments does not impropery depend on connotations of the word "god" found in his coinage "God-like" (the latter being only short hand for the property described by the bolded phrase above).

Nothing more. If JAK now agrees with me then fine--but I thought that's what he meant when he said that definition1 "assumed".

JAK also said that definition 1 was a claim. But it isn't.
In logic and mathematics, a phrase such as

Definition. An X is an A that has property B (notice that the label "Definition plays a role here).

means

We definite the word X to be a descriptive word that refers to, by definition, anything that is an A and also has property B.

See! Its not a claim. This is a simple and ubiquitous convention that JAK seems unaware of.

Example:
Definition: A prime is a positive integer that is divisible only by 1 and itself.

That's not a claim, its a definition just as is Godel's definition of property G (God-like). Notice that prime had several common meanings and connotations before this definition was made but these don't enter into any number theory argument.


Q. Isn't there is something intuitively wrong with any argument for a God concept, which claims to provide some sort of reliable conclusion.

Ans. I feel the same way, especially if what is meant by the word God is anything like what most people imagine. To demonstrate such a God, we would need empirical evidence and plenty of it (it's and extraordinary claim).

Godel tries to circumvent this by implicit use of a platonic ontology but in the end it isn't convincing.



postscript:

As for why JAK gets attacked, it could have something to do with the fact that he writes long tedious hit and miss analyses that are full of misguided accusations of fallacies and other things that show he is sort of cutting and pasting ideas often just beyond his grasp. For one example, how does one not see that defintion1 is a definition, not a claim?? It's infuriating (but made my professor friend laugh out loud).

Even if the identification of fallacies is done correctly (a big if in JAK's case), the overuse of fallacy accusations is also stylistically poor.

I agree with what Gad wrote once:

http://gadianton2.tripod.com/index.blog?from=20060115

[_marg]

Excuse the intrusion of a know-nothing preschool teacher to this discussion but:


Tarski has already addressed that the definition is not a claim and already explained why.

I don't have the time to address every post, by just anyone who pops into the discussion for a brief giving of an opinion. And you know by now Jersey Girl what I think about your opinions.

I'm happy for you that you understand Tarski's post. Good for you.

I don't. The def'n of a def'n is that it is a statement of a meaning of a word.
The def'n of a claim is it is a statement of something as fact. Now it might be that in math one can draw a clear distinction, but it seems to me looking at the def'n of those 2 words that a def'n is a distinction of a particular sort of claim.

Let's look at the def'n of a teacher. A teacher is a person who educates. That is also a claim. It is a statement of fact isn't? And that is the def'n of a claim.

Tarksi today I likely won't have time to respond to your post. Just giving you a heads up.

[_marg]

Even if the identification of fallacies is done correctly (a big if in JAK's case), the overuse of fallacy accusations is also stylistically poor.

However Tarski, when one discusses with another it is assumed that both are intellectually honest, sincere in attempting to understand each other and reach consensus of conclusion. The reason for pointing out fallacies to another in a discussion is for short-hand explanation for why one doesn't accept the claims or counter arguments of the other person. When fallacies are consistent and they are pointed out and the other person doesn't stop, it indicates disingenuous intent. Theoretically a sincere individual will stop their fallacious argumentation techniques when it is correctly pointed out. If they don't it indicates the argument/discussion can not proceed fruitfully to conclusion. At that point just pointing out the fallacies is all that is worthwhile, if any further words are worthwhile. It is up to the reader/audience and the other person in discussion to appreciate whether or not the fallacies pointed out are justified

Gad started out as being the expert on fallacies, telling JAK he didn't know what begging the question was. JAK pointed out Gad's fallacious argumentation techniques, ad hominem for example and Gad didn't stop. So not only is it is obvious to JAK, I'm sure, but to those who read the thread and appreciate argumentation, Gad has another agenda besides honest sincere discussion.

Now with regards to ad hominem fallacy, you had explained your perspective and I believe I disagreed with it. It is fine to point out incompetence when it is demonstrated to have a bearing on the argument, but to keep pointing it out and it's not been demonstrated adequately indicates..fallacious argumentation, diversion away from issues and onto the person instead.

[_marg]

Tarski with regards to def'n versus claim.

Isn't it true that with Euclidean geometry a definition of parallel lines is that they never meet?

But isn't it also true that holds only in the system of Euclidean geometry with plane surfaces?

So the def'n of parallel lines is a claim that holds true within Euclidean geometry but not outside the system.

[Guest]

Here are three proofs. in formal logic, they are all perfectly identical. Hopefully some of the motivation for making definitions is apparent.

axiom 1: pears and peaches and strawberries
axiom 2: bannas and apples and grapes and cherries and watermelon
axiom 3: if joan likes pears and peaches and strawberries and bannas and apples and grapes and cherries and watermelon then steve will invite her on a picnic and they will eat pears and peaches and strawberries and bannas and apples and grapes and cherries and watermelon
axiom 4: joan likes pears and peaches and strawberries and bannas and apples and grapes and cherries and watermelon
theorom 2: steve will invite joan on a picnic and they will eat pears and peaches and strawberries and bannas and apples and grapes and cherries and watermelon.

------

axiom 1: pears and peaches and strawberries
axiom 2: bannas and apples and grapes and cherries and watermelon
definition 1: let the word "fruit" take the place of "pears and peaches and strawberries and bannas and apples and grapes and cherries and watermelon"
axiom 3: if joan likes fruit, then steve will invite her on a picnic and they will eat fruit.
axiom 4: joan likes fruit
theorom 2: steve will invite joan on a picnic and they will eat fruit.

------


axiom 1: pears and peaches and strawberries
axiom 2: bannas and apples and grapes and cherries and watermelon
definition 1: let the word "chocolate" take the place of "pears and peaches and strawberries and bannas and apples and grapes and cherries and watermelon"
axiom 3: if joan likes chocolate, then steve will invite her on a picnic and they will eat chocolate.
axiom 4: joan likes chocolate
theorom 2: steve will invite joan on a picnic and they will eat chocolate.

[_Tarski]

Tarski with regards to def'n versus claim.

Isn't it true that with Euclidean geometry a definition of parallel lines is that they never meet?

But isn't it also true that holds only in the system of Euclidean geometry with plane surfaces?

So the def'n of parallel lines is a claim that holds true within Euclidean geometry but not outside the system.

No that's is not correct. It is an axiom or postulate that does not hold true in hyperbolic or spherical geometries. In axiomatic geometry parallel is defined. It is not a claim. If parallel is defined in terms of non-intersection then in spherical geometry there are no distinct parallel lines and in hyperbolic geometry there is more than one line parallel to given one and through a fixed point not on the first line.

In either case, the definition of parallel stands and is not a claim. We make the definition and then draw conclusions from the axioms etc.
A claim needs a proof. A definition does not. The only condition on a definition is that it be unambiguous.

[_Tarski]

Excuse the intrusion of a know-nothing preschool teacher to this discussion but:


Tarski has already addressed that the definition is not a claim and already explained why.

I don't have the time to address every post, by just anyone who pops into the discussion for a brief giving of an opinion. And you know by now Jersey Girl what I think about your opinions.

I'm happy for you that you understand Tarski's post. Good for you.

I don't. The def'n of a def'n is that it is a statement of a meaning of a word.
The def'n of a claim is it is a statement of something as fact. Now it might be that in math one can draw a clear distinction, but it seems to me looking at the def'n of those 2 words that a def'n is a distinction of a particular sort of claim.

Let's look at the def'n of a teacher. A teacher is a person who educates. That is also a claim. It is a statement of fact isn't? And that is the def'n of a claim.

Tarksi today I likely won't have time to respond to your post. Just giving you a heads up.

Maybe this will help Marg.

There is nothing at all wrong with the following definition. It is not wrong in formal logic if it is used properly:

Definition. A God-frump is a cat that can speak English.

The definition is just fine (at least to the extent that the ability to speak English is not thought to be ambiguous).


In formal logic and in mathematics we often give novel definitions of words. Some of those words already have different definitions which may be similar or have nothing to do with the new definition.
If a definition is given in a deductive proof, it is to be taken as if it were the very first time it has ever been defined. Often it literally is the very first time that the definition is given even if the spelling is the same as an already extant word. Within the scope of the argument, that definition rules and is independent of any definitions you may find in a dictionary.

The following is also perfectly fine if is shows up in a deductive argument:

Definition: A God is a is a prime number larger than 7.

or

Definition: A hat is said to be god-like if it is grey and has mass greater than a gram.

Useless perhaps, but logically OK.

One gives the definitions. Then once it is clear that the definitions are not ambiguous, the argument proceeds and any invalidity in the argument has to be in how the definitions and axioms are used, not in the definition itself.

Now look again at Godel's definition of "god-like". Does it sound silly or apropos? It doesn't matter as long as it is clear. What he does with the definition is another matter entirely.

Now what about claims? Well suppose that we take the definition of God-frump given above.
Now "god-frump" can feature in a claim. For example.

Claim: A god-frump is always larger than a pea.

Perhaps the claim is actually true. Once the definition has been established certain things may be true claims and others false within the context of the deductive argument.

Marg and JAK, perhaps you think all words just have God given definitions and when someone states the definition that are claiming that this is the true definition. That would be quite odd.