The Roles of Logic and Science in Questions of Theology

[_Tarski]

.............
.

Well, Marg, lets put it this way. If what you say about no new information being gain by deductive means is true then in all of the history of mathematics, no new information has come to humanity.

If so, then nothing is learned by

1. the proof of the Atiyah Singer Index Theorem,
2.the discovery of non-euclidean geometry,
3. the proof of Fermat's Theorem
4. the singualrity theorem's of Penrose and Hawking,
5. the existence on uniqueness theorems of partial differential equations,
6. even perhaps Godel's incompleteness theorem itself.
and much much more.

Indeed, all of pure mathematics would be a joke.

Also, we should not care in the least whether the Riemann hypothesis is true or not.

We would be fools for saying that nowhere differentiable functions exist!
We should not think that any new knowledge could be gained by thinking about prime numbers.

Silliness no?


Oh but you may say (and Dr. Hill may say) "yes but the information is already contained in the analytic universe of axioms and definitions. The truth of a theorem is implicit in the mathematical structure!!""

I would respond as follows:
So? The information about the physical world is also already contained in the physical universe. In either case we mortals must do work to discover the truths, be they analytic or not.

(Someone needs to read some critique of pure Kant LOL-----look here http://plato.stanford.edu/entries/analytic-synthetic/#3)
(That was an inside joke for gad)

Here is what we can say:

No new knowledge about the physical world can be gained by purely analytic means. Physical experiment is necessary.

This is a much more reasonable statement. Similar statements about other more soft sciences such as sociology should also be true.

I should love to talk to Dr. Hill about this myself. Where does he teach?

Edit to add, after talking to Gad I now feel that Prof. Hill would agree with me after all on most points. Marg is reading him naïvely and he may have not been very clear at some points which she read.

[_Tarski]

More...

I implied above that Marg may be misrepresenting or at least misunderstanding Dr. Hill. Gad has pointed out some evidence to this effect.
Indeed, a close reading leave one with the impression that Dr. Hill himself is not attacking analytic truths as trivial.
Thank goodness. It's only Marg.

Gad, go ahead and make a comment on this.

Note that right from the start, we have nearly all agreed that Godel's theorem stands little chance of proving the existence of any God in the sense theists seem to want?
Why? because for one thing, most folks should like this God to be connected to the physical universe. Here we certainly do need more than deductive argument!
We all agree with that. It is amazing how often I feel like someone is foisting on me an opinion about the proof that I do not hold. I am not convinced by the proof and am very very sure that it provides no proof of God.

[_Gadianton]

According to Prof. Hall and I agree with him, all arguments which are analytic a priori are question begging and presume the subject being argued for

Marg,

I'm not going to call you insane because Prof. Hall didn't write his little piece very clearly. I had to go back and read it three or four times (how many times did JAK read Godel's work before jumping to plagiarizing sources to defame it?) and he is NOT saying what you think he's saying. He is not saying all analytic a priori arguments are circular. Tarski's response to that simplistic suggestion is spot on with, especially, pure mathematics. I actually agree with Prof. Hall. The part in bold, you are misreading from prof. Hall.

[_Tarski]

According to Prof. Hall and I agree with him, all arguments which are analytic a priori are question begging and presume the subject being argued for

Marg,

I'm not going to call you insane because Prof. Hall didn't write his little piece very clearly. I had to go back and read it three or four times (how many times did JAK read Godel's work before jumping to plagiarizing sources to defame it?) and he is NOT saying what you think he's saying. He is not saying all analytic a priori arguments are circular. Tarski's response to that simplistic suggestion is spot on with, especially, pure mathematics. I actually agree with Prof. Hall. The part in bold, you are misreading from prof. Hall.

Thanks Gad.

I will make one more vague concession to Marg. The feeling she has that the reason that Godel's proof of God fails has to do with a missing connection between the purely analytic world of deduction with the everyday world where God is supposed to reign, is a correct intuition.

However, this does not mean that Godel simply assumes that God exists in his first and stipulative definition. It also does little to analyze the structure of the argument at the formal level (some even suggest that it is valid but not sound---I'm not even sure about that much).

Finally, any argument or claim that seems to trivialize deductive thought and mathematical "reality" will get my goat.
If I think you are dismissing mathematics as such, you can bet I will get riled up a bit.

[_Gadianton]

According to Prof. Hall and I agree with him, all arguments which are analytic a priori are question begging and presume the subject being argued for

Marg,

I'm not going to call you insane because Prof. Hall didn't write his little piece very clearly. I had to go back and read it three or four times (how many times did JAK read Godel's work before jumping to plagiarizing sources to defame it?) and he is NOT saying what you think he's saying. He is not saying all analytic a priori arguments are circular. Tarski's response to that simplistic suggestion is spot on with, especially, pure mathematics. I actually agree with Prof. Hall. The part in bold, you are misreading from prof. Hall.

Thanks Gad.

I will make one more vague concession to Marg. The feeling she has that the reason that Godel's proof of God fails has to do with a missing connection between the purely analytic world of deduction with the everyday world where God is supposed to reign, is a correct intuition.

However, this does not mean that Godel simply assumes that God exists in his first and stipulative definition. It also does little to analyze the structure of the argument at the formal level (some even suggest that it is valid but not sound---I'm not even sure about that much).

Finally, any argument or claim that seems to trivialize deductive thought and mathematical "reality" will get my goat.
If I think you are dismissing mathematics as such, you can bet I will get riled up a bit.

I think prof hall's point reduces to the "existence is not a predicate" argument and that if the ontological argument makes this mistake, as some (Kant) have said, then that's where he goes wrong. Godel of course, was aware of this potential problem, whether he overcame it is another story. Hall, by the way, uses the qualification "If"...

It was Gödel who tackled the problem of possible existence head on. Before we consider his proof, it should be pointed out that much of Gödel's argument is inspired by the writings of Leibniz on the subject. Leibniz argued that the weakness of Anselm's argument is in the statement of God's possible existence. Gödel's argument follows Leibniz in bolstering this step with additional argumentation. Reader's who are familiar with Leibniz's ontological argument may find echoes of Leibniz's method of affirmative simples in Gödel's work.

http://www.stats.uwaterloo.ca/~cgsmall/ontology1.html

Christopher Small, who is cited by SEP has a couple very good resources, I'm going to wager the best on the net, for understanding Godel's argument. He has another one more technical than this website. As for my gut feelings, well, i feel like i've encountered this kind of arguing in other places for other things. The SEP cites Bertrand Russell as saying that most of us aren't convinced by the ontological argument, but it's hard to say what exactly is wrong with it. It's hard to get more of a die-hard logician and atheist than Russell. And I wonder, if maybe the general thrust of the argument works and the problem lies with the comittment to realism. In other words, if a strong version of realism is true (i know still a lot of ambiguity there) then something like this follows as a fortuitous consequence. If I can ever get my thoughts together on that i'll do a post on it. but it's something i've thought about a long time (unfortunately) and not in context of the ontological argument.

[_marg]

Ok Tarksi and Gad,

I know you have written some replies, which again I'm going to have to say that atm, I don't have the time to address and at this point I haven't read.

I have transcribed one particular lesson of Prof hall's lecture..it's about 30 minutes broken up into 6 minute sections. Due to copywriting I won't keep it up, perhaps only for a week.
I simply want you to be aware of what he says. If you disagree with anything that's fine point it out. If you think I or JAK are in disagreement with him ..then point that out if you wish. I will not put the whole lecture up at once or today. I'll put up part 1 & 2 for now, essentially the intro. I'm sure it will be a quick read and nothing you probably don't all ready know but at least you know where he's coming from and consequently where I'm coming from and why I have respect for the reasoning and facts he offers. And I have to say that ultimately in my opinion, JAK has the same perspective.

Why Ontological Argument is said to fail.



Section 1 & 2

Point made in last lecture, validity matter of form. That if the form of an argument is right then its valid. I can’t move forward however without pointing out a corollary that goes with that. In order to be successful an argument needs to be more than valid. Needs to be not only valid, it needs also to be sound. And in order for an argument to be sound, it must not only have a valid form but the premises of the argument themselves must be true. From true premises if you argue validly you’ll get nothing but true results. But from suspect premises or corrupt premises you may argue perfectly validly but still not get anywhere. Logicians make this point this way. They say validity is truth preserving. Validity, good form will keep whatever truths you start out with but it will not turn falsehoods into truths. Bear in mind then as we begin to look at Anselm's argument and at Descartes version of that argument, that we might not only question whether or not it’s form is valid, we might also question whether or not its assumptions are true. And if indeed we came to the conclusion it’s assumptions were not true we might then come to the conclusion that the argument as whole is unsuccessful even though its form is okay.

Anselm has articulated an argument for us, about the fool having said in his heart there is no God and of necessity then the fool is saying that which exists does not exist. That even to deny the existence of God affirms the existence of God because of the very nature of god-hood itself..i.e. to be that which no greater can be conceived.

Well the sun had hardly set on Anselm's publication of this argument before a local monk named Gaunlilo said basically wait a minute this isn’t going to work. Imagine ..start with a different fool this time , not about matters theological but about matter tropical. A fool about vacations and garden spots and so on. And our fool says this time Gaunlilo is going to say there is no perfect tropical island. Every single one is flawed in some way. But now wait. His statement is intelligible, therefore the descriptive phrase tropical island that must be intelligible. In order for the phrase perfect tropical island to be intelligible it must refer. Now either when he says there is no perfect tropical island he is referring to the perfect tropical island or he’s referring to something else, who knows what. Let’s presume for the moment he is referring to the perfect tropical island. Consider any 2 islands broadly speaking the same. The are alike in every conceivable respect except for the misfortunate fact that this one’s real, this one’s a figment of my imagination. Which one is more perfect? Which one is better? Well the one that exists. A perfect tropical island has to exist. A tropical island that didn’t exist wouldn’t be perfect. So the fool who says there is no perfect tropical island is either referring to the perfect tropical island which does in fact exist in which case he’s contradicting himself in saying that which exists does not exist or he’s just misreferring off there, skating around in kookoo land somewhere. Goodness knows what he’s talking about. He’s either engaging in nonsense or he’s contradicting himself. There you go. So the perfect tropical island has to exist. And then Gauniloo does a slow one- two- three and says but that’s absurd. We can’t just think islands into existence. Whether there is or there is not a perfect tropical island out there is an issue that can be settled only by going and looking and seeing.

(Prof Hall goes into another example dealing with money which I’ll skip..other than his point you can’t think perfect money into existence if you could, what a wonderful way to think your way to wealth)

It’s not going to work. Now the puzzling thing is to figure out why it doesn’t work. Because for all of the overtones of word magic that sound like they are going on here, there’s a part of us that says well what went wrong. Because clearly a triangle defined this way has to have these properties. Clearly a God defined this way would have to exist. A perfect $ 1,000 bill described in my billfold if there were such a thing, there would have to be one. An imaginary one is imperfect in innumerable ways. Something is wrong. Well the history of philosophy since Anselm and Gaunilo’s time has been in part a history of people lining up to say “let me tell you what’s wrong with the ontological argument” There have been a whole bunch of different people and they have argued a whole bunch of different ways as to what’s wrong with it. I want to mention just a few of those and a couple of them I’ll gloss out with some length but others I will move over fairly rapidly and then I want to see if there is anything we can salvage out of the ontological argument that does work. And I’ll tell you in advance I think there is something very important that we can salvage out of the ontological argument. Someplace where it does work.

[_Tarski]

Ok Tarksi and Gad,

I know you have written some replies, which again I'm going to have to say that atm, I don't have the time to address and at this point I haven't read.

I have transcribed one particular lesson of Prof hall's lecture..it's about 30 minutes broken up into 6 minute sections. Due to copywriting I won't keep it up, perhaps only for a week.
I simply want you to be aware of what he says. If you disagree with anything that's fine point it out. If you think I or JAK are in disagreement with him ..then point that out if you wish. I will not put the whole lecture up at once or today. I'll put up part 1 & 2 for now, essentially the intro. I'm sure it will be a quick read and nothing you probably don't all ready know but at least you know where he's coming from and consequently where I'm coming from and why I have respect for the reasoning and facts he offers. And I have to say that ultimately in my opinion, JAK has the same perspective.

Why Ontological Argument is said to fail.



Section 1 & 2

Point made in last lecture, validity matter of form. That if the form of an argument is right then its valid. I can’t move forward however without pointing out a corollary that goes with that. In order to be successful an argument needs to be more than valid. Needs to be not only valid, it needs also to be sound. And in order for an argument to be sound, it must not only have a valid form but the premises of the argument themselves must be true. From true premises if you argue validly you’ll get nothing but true results. But from suspect premises or corrupt premises you may argue perfectly validly but still not get anywhere. Logicians make this point this way. They say validity is truth preserving. Validity, good form will keep whatever truths you start out with but it will not turn falsehoods into truths. Bear in mind then as we begin to look at Anselm's argument and at Descartes version of that argument, that we might not only question whether or not it’s form is valid, we might also question whether or not its assumptions are true. And if indeed we came to the conclusion it’s assumptions were not true we might then come to the conclusion that the argument as whole is unsuccessful even though its form is okay.

Anselm has articulated an argument for us, about the fool having said in his heart there is no God and of necessity then the fool is saying that which exists does not exist. That even to deny the existence of God affirms the existence of God because of the very nature of god-hood itself..I.e. to be that which no greater can be conceived.

Well the sun had hardly set on Anselm's publication of this argument before a local monk named Gaunlilo said basically wait a minute this isn’t going to work. Imagine ..start with a different fool this time , not about matters theological but about matter tropical. A fool about vacations and garden spots and so on. And our fool says this time Gaunlilo is going to say there is no perfect tropical island. Every single one is flawed in some way. But now wait. His statement is intelligible, therefore the descriptive phrase tropical island that must be intelligible. In order for the phrase perfect tropical island to be intelligible it must refer. Now either when he says there is no perfect tropical island he is referring to the perfect tropical island or he’s referring to something else, who knows what. Let’s presume for the moment he is referring to the perfect tropical island. Consider any 2 islands broadly speaking the same. The are alike in every conceivable respect except for the misfortunate fact that this one’s real, this one’s a figment of my imagination. Which one is more perfect? Which one is better? Well the one that exists. A perfect tropical island has to exist. A tropical island that didn’t exist wouldn’t be perfect. So the fool who says there is no perfect tropical island is either referring to the perfect tropical island which does in fact exist in which case he’s contradicting himself in saying that which exists does not exist or he’s just misreferring off there, skating around in kookoo land somewhere. Goodness knows what he’s talking about. He’s either engaging in nonsense or he’s contradicting himself. There you go. So the perfect tropical island has to exist. And then Gauniloo does a slow one- two- three and says but that’s absurd. We can’t just think islands into existence. Whether there is or there is not a perfect tropical island out there is an issue that can be settled only by going and looking and seeing.

(Prof Hall goes into another example dealing with money which I’ll skip..other than his point you can’t think perfect money into existence if you could, what a wonderful way to think your way to wealth)

It’s not going to work. Now the puzzling thing is to figure out why it doesn’t work. Because for all of the overtones of word magic that sound like they are going on here, there’s a part of us that says well what went wrong. Because clearly a triangle defined this way has to have these properties. Clearly a God defined this way would have to exist. A perfect $ 1,000 bill described in my billfold if there were such a thing, there would have to be one. An imaginary one is imperfect in innumerable ways. Something is wrong. Well the history of philosophy since Anselm and Gaunilo’s time has been in part a history of people lining up to say “let me tell you what’s wrong with the ontological argument” There have been a whole bunch of different people and they have argued a whole bunch of different ways as to what’s wrong with it. I want to mention just a few of those and a couple of them I’ll gloss out with some length but others I will move over fairly rapidly and then I want to see if there is anything we can salvage out of the ontological argument that does work. And I’ll tell you in advance I think there is something very important that we can salvage out of the ontological argument. Someplace where it does work.

Marg,

This portion of the lecture seems quite reasonable and, I might add, familiar--basic stuff here. He is warming up to exactly the kinds of issues that are familiar to anyone who has studied up on the various ontological arguments before. There is much to discuss here but...but...so far as I can see,
nothing said in that couple of paragraphs contradicts any point either I or Gad has made. It also doesn't lend support to any of the mistakes that JAK has made.
In particular, there is nothing that lends support to the ridiculous idea that the stipulative Definition 1 either assumes God or inevitably inserts specious connotations inhering in the word "God" as it is commonly used. Godel was no fool. In any case, if such connotations had inappropriately played a role it would have been at a later point in the argument, not in a purely stipulative definition where the word "god-like" is coined. But Godel made no such mistake. Rather, he is vulnerable on several other points-much more subtle points.

Having had experience with previous ontological arguments, the first thing we should ask is whether Godel has circumvented the usual problem of whether existence is a proper predicate or not. Secondly, one looks for terms that are not given clear stipulative definitions. Well, there is such a word but it isn't god-like. Namely, it is the word "positive". Godel appeals to common moral intuitions for that one. Perhaps he feels that if one can get from assuming an appropriate notion of positivity to the existence of God (or in his technical sense a god-like being) then that would be a sort of progress. Indeed, the leap to positivity looks shorter at first so it might have been progress. Again Godel is no fool.
Perhaps, but I expect that the first issue about existence being a property which we may predicate of an entity is still lurking in a more subtle form. Secondly, I think that most thinkers in this age that are not committed to some kind of Platonism will have trouble with the idea that there is a notion of positive that is up to the task. That is, there is likely no notion of "positive" that is precise, objective, and renders the axioms in which it features, true.

But in the end few are convinced by this more sophisticated version of what is probably Liebnitz' argument (I think Godel says as much).
As always, such deductive arguments for the existence of what is hoped to be a real active force in the everyday world attempt to perform a sort of implausible miracle.

[_marg]

Tarski,

My time has been spent transcribing. You had asked how the prof deal with a priori analytic math argument which you say are fruitful. I'll put up both (3 & 4) and (5 & 6) It is (5 & 6) which addresses this in particular. I quickly glanced at the latest reply of your in which you mentioned that the def'm doesn't assume anything. I disagree that prof Hall doesn't address this. It's subtle but there. I have discussed previously the concept. That things have to refer to something. Non existent thing do not have properties. Anyhow, just as you feel you can not explain any further on the point of def'n presuming a thing. I feel I can elaborate much more either on that. Prof Hall does address it though, but not explicitly. And math and symbolic logic may deal with properties on non existent things, but conceptually non existent thing have no properties.

Take a look at section 5 & 6...with regards to your question posed to me in a few posts back.

Section 3 - 4

Prof Hall goes into a brief bio of Kant, as well as what the time period was like at time 1776. Critique of Pure Reason published.

Kant says look, an argument like this would work for any property. A perfect $1,000 bill will have to have certain properties. Engraving, patterns, numbers in certain sequences, made on right kind of paper. You can infer any number of things about the properties that something would have to have in order to be a perfect something of its kind. The one thing Kant says you can not argue is that it has to exist because existence Kant says simply isn’t a property. That is an odd statement to make, but I think he’s right. Existend if you like is the arena called for in order for any properties to occur. Things that don’t exist don’t have properties. Imaginary things have imaginary properties. Real things have real properties. In order for something to have real properties it has to really exist. But the existence itself, is not just a property of the thing it is the affirmation that there is such a thing such that it might have properties of one sort or another. As far as Kant was concerned that’s all that needed to be said. It would work for properties, existence isn’t a property, let’s move on.

Now let me simply make a historical connection and a footnote here and I will tell you where to go if you want to pursue this. But I’m not going to pursue it because if I did it would take us 3 or 4 weeks. Simply to unravel it and we’d have to begin with my teaching you modal logic, but that’s not enough I’d have to go back to Richmond to learn it myself to teach it. Modal logic is not my cup of tea. I know a little bit about it. I have a very good colleague in my dept. who is very good at it.

Model logic is not the logic of “if this is the case then that follows, if such and so is the case then that is entailed. Model logic is the logic of possibility and necessity. Model logic has to do with if certain things are possible then certain things necessarily are not impossible, or if something is necessarily not impossible then it must necessarily be the case etc.It’s the logic of necessity.

It’s very very tricky and it doesn’t behave exactly the way ordinary everyday logic behaves. In my own judgement it is not nearly as well mapped as the ordinary everyday logic of true and false descriptions either. But that is simply the observation of a non expert.

Some modal logicians especially A. Plantinga insist that Gaunulo’s interpretation of Anselm’s argument and Kant’s interpretation of Anselms’ & Descartes argument simply misconstrues what is going on. That Anselm is not talking about mere existence Anselm is talking about necessary existence. Anselm is talking about a modality of existence and Plantinga says when we start talking about modalities of existence we are talking about properties and what Kant has to say on the topic is irrelevant.

Plantiinga is firmly convinced that if you take out all references to existence on the ontological argument and replace with necessary existence instead ..that the ontological argument will then go. Valid absolutely secure as far as its logic is concerned.

For balance and I’ll give you the name of another who has done a lot with modal logistics, a man named Peter Geach…for the record where A. Plantinga is a devote commited Christian in the tradition of the American Reform Peter Geach is a devote and commited Christian in the RC tradition. The argument between these 2 is not an argument between believers and disbelievers at all. Geach’s response to Plantigna’s use of the modal logic is “this is absurd”. Anyone who really understands what is going on in modal logic realizes you just can’t make it do that.

This is beyond your present instructor. I am quite content for the modal logicians to work this through and the day will come that one or anotherof them, I’m quite sure will reach closure.

Clearly however, Anselm’s and Descartes arguments in the original form to the affect that the existence of God can be inferred from the very def’n of god, that runs into trouble on Kant’s claim. That existence simply is not a predicate.

Now I want to shift gears to a different line of reply to this kind of argument.

You’ll remember at the beginning of previous session I talked a little bit about the difference between a priori and a posteriori arguments. Arguments that work with reason alone and arguments that work with reason funded by experience. Keep that distinction apriori a posteriori in mind and let me add another distinction along with it, so that I can draw an inference. I want to distinguish between what I call analytic statements on the one hand and synthetic statements on the other. Calling on a very ancient and honorable philosophically tradition called the analytic synthetic distinction, which I grant you has been called into question by Quine and other in 20th century but we don’t need to go there. That’s not essential to our immediate task. The analytic synthetic distinction boils down to this. There are certain sentences, predicative sentences “All swans are either white or black, all humans are mortal etc. where in order to figure out whether or not the predication is being made, the predication of mortality to humans, the predication of white and black to swans, in order to figure that out involves let me call it , field work. The proposition puts together or synthesizes new information. The proposition says more than what is included in the very def’n of its subject term.
Being either white or black, being mortal is not part of the def’n of swan or part of the def’n of human being, necessarily. On the other hand suppose I said a bachelor is an unmarried male of marriageable age or all bachelors are unmarried. Well if the def’n of bachelor is unmarried male of marriable age, then the statement all bachelors are unmarried is analytic. That is you may discover the truth of the predication by simple analysis of the meaning of the subject.

------------------------------------------------------------------------------
Section 5 & 6
Analytic you can discover the truth by analysis alone
Synthetic you can discover the truth by field work because it is synthesizing a new truth an independent truth. A priori, a posteriori, known by reason alone, known only by availing yourselves of experience.

Consensus in 20th century is that any statement that can be known a priori, independent of experience must of necessity be analytic. That is we must be able to discover its truth by analysis of the subject term. That any statement that is synthetic, any statement that delivers new information is of necessity a posteriori. It’s going to use reason sure, we’d be strapped to do anything without reason. It’s going to use reason, but it’s going to have to use reason working with the fuel, the raw material provided by experience.

Point: If the ontological argument is a priori and if truth of conclusion of ontological argument is discovered by analysis of what is contained in the subject term alone, which sound like exactly what Anselm was trying to say, let’s just unpack the very notion of God and we’ll see that into that very notion is existence. What we have here is a circular or question begging argument. It presumes the very thing it sets out to prove.

Now you may remember very early on in our discussion we talked about evidence in an argument needed to be relevant to what you are trying to prove,yes, but if the evidence in that argument was the very thing you were trying to prove, that was just a little bit too relevant for comfort.

And so the point we are making here in terms of a priori arguments being analytic is that a priori analytic arguments are sterile as far as finding out anything beyond what is presumed in the very assumptions of the argument itself.

Let me illustrate that very briefly by going to Descartes. You’ll remember and the beginning of this lesson that there are 2 ways to attack an argument.

You can attackit on grounds it is not valid, something’s wrong with form, you can attack it on grounds that it is not sound, one or another of the premises is not true.

Now with Descartes argument about triangles, Descartes begins with the notion that the axioms of Euclidean geometry are descriptively true of the world. And he then reasons his way to the conclusion that the sum of the interior angles of any real triangle in the real world has to add up to 180 degrees because we can deduce that from Euclid’s premises and Euclid’s premises are self evident. Euclids premises are knowable a priori. Quite honestly contemporary mathematicians don’t look at it that way, anymore. Indeed they don’t even call the first principles of Euclidian geometry axioms anymore , they wouldn’t dream to say they are self evidently true of the real world. They are simply definitional truths for this conventional system of geometry. And yes, every Euclidean triangle will have its interior angles add up to 180 degrees. Because that is an entailment of the very def’n of Euclidean triangles in Euclidean space using Euclidean axioms or postulates or assumptions but replace those Euclidean assumptions with alternate ones from other mathematicians like Riemann or Lobochevsky and you wind up with triangles in those conventional systems that add up to more than 180 degrees or less than 180 degrees. It all depends on where you start. It all depends on what’s packed into the assumptions. The arguments in a word are circular. Now it’s a marvelous and good and useful thing that Euclidean geometry happens to apply to the world. And it does very nicely. But is that a self evident truth? No The only way to find out whether the world is Euclidean or not is to go and look and see. And in fact as we have mentioned before certain parts of the world subject to intense gravitional forces and similar kinds of things behave in non euclidian ways and we would find consequently that in those parts of the world Euclids argument is not any good not because it is invalid but because it’s assumption are false. That the axioms out of which the Euclidian conclusions are milked are not true in the neighbourhood of a white dwarf star although true in my backyard.

What’s the point, The point is that an apriori analytic argument can tell you what is entailed by a set of assumptions but it can not tell you whether the assumptions themselves are true. So analytic apriori arguments, it is generally held these days are sterile as far as a way finding out what is in fact the case. They are fruitful as a way of finding out what is implied by certain assumptions that we make.

I want to end today with a quick sketch of another response to this argument one which salvages an important point from it.

You could argue that the bottom line of both Anselms and Descartes argument is that all gods exists..a universal affirmative. That might not sound too plausible on its face immediately but there is an equivalent form that says exactly the same thing by means of obversion. That is obviously true, self evidently true. “ Nothing that doesn’t exist is a god.” Nothing that doesn’t exist deserves worship. It is folly to waste worship on non existent things. But that’s all in the world the statement that all gods exists says. It’s genuinely a hypothetical on a modern understanding of how class logic works. To say all gods exist is to say for any x if it is a God, then it exists. And if it doesn’t exist then it’s not a god. On those grounds, on that interpretation Anselm and Descartes are right , the bottom line of their argument can validly be drawn. Each and everything that deserves to be worshipped exists. Now how many are there? That’s the catch. Is there even one. The argument doesn’ t tell us. The argument simply tells us if you are wasting your time worshipping non existent things, you are genuinely wasting your time. I like to think of the ontological argument then as an argument against idolatry rather than an argument for the existence of god.

[_Tarski]

Tarski,

My time has been spent transcribing. You had asked how the prof deal with a priori analytic math argument which you say are fruitful. I'll put up both (3 & 4) and (5 & 6) It is (5 & 6) which addresses this in particular. I quickly glanced at the latest reply of your in which you mentioned that the def'm doesn't assume anything. I disagree that prof Hall doesn't address this. It's subtle but there. I have discussed previously the concept. That things have to refer to something. Non existent thing do not have properties. Anyhow, just as you feel you can not explain any further on the point of def'n presuming a thing. I feel I can elaborate much more either on that. Prof Hall does address it though, but not explicitly. And math and symbolic logic may deal with properties on non existent things, but conceptually non existent thing have no properties.

Take a look at section 5 & 6...with regards to your question posed to me in a few posts back.

Section 3 - 4

Prof Hall goes into a brief bio of Kant, as well as what the time period was like at time 1776. Critique of Pure Reason published.

Kant says look,.....
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. The argument simply tells us if you are wasting your time worshipping non existent things, you are genuinely wasting your time. I like to think of the ontological argument then as an argument against idolatry rather than an argument for the existence of god.

Marg,
Once again this is all very very basic philosophy. I can't find much to argue with in there except for a couple points that might leave the wrong impression.

Some points:

1. The original ontological arguments did indeed fail because of the problems with treating existence as a property. It is also true that if one just asserts that a God by definition must exist and that somehow that means there does exist a real God then I agree that this may be described as a kind of begging of the question--that is assuming what was to be proved. However, it is not simply explicitly assumed as part of the definition of God. In any case, Godel's version is more subtle.
I agree that drawing conclusions by analyzing the very meaning of a single term such as bachelor does not give new information.
All basic familiar stuff here.

But....

2. When one takes the whole set of the Frankel-Zermelo axioms and searches for and tries to prove subtle theorems on a variety of topics such as differential geometry, real analysis, number theory etc. all of which are based on the F_Z axioms, then this is not a trivial matter of unpacking the meaning of a term and we see that we can truly discover deep things and add new information the the pool of human knowledge. That is what mathematics is.
Similarly for logic proper.

3. Godel was well aware of all of these basic things that we have discussed and that Prof. Hill has discussed. Godel tries an approach that is more sophisticated. Even if it ultimately fails notice that in definition 1 Godel does not assume that the very definition of God includes existence. He tires to show that necessary existence is a property that must adhere to anything satisfying his definition of "god-like".
Indeed, it is at least conceivable to me that the definition as stated might actually contain the seeds of destruction in the sense that we may be able to construct an argument that there cannot exist any God-like beings at all (I imagine such an argument to be flawed but just as convincing as godel's attempt to prove the opposite).

In any case, we don't get into the traditional trouble until we try to include existence into the list of properties with which we reason since existence as a property is problematic. Now Godel uses "necessary existence" instead of existence and I guess the idea is that under certain notions about the way modal logic works, this is supposed to make it all OK. I am skeptical to say the least!!

But this trouble happened later on when axiom 5 connected the notion of positive with necessary existence. We are led to ask--is he then saying that necessary existence is a positive property? He has made that an axiom so it really is assumed! Well if so, then we had better be darn sure that necessary existence can be a property at all! We had also better give good reasons why this should be taken as an axiom. Now we are in real trouble.

But notice that the trouble was not that Definition 1 simply assumes the existence God just because those three letters (g-o-d) are being used. Godel has cleverly separated the technical notion of god-like from the notion of necessary existence by use of the intervening notion of positivity. As for necessary existence, he assumes it is not only a property but a positive property!
Godel also never allows any connotations of the word "god" to create an invalid step in the formal logic. This is another thing JAK gets wrong in my opinion. Godel does not covertly import anything that he hasn't already admitted to be an assumption by virtue of it's inclusion in the list of axioms (axioms are indeed assumed--he knows this). Godel knows that if the axioms are not true or are somehow self contradictory then his argument is not sound.

[_Gadianton]

When it comes to non-trivial topics, every professor is going to have his own unique spin and prejudices, I agree with Tarski that Hall does a decent job. When I took history of phil at BYU, it was basically taught with no further explanation that Kant refuted Anselm and put into question that whole tradition with his "existence is not a predicate" observation. So Hall actually gives far more than I got in my history of phil class.

It’s very very tricky and it doesn’t behave exactly the way ordinary everyday logic behaves. In my own judgement it is not nearly as well mapped as the ordinary everyday logic of true and false descriptions either. But that is simply the observation of a non expert.

I think this is correct. Not that I'm an expert either, but modal logic always leaves more ambiguity. However, you have to consider the tradoffs. I mean, modal logic wasn't invented and developed just to try and sneak in questionable religious beliefs. The problem with formal logic in general, is that it's not easy at all to tackle real-life issues by breaking them down to propositional logic. Math is about the only really useful thing formal logic can provide deep insights on, once you get past the basic formal rules and fallacies that might apply in broad ways to real life. One very important problem with normal logic is the inability to deal with counterfactuals (C. Small deals with this too in the link I provided). Talking about how things could have been, but aren't, is an important part of everyday reasoning and it's nearly impossible to do it with normal logic. So it might not be as well-mapped, but what else are you going to do?

And so the point we are making here in terms of a priori arguments being analytic is that a priori analytic arguments are sterile as far as finding out anything beyond what is presumed in the very assumptions of the argument itself.

This is the most problematic statement I think he makes. Tarski has already responded appropriatly. Most of the battle might actually be finding out the right assumptions whether you're project is analytic or synthetic. Look at Bertrand Russel's principia mathematica (the 900 page book godel rendered obsolete in 15 pages). we already know what math is, we already know how to count, no one disagrees with the usefullness of basic arithmetic, so largely, trying to formalize arithmetic into logic calls for working backwards, assuming arithmatic is true and really trying to figure out the most basic assumptions one needs to make to get arithmatic and then figure out how to get from point A to B. So it's great to say that you can't find out anything beyond the assumptions with analytic arguments, but getting the assumptions right is a big discovery in the first place. then you can go on to the horribly hairy practical problems like proving fermat's last theorom as tarski mentioned. Hell, let's go further, Tarski can say far more on this than I can, but a huge sweep of pure mathematics that otherwise wouldn't have any practical relevance seems to get utilized towards subjects like string theory in theoretical physics. Even simple counter-intuitive concepts that are purely mathematical like the square root of an imaginary number end up having huge importance for standard engineering.

Tarski also noted that "synthetic" arguments could also be seen as "tautologies" if we want to say that about what we know analytically. If I understand logical positivists, well, mainly Rudolf Carnap right, then we can see how Tarski's point becomes true very literally. Carnap surely divided between analytic and synthetic. And I believe he ultimately saw the possibility of getting a finite number of observation axioms that could then be used to derive all the laws of the universe (not in the immediate future). If Carnap should ever be proved right, then we'd say all of physics is also a tautology. But just like Godel struggling to find rock-bottom axioms that work to his ends, Carnap's project would require some tremendous insights in order to figure out what the most basic observation sentences should be. In other words, most certainly, to formalize all of physical reality if such a thing is possible would be a project in reverse, the "assumptions" wouldn't reveal themselves until the bitter end.

why would anyone want to formalize anything, by the way, if to do so guaranteed the project as a logical fallacy?