Might be.

The analogue you want is the jump from there being no highest number to a number greater than any assignable quantity - to infinity, and beyond! You want to jump from something greater than anything to something greater than greatness...

And the suggestion is that there need be no such thing. But also, that g:=ix¬(∃y)M(y,x) does not give an definition in line with this second account.

No. Kids will ask wha the highest number is. Takes them a while to see that there isn't one. — Banno

Okay, so you're not actually objecting to step (2) of the proof?

Notice that the existence (as a thought) of such an individual is here just assumed. — Banno

Klima is explicit that step (2) is a supposition and that step (1) is a definition, so I'm not sure what you're attempting to disagree with.

What a mess. So god is not the thing greater than everything, but the thing greater than the thing greater than everything. — Banno

We can come back to this, but you seem to be missing the ampliation entirely. The key point of the paragraph you here quoted is the ampliation on "thought," so the fact that your assessment leaves out thought entirely is strong evidence of a misinterpretation. This common misinterpretation is precisely why Klima included that paragraph along with the buildup on ampliation.

Might be. — Banno

Well can you go back and fix your misrepresentations of Klima? If you are going to call his argument "ugly," at least give his argument instead of some weird symbols that do not occur in his paper.

Okay, so you're not actually objecting to step (2) of the proof? — Leontiskos

Well, not yet. One at a time.

I did fix the ugly: g:=ix¬(∃y)M(y,x). I asked you if it was acceptable, and did not yet get a reply.

I'm gonna Pontifications from 30,000 feet again. The generic flaw in ontological arguments is that if they are valid then they assume the conclusion somewhere in the argument. The task for the logician is to find out where.

They must do this because existence cannot result from a deduction. It can only be presumed, either in the argument or in the interpretation.

For the theist, the assumption is often trivial, even self-evident. But not for others.

So the argument will not be of much use in convincing non-theists. As here. But on the other hand, it also does not disprove that god exists, and it may be of use in showing god's nature to theists.

From were I sit it looks to be another example of trying to put the ineffable into words, and getting tongue-tied.

I did fix the ugly: g:=ix¬(∃y)M(y,x). I asked you if it was acceptable, and did not yet get a reply. — Banno

Your misrepresentation is still there: (1) g=dfix.~($y)(M(y)(x))
(As well as the other lines of the proof where similar problems occur)

That post of yours is the first place in the thread where Klima's formalization of Anselm's proof occurs, which is why I would like it to be accurate. It is a thread on Anselm's proof, after all.

The generic flaw in ontological arguments is that if they are valid then they assume the conclusion somewhere int he argument. — Banno

Are you just saying that ontological arguments beg the question? This is a common charge that Klima is aware of. But it must be demonstrated that someone has begged the question. It can't just be asserted.

So the argument will not be of much use in convincing non-theists. — Banno

I am amused that you claim to have read the paper.

But Banno, if you want to do analytical philosophy, this is a thread for it. That's why I made it - because all these folks think they want rigorous analytical philosophy. Well, this is it. It requires reading, patience, careful thought and interpretation.

Klima is explicit that step (2) is a supposition and that step (1) is a definition, so I'm not sure what you're attempting to disagree with. — Leontiskos

To be sure, it is not clear that the definition g:=ix¬(∃y)M(y,x) can be made coherently, any more than can "Let G be the number bigger than any other number".

Your misrepresentation is still there: (1) g=dfix.~($y)(M(y)(x)) (as well as the other lines of the proof where similar problems occur). — Leontiskos

You seem to be talking past me.

a single question, yes or no: is
g:=ix¬(∃y)M(y,x)
a good representation of line 1? Or do we need to use mathjax?

On my computer screen Klima's html version reads as follows:

(1) g=dfix.~(∃y)(M(y)(x))

Or if we look at the official book chapter, linked in the OP:

g =_df ix.~(∃y)(M(y)(x))

(where in both cases i = the descriptor)

g:=ix¬(∃y)M(y,x) — Banno

That is what Klima writes, is it not?

I am wondering why <this post> of yours is misrepresenting Klima? Why does it contain symbols and steps that do not appear in Klima's paper? Don't you think we should represent his argument accurately?

Well, I'd like to talk about the argument rather then the formatting. Can we move on?

Maybe you could reply to what I said about (1).

I was wrong about the paper. Sorry for being so stubborn and impatient, and for unnecessarily derailing the thread. — Banno

Cool, thanks Banno. I guess we're on the same page that quoting someone accurately or inaccurately makes no difference. Syntactical "formatting" is just a sideshow. Obviously you won't mind that I changed some of the "formatting" of your post. :up:

Maybe you could reply to what I said about (1). — Banno

Perhaps I can help.

Following the analogue, the first transfinite number is

ω:=min{x∣x is an ordinal and ∀n∈N,n<x}

You need something like this, but with g for ω. But notice that ω is an ordinal, and is define as greater than any natural number. This avoids the contradiction that would result if ω were defined as greater than any other ordinal, or as a natural number greater than any natural number.

So you can't just write g:=ix¬(∃y)M(y,x) without a problem, becasue it may be that there is no greatest individual. You need god to be something else, not an individual or not a part of the domain or something, to avoid shooting yourself in the foot.

But if you manage that, you have the analogue of the transfinite numbers - no sooner have you defined g as the greatest, and then you can bring to mind something greater than g, and the problem repeats itself.

So even as there is good reason to think that it is not possible to make sense of "the largest number", it is difficult to see how to make sense of "the greatest individual".

Consider an analogous argument defining the highest number as that number which is higher than any other number. The definition is fine, except that there is no such highest number.

Gaunilo of Marmoutier took this approach by positing an "island greater than which none can be conceived," in order to try to show that Anselm's argument can be used to demonstrate the existence of all sorts of things. But Guanilo's argument is generally seen to fail even by critics of St. Anselm. As with number, there seems to be no maximum for how great and island can be. Just in terms of size, it can always get bigger. But there do seem to intrinsic limits for those properties Anselm associates with God. Perfect knowledge is knowing everything; one does not make their knowledge more perfect by knowing more than all there is to know. Moral perfection is not a quantity, etc. There are intrinsic maximal perfections inherit in these concepts.

This is not to say people haven't brought up challenges to these properties (e.g. that it is contradictory for a being to be both omniscient or omnipotent, etc.), they have. But "more omnipotent than omnipotent," doesn't make sense.

The generic flaw in ontological arguments is that if they are valid then they assume the conclusion somewhere in the argument. The task for the logician is to find out where.

Well, if the issue is that the conclusion must be contained in the premises, that's a problem for all deductive arguments. Hintikka's ol' scandal of deduction. What is being assumed here is the existence of a being of thought. No need to look too hard. The argument is meant to demonstrate that such a being must exist simpliciter if it exists as ens rationis.

But I think real problem for ontological arguments is that they are unconvincing. I don't think anyone has been converted by an ontological argument, or that many people of faith feel their faith significantly bolstered by such arguments. And indeed, there are also atheist logicians who have allowed that modifications of Anselm, Gödel's proof, etc. seem to work and have premises that seem innocuous enough, but are nonetheless not even remotely convinced.

Gaunilo of Marmoutier took this approach — Count Timothy von Icarus

Close, perhaps. This objection is specific to the argument at hand. The intrinsic limit needed is missing from g:=ix¬(∃y)M(y,x), which is "God is defined as the thought object x such that no y can be thought to be greater than x", and the objection is not that anything might fit this, as that nothing might fit this. The question is, is the idea of such an object coherent? It's analogous to defining a number x such that no number y can be greater than it. There an be no such number.

It doesn't help to say that there may be intrinsic limits to god's greatness, becasue of the way (1) is set out.

if the issue is that the conclusion must be contained in the premises, that's a problem for all deductive arguments. — Count Timothy von Icarus

Quite so. It would be a surprise if an argument could demonstrate the existence of something ex nihilo, as it were. And yes, what is assumed is a being of thought. But what supposedly pops out of the algorithm is something else. The move from ens rationis to ens reale only works if we already accept that "existing in reality" is a necessary property of the greatest conceivable being.

We can see this more clearly in free logic, taking the inner domain as ens reale and the outer domain as ens rationis. Stealing from the SEP article, the theist would need an argument of the form:
$Ti, \forall x(Tx\rightarrow E!x) \vdash E!i$
...where Ti might be the assumption that god is the greatest possible thought object, and E!i that god exists in reality. But such arguments are invalid.

I think it's worth taking a moment to say something here:

I'm gonna Pontifications from 30,000 feet again. The generic flaw in ontological arguments is that if they are valid then they assume the conclusion somewhere in the argument. The task for the logician is to find out where.

They must do this because existence cannot result from a deduction. It can only be presumed, either in the argument or in the interpretation. — Banno

To be sure, it is not clear that the definition g:=ix¬(∃y)M(y,x) can be made coherently... — Banno

The trouble with the 30,000 foot view is that everyone is right in their own book at 30,000 feet, as it's just a matter of so-called (see my bio quote from Hadot on this point). Thus the atheist sees an argument for God's existence and he knows it must be wrong. He sees the conclusion and he infers things about the premises. All he is doing is begging the question (even though it is sometimes practical to beg the question).

The same sort of thing is happening here:

...it is not clear that [it] can be made coherently... — Banno

Okay, but that sounds like a hunch, much like, "It doesn't smell quite right to me." "It's not clear it can be made coherently." At this point the engagement with the text is minimal (and I will get to the elaboration). "Not clear it can be made coherently," is not a substantive objection to a premise.

The generic flaw in ontological arguments is that if they are valid then they assume the conclusion somewhere in the argument. — Banno

This is also very similar to the question-begging atheist:

1. All valid ontological arguments beg the question
2. This is a valid ontological argument
3. Therefore, this begs the question

But how does the inductive (1) get to be so strong? And even beyond that, what is "an ontological argument"? As the very first sentence of Klima's introduction implies, that whole label is anachronistic. Certainly Anselm would wonder how one can know that a whole bundle of loosely-affiliated arguments are known to be faulty a priori.

Similarly, the argument, "Some beings of reason are not beings (simpliciter), therefore this being of reason is not a being (simpliciter)," doesn't cut. Klima acknowledges that not all beings of reason are beings. Why think that Klima's (g) is relevantly similar to the idea of a largest number in the first place?

So there is not a lot of rigor in these blanket approaches, and this is why I want to get away from the 30,000 foot view. Luckily, Klima helps us get down to concrete points.

Yeah, all that, perhaps, but I also gave a very specific critique of (1) in the argument.

At least Tim tried to address it.

And again you misrepresent what I said. I did not claim all ontological argument beg the question. IF the argument is valid, and it shows that something exists, then that must be assumed in the argument somewhere. That's how logic works. The problem isn’t just that the argument assumes its conclusion, since as Tim pointed out all valid deductive arguments do that. It's that it does so in a way that makes the supposed proof of existence trivial. The argument becomes "God exists therefore god exists".

So you can't just write g:=ix¬(∃y)M(y,x) without a problem, becasue it may be that there is no greatest individual. — Banno

It sounds like you're saying that we can't have a being of reason if it isn't a being. Or in other words: we can't think of what doesn't exist. "X doesn't exist, therefore we cannot think of it." This is what section 1 addresses.

But of course Klima has no premise which says that there is a greatest individual.

But if you manage that, you have the analogue of the transfinite numbers - no sooner have you defined g as the greatest, and then you can bring to mind something greater than g, and the problem repeats itself.

So even as there is good reason to think that it is not possible to make sense of "the largest number", it is difficult to see how to make sense of "the greatest individual". — Banno

So you are disputing (3), then? Because that is precisely the premise that bears on how the "greater than" predicate cashes out.

-

IF the argument is valid, and it shows that something exists, then that must be assumed in the argument somewhere. That's how logic works. The problem isn’t just that the argument assumes its conclusion, since as Tim pointed out all valid deductive arguments do that. — Banno

Then I will quote this for the second time today:

(Some of my own philosophical arguments have been accused of something very like ‘begging the question’ – I concede the phrase was not used – simply because they were formally valid arguments for a conclusion the accusers thought was false. Their reasoning seems to have been something like this: if the conclusion of an argument can be formally deduced from its premises, then that conclusion is, as one might put it, logically contained in the premises – and thus one who affirms those premises is assuming that the conclusion is true. As R. M. Chisholm once remarked when confronted with a similar criticism, ‘I stand accused of the fallacy of affirming the antecedent.’) — Peter van Inwagen, Begging the Question

(The quote is from a book on ontological arguments.)

The argument becomes "God exists therefore god exists". — Banno

Do you say that such a thing is begging the question, or not?

The move from ens rationis to ens reale only works if we already accept that "existing in reality" is a necessary property of the greatest conceivable being. — Banno

But the proof at hand does not assume that, and it nevertheless succeeds in drawing the conclusion. It does not assume that "existing in reality" is a necessary property of the greatest conceivable being. There is certainly no premise to that effect. So you have to deal with the proof. With the paper. If the paper is right then the theory you have on paper turns out to be wrong.

(I think a lot of this comes back to the way you simply overlook Klima's "ampliation".)

But I think real problem for ontological arguments is that they are unconvincing. I don't think anyone has been converted by an ontological argument, or that many people of faith feel their faith significantly bolstered by such arguments. — Count Timothy von Icarus

I actually know philosophers who find the argument convincing, but they lack prejudice in an abnormal way. Someone without prejudice who encounters an argument that they cannot find fault with will accept the conclusion, or at least be greatly troubled by it. But that's rare.

I haven't generally found Anselm's argument convincing, but there are presentations which are undeniably beguiling.

It sounds like you're saying that we can't have a being of reason if it isn't a being. Or in other words: we can't think of what doesn't exist. "X doesn't exist, therefore we cannot think of it." — Leontiskos

No.

And so far I am only looking at premise (1), no further. We can go on when this bit has been understood.

But the proof at hand does not assume that — Leontiskos

Yeah, it does, and that can be shown. But you wanted small steps.

you simply overlook Klima's "ampliation" — Leontiskos

Not at all. I address it quite specifically:

Trouble is, that is not what g:=ix¬(∃y)M(y,x) says. God is still a thought object, albeit the greatest thought object. — Banno

One of the points I made is that Klima does not make use of the "ampliation" in (1), and he ought. The point was repeated and expanded, here:

Following the analogue, the first transfinite number is

ω:=min{x∣x is an ordinal and ∀n∈N,n<x}

You need something like this, but with g for ω. But notice that ω is an ordinal, and is define as greater than any natural number. This avoids the contradiction that would result if ω were defined as greater than any other ordinal, or as a natural number greater than any natural number.

So you can't just write g:=ix¬(∃y)M(y,x) without a problem, becasue it may be that there is no greatest individual. You need god to be something else, not an individual or not a part of the domain or something, to avoid shooting yourself in the foot.

But if you manage that, you have the analogue of the transfinite numbers - no sooner have you defined g as the greatest, and then you can bring to mind something greater than g, and the problem repeats itself.

So even as there is good reason to think that it is not possible to make sense of "the largest number", it is difficult to see how to make sense of "the greatest individual". — Banno

Gaunilo of Marmoutier took this approach by positing an "island greater than which none can be conceived," in order to try to show that Anselm's argument can be used to demonstrate the existence of all sorts of things. — Count Timothy von Icarus

Yes, and I actually think Klima's interpretation vindicates Anselm's reply to Gaunilo. I added a link to Anselm's Proslogion <here>, and the header will get you to the appended parts with Gaunilo.

But in my opinion Banno is doing something a fair bit different. He is saying something like, "There is no greatest-number-concept; and a greatest-thought-concept is a lot like a greatest-number-concept; therefore there probably is no greatest-thought-concept; and therefore Klima/Anselm is not allowed to define God after the manner of a greatest-thought-concept." Or similarly, "A child might think there is a greatest number, but there is not a greatest number; therefore the child never had the concept of a greatest number in the first place." Banno is engaged in a form of concept denial, which he would need to flesh out.

(And it is worth noting that Banno's objection is much closer to Russell and Quine than Gaunilo's is.)

Banno is engaged in a form of concept denial, which he would need to flesh out. — Leontiskos

So you want me to flesh out your concept of god for you.

I don't think so.

So you want me to flesh out your concept of god for you. — Banno

Your objection relies on the idea that some concepts cannot exist even as beings of reason (entia rationis). If you can't flesh out that idea then the objection goes nowhere, given that the whole thrust of section 1 is that for Anselm a being of reason need not be a being (simpliciter).

Your objection relies on the idea that some concepts cannot exist even as beings of reason — Leontiskos

Yep. Concepts that contradict themselves. Like "The largest number". That's what I explained previously. If your argument is to hold, you have to show that "the greatest thingie" or whatever is not of this sort.

And so far I am only looking at premise (1), no further. We can go on when this bit has been understood. — Banno

The problem with objecting to the two-place predicate M()() in premise (1) without looking at premise (3) is that premise (3) is the crucial place where that predicate is actually doing work (and it is therefore the locus for understanding the predicate). You are effectively objecting to a possible way that M()() might be used, and the response is, "The place where Klima uses it is premise (3), and if his usage in premise (3) does not contravene your stricture on a possible way that it cannot be used, then the objection to this possible misuse of M()() has nothing to do with Klima's formulation of Anselm's proof."

One of the points I made is that Klima does not make use of the "ampliation" in (1), and he ought. — Banno

That's a remarkable claim. Why don't you think he is making use of ampliation in (1)? And how ought he have made use of it?

Yep. Concepts that contradict themselves. Like "The largest number". — Banno

Why does "the largest number" contradict itself? It seems to me that ω produces an infinite loop, not a contradiction.

The problem with objecting to the two-place predicate M()() in premise (1) without looking at premise (3) is... — Leontiskos

If you want to raise your own objection, go ahead. I've raised mine, with (1), and you have yet to address it.

Why don't you think he is making use of ampliation in (1)? — Leontiskos

I explained that, with the comparison to infinity and transfinite numbers given then quoted above. TO achieve the desired ampliation one needs to go a step past g:=ix¬(∃y)M(y,x), just as one can't get to infinity by iteratively picking the next highest number.

I'm sorry you are not following this, but that's the third time I've made the point.

Will someone be good enough to provide as an aid to navigation a simple proposition expressing exactly what they think Anselm proves? And the same service for Gyula Klima's paper?

Two from the conclusion of Klima's paper:
1) "Anselm’s argument, therefore, can be compelling only for those who are willing to make by his description constitutive reference to God, that is, whose “universe” of thought objects already contains a thought object than which, they think, nothing greater is thinkable. This willingness, however, cannot be enforced by Anselm’s argument on anyone whose “universe” of thought objects does not contain such a thought object.

2) "Indeed, in general, this kind of concept-acquisition seems to be essential for mutual understanding between people conceptualizing the world (and what is beyond) differently, thereby being committed to radically different “universes” of thought objects. Unless one is able to learn to think and live with the concepts of another person and the thought objects constituted by them, one will always fail to have a real grasp on the meaning of the other person."

And these intersect with R.G. Collingwood's understanding of Anselm's proof as a metaphysical argument, which in Collingwood's terms means simply that Anselm's conclusions are consistent with the presuppositions that he started with, and on which basis his argument unassailable. And Anselm himself is clear on this point, even from the beginning of his discourse: "In this brief work the author [Anselm referring to himself] aims at proving in a single argument the existence of God, and whatsoever we believe of God.... The author writes in the person of one who contemplates God, and seeks to understand what he believes."

As to his argument as science or logic, a modern reader must inevitably turn away from it as being nonsensical. Terms aren't well-defined; too much granted/given. And it leaves the question as to whether the God demonstrated is a one or a many. That is, as he is a product of conception, whose conception? And if all you've got is existence, then nothing of quiddities. And to be sure, the "acid test" of what the fool believes, is based on what the fool believes, not what he knows or does not know.

Will someone be good enough to provide as an aid to navigation a simple proposition expressing exactly what they think Anselm proves? — tim wood

Anselm's proof is for the conclusion that God "has to exist also in reality."

And the same service for Gyula Klima's paper? — tim wood

In order to understand what a paper contains one must read it. That's what we are doing. We are reading the paper. We are on section 2 of 5. Once we finish the paper we will be positioned to answer the question of what the paper is about. You can't say what a paper is about before you have read (and understood) it.

So I would be happy to talk about your first question regarding Anselm's proof, but as to your second question, I do not think we are yet positioned to answer it. In fact the second question ignores the OP and seeks an understanding of the paper before we have even moved on to section 3. I think it is good for philosophers to take their time in this way - to not draw their conclusions until all of the arguments and sections have been examined. Until all of the pages of the book have been read. In any case, that's what I want to do in this thread.

Let’s look at ampliation in relation to Banno’s objection:

But that it was essentially the same conception of reference that was at work in his mind when he formulated his arguments in the Proslogion is clearly shown by his insistence against Gaunilo that his crucial description “that than which nothing greater can be thought of” is in no way to be equated with “greater than everything”. It is precisely the ampliative force, recognized as such by 12th-century logicians, that is missing from the latter, and is missed from it, though not described as such, by Saint Anselm in his response to Gaunilo’s objection. — Gyula Klima, St. Anselm's Proof - Section 1

Let’s consider three different options with respect to the greatest number:

• First: "The greatest number"
• Second: "The greatest number one can think of"
• Third: "The number than which no number can be thought to be greater"

For the moment let’s stick with the first and second options.

So suppose @Banno and @Count Timothy von Icarus are on a game show where they are asked a question, and they both have to answer the question within two seconds. If they were asked a question about the first option, “What is the greatest number?,” there would be no answer.

But what if we take a particular instantiation of the second option? “What is the greatest number you can think of?” With two seconds to answer, @Banno says x and @Count Timothy von Icarus says y. In fact as long as x != y one of the two numbers will be greater than the other, and either @Banno or @Count Timothy von Icarus will have won the round.

Similarly, children (or adults too) might play the game, “What is the greatest number one can think of?” We can imagine the dialogue:

• One hundred
• One million!
• One billion
• One billion plus 1
• One billion times one billion
• 2 undecillion (the number of rubles that Russia fined Google)

Eventually someone might offer an analogy as an answer to the question: < x:∞ :: 0.999… : 1 >
(Whether or not we think this makes sense)

Similarly Banno offers the following, a worthy candidate:

ω:=min{x∣x is an ordinal and ∀n∈N,n<x} — Banno

Now in the game show and in each of the children’s answers, the concept, “The greatest number one can think of” is operative. That is precisely the concept they are using to formulate their answers. So the idea that there is no such concept looks to be mistaken.

The fact that “thought” is incorporated into option 2 in a way that it is not incorporated into option 1 is a form of ampliation. “Thought” is part of the option itself. To talk about option 1 instead of option 2 would be a form of equivocation which avoids the ampliation. Indeed, option 2 represents a concept which produces determinate answers when engaged, but which has no determinate answer of itself. Nevertheless, each of the determinate answers it produces when engaged does have a form of determination qua the thought of the engaging individual (namely it will represent something like a personal limit on number knowledge).

Now suppose someone believes that they have a proof (say, from mathematical induction) that there is no greatest number (or else greatest prime, which is more fun). In that case they will believe that option 1 represents a contradiction (via their proof), but the question of the status of the concept is still an open question (given the fact that not everyone possesses such a proof, valid or invalid).

Concepts that contradict themselves. — Banno

But you know full well that you haven't demonstrated a contradiction:

good reason to think that it is not possible — Banno

Good reason != contradictory proof

Will someone be good enough to provide as an aid to navigation a simple proposition expressing exactly what they think Anselm proves? And the same service for Gyula Klima's paper? — tim wood

Anselm thinks he proves that the very idea of god shows that He exists. He's mistaken. Klima realises this, but still sees a use for such arguments in explaining to non-theists how theists think about the world. He is specifically advocating not becoming involved in the sort of discussion now occurring here, that the parties 'should not seek sheer “winning” in a debate'.