Now like a child with an attention disorder, the crank asks me whether the members of a set are abstractions or concretes, after I explicitly said that they can be either, and I gave explicit examples. Is the crank not able to read?

I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same. — TonesInDeepFreeze

The crank mindlessly replies "Right, continue in your violation of the law of noncontradiction."

The crank is so mentally deficient that he can't see that it's not a contradiction that "there are 24 orderings of a set" does not imply "all those orderings are the same". It's an incorrect implication, because, indeed it is a contradiction to say that 24 different orderings are all the same ordering. It seems it is in the crank's imagination that, somehow 'different' implies 'same'. That's his problem, not mine, since indeed, for me 'different' does not mean 'the same'.

The crank's illogic and utter obtuseness are not less than stunning. And in a philosophy forum!

The crank says, "For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two."

Stop lying, crank.

This deserves to be especially highlighted:

The crank says, "[Tones In Deep Freeze] has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same"

That is yet another flat LIE from the crank, and a really stupid lie.

I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same.

The cranks daily posts rank garbage on a ... PHILOSOPHY forum!

The sophist crank says, "the principle you stated, the elements of a set are not things".

I never said any such thing. I've said the opposite. — TonesInDeepFreeze

Then the sophist crank says "I know you never said such a thing. You mix up physical objects and mathematical objects as if there is no difference between them, and as if the law of identity would apply to both equally."

That's another LIE from the crank.

That the law of identity applies to both numbers and rocks does not entail that there is no difference between numbers and rocks! It does not entail that there is no difference between abstractions and concretes. The laws of traffic apply to both domestic vehicles and foreign vehicles, but that doesn't entail that there's no difference between domestic vehicles and foreign vehicles! The crank can't reason successfully in even the most basic ways!

Moreover, I did not say that an element of a set cannot be a concrete thing. The set of pencils on my desk has only concrete things as members.

The sophist crank is as usual abysmally confused and making false claims about what I've said.

/

When I first used the term 'sophist crank' I knew I was indulging redundancy', since cranks are by nature sophists. But I've been doing it anyway, to stress the point. It's clear enough by now, though it's been clear enough about him for years.

/

Then the crank, in his usual manner of self-serving sophistry, misconstrues @fishfry. fishfry didn't contradict that the law of identity is different from the identity of indiscernibles.

/

The crank says that the bandmates in the Beatles don't provide for a set. But they do, as they provide for the set {George, Ringo, John, Paul}. The crank can't understand what even a child can understand.

{the pencil on my desk, the pen on my desk} is a set whose members are of concretes and it has two orderings.

{1, 2} is a set whose members are mathematical objects and it has two orderings.

And even if we demurred from saying that such things as number are abstract objects, then still the principle that there is more than one ordering of a set obtains, since we may adduce sets whose members are concrete objects.

So, what example would the crank give of a set with more than one member? Whatever example the sophist gives, that set has more than one ordering.

Then the crank says "a set is a mathematical structure". That is an example of arguing by mere insistence on one's personal definition. Typical sophistry. Of course, one may stipulate any definition one wants to stipulate. But that carries no argumentative import in context of use of the word with a different definition. In mathematics and even in everyday life, the word 'set' is not ordinarily used to mean 'a structure'. However, mathematics does also address the notion of structure, and provides rigorous definition; but the crank, in his obdurate willful ignorance knows nothing about that, as he knows nothing about the mathematics he incessantly gets completely wrong.

Then the crank points out that I said the set has 24 orderings and that I did not say it has 24 possible orderings. That is, typically, an inane objection by the crank. (1) Extensional mathematics does not use intensional modalities. (But there are systems of intensional mathematics too.) (2) Even if we do speak instead of 'possible orderings', any particular one of those possibilities is not the only possibility, so it is still not THE ordering. It is merely one of the "possible" orderings chosen for our consideration. Any other "possible" ordering could be chosen and then, following the crank's notion, it would have to be considered to be THE ordering. So there would a different THE ordering depending on which ordering we happen to choose for consideration, which is still incoherent.

So, even deferring to the crank's insistence about "possible", which of the 24 possible orderings of the set whose members are the bandmates in the Beatles is THE ordering of that set?

I've given the crank the following information about half a dozen times already, but like the horse led to water who will not drink, the crank will not think (apologies to D. Parker):

Yes, we can specify a particular ordering of a set and refer to that set vis-a-vis that specified ordering. For example, let B be the set whose members are all and only the bandmates in the Beatles, and let R be the ordering of B alphabetically by first name. Then we have the STRUCTURE <B R>. That accords with the notion of a set along with a particular ordering.

The crank says, "there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time.

At this exact moment of time, there are two orderings of the set of writing tools on my desk:

{<pencil pen>}

and

{<pen pencil>}

At this exact moment of time, there are two orderings of the two kids on the playground:

{<Joe Maya>}

and

{<Maya Joe>}

/

Then the crank goes on with yet more confusions. Reading his posts, I am reminded of a character in 'The Office' saying about the on and on, full of it fool, Michael Scott, "Where's the off button on this moron?"

They are not inferences but independent premises and might both be true. — Michael

You wrote in the argument:

If take A(x) to mean something like "x is male" then both (2) and (3) are true. — Michael

[emphasis added]

So in my first post I captured that implication.

And in my second post I gave a version in which instead they are premises:

(2) pEx(Fx & Ax) ... premise

(3) pEx(Fx & ~Ax) ... premise — TonesInDeepFreeze

I can't argue with you about anything — fishfry

Then you can't argue with me that you can argue with me.

It could be fixed this way:

(1) E!xFx ... premise

(2) pEx(Fx & Ax) ... premise

(3) pEx(Fx & ~Ax) ... premise

(4) {(1), (2), (3)} is consistent

(5) pnEx(Fx & Ax) -> nEx(Fx & Ax) ... theorem

(6) pnEx(Fx & ~Ax) -> nEx(Fx & ~Ax) ... theorem

(7) {(1), (2), (3)} |/- pnEx(Fx & Ax) ... (1), (3), (4), (5)

(8) {(1), (2), (3)} |/- pnEx(Fx & ~Ax) ... (1), (2), (4), (6)

https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1(~9~7x(Fx~1Ax)~1~9~7x(Fx~1~3Ax))))~5~9~8~7x(Fx~1Ax)||universality

If I haven't made any mistakes here:

At least for me, this is more exact and clear:

(1) E!xFx ... premise

(2) pExAx ... premise

(3) pEx~Ax ... premise

(4) {(1), (2), (3)} is consistent

(5) pE!x(Fx & Ax) ... (1),(2)

(6) pE!x(Fx & ~Ax) ... (1),(3)

(7) pnEx(Fx & Ax) -> nEx(Fx & Ax) ... theorem

(8) pnEx(Fx & ~Ax) -> nEx(Fx & ~Ax) ... theorem

(9) {(1), (2), (3)} |/- pnEx(Fx & Ax) ... (1),(4),(6),(7)

(10) {(1), (2), (3)} |/- pnEx(Fx & ~Ax) ... (1),(4),(5),(8)


* But the inferences at (5) and (6) are invalid (according to the validity checker).

https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1~9~7xAx))~5~9(~7x(~6y((Fy~1Ay)~4x=y)))||universality

https://www.umsu.de/trees/#(~7x~6y(Fy~4x=y)~1~9~7xAx)~5~9~7x~6y((Fy~1Ay)~4x=y)

"There exists exactly one falcon, and it possible that there exists a non-falcon" doesn't entail "It is possible that there exists exactly one falcon that's a non-falcon".

and

"There exists a falcon, and it possible that there exists a non-falcon" doesn't entail "It's possible that there exists a falcon that's a non-falcon".

But

{(1), (2), (3)} |/- pnEx(Fx & Ax)

and

{(1), (2), (3)} |/- pnEx(Fx & ~Ax)

are correct anyway (according to the validity checker). Just not by your argument.

https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1(~9~7xAx~1~9~7x~3Ax)))~5~9~8~7x(Fx~1Ax)||universality

* I don't see the relevance of this to your specific argument:

https://www.umsu.de/trees/#~9~7xP(x)~5~9~8~7xP(x)||universality

What you want to prove is not just that that formula is invalid but to prove:

{(1), (2), (3)} |/- pnEx(Fx & Ax)

and

{(1), (2), (3)} |/- pnEx(Fx & ~Ax)

Those are correct (according to the validity checker). Just not by your argument.

I don't know which of my posts or comments you are commenting on.

In a recent post, I said that I don't understand the proof at the proof generator.

I'm not stating a criticism of Michael's posts. I'm just trying to figure them out.

Thanks.

I'm saying that I'll take your latest note and incorporate it as I go over your argument again. Not waiting on you.

I don't understand. You said a certain formula is valid in S5. The proof generator shows a deduction of the formula. But I can't make sense of the deduction at the lines I mentioned. The proof generator makes no mention of exemplification and positive. Bringing in exemplification and positive does not address my points. And I'm not even talking about Godel. I'm just looking at certain claims about what is derivable in S5, as those claims don't invoke exemplification or positive.

I don't understand that proof.

Where can I see a specification of S5 extended to a deduction calculus with quantifiers?

I don't know what deduction in S5 permits:

inferring line 4 from line 3. (~nQ does not imply ~Q)

inferring line 5 from line 2. (pQ does not imply Q)

line 6 from line 5 is existential instantiation applied to a modal formula, but S5 is only a modal propostional logic.

I hope it won't be too long that I'll have time to resume going over your argument with the emendations.

My questions were here:

https://thephilosophyforum.com/discussion/comment/914470

Your response was to switch to a different description of your idea.

We could start with the first question:

Am I correct that by "we cannot assume pEx(nPx) is true for any logically consistent Px" you mean "For all consistent Px, we have that pEx(nPx) is not logically true"? — TonesInDeepFreeze

We can assume anything. So I take it "cannot assume" is colloquial for something more logically definite. Thus my question above.

Also, you have a modal operator after a quantifier. I don't think S5 can do anything more with that than to regard the quantified formula as just a sentence letter, so S5 sees pEx(nPx) as just pQ.

Interesting. I hope I didn't bury the lede. I'm not all up about sarcasm. Rather, what I find important is (1) striving not to misrepresent a poster's remarks and to stand corrected when it is pointed out that one has; and (2) not to argue by ignoring key counter-arguments and explanations; not to just keep replying with the same argument as if the other guy hadn't just rebutted it.

I don't ask for apologies. But it's okay if you want to give them. But you embed into your apologies yet more items that I feel deserve response. Your apologies themselves are snarky; "sins imagined" e.g. I don't even object to snark, except it's your way of ostensibly apologizing while still turning it back on me.

If I misconstrue someone's math or philosophy points, especially to mischaracterize them, then if the person calls me on it or I discover it myself, before posting back to that person again, I should post my recognition of my mistake. That's my ethos. Yours might be different. But I will stick with my prerogative to reply when I like.

And to answer your question: No, I definitely do not have any interest in "picking fights" and I find no value in fighting for the sake of fighting. But I do find value in posting disagreements and corrections, whether regarding the math and philosophy or regarding the personal specifics of the posting interchanges. In various thread, you have posted a lot of inaccuracies and misconceptions about math, and now lately about me. I respond to that.

Zeno's paradox concerns analysis of an actual physical event. Thomson's lamp concerns analysis of a hypothetical state-of-affairs. One difference is that with Zeno's paradox, we read a conclusion that a certain fact is impossible, which is impossible. With Thomson's lamp, according to Thomson, there is a derivation of a contradiction; but that comes from non-factuals. Are there other crucial differences between Zeno and Thomson?

When possibility is part of the analysis, the analysis can get complicated. We should be careful that our inferences regarding the modalitiy are proper.

@Michael

I've not gone back to review all that's been said in this thread, and I need to catch up to your replies, but starting again from the beginning of your argument.

I surmise that the reason you put your argument in numbered steps is so that it can be seen to be airtight.

Is your argument intended to be Thomson's argument?

You have mentioned different conclusions you draw:

(1) The conditions (the premises) of the lamp are inconsistent.

(2) Supertasks are impossible. (But can we infer from the impossibility of Thomson's lamp that all supertasks are impossible?)

(3) Time is not continuous. (I've suggested that what you actually seem to dispute is that time is not densely ordered (infinitely divisible), which is a stronger claim.)

(4) Benacerraf is wrong.

Here's Thomson's statement of the problem:

"There are certain reading-lamps that have a button in the
base. If the lamp is off and you press the button the lamp goes
on, and if the lamp is on and you press the button the lamp goes
off. So if the lamp was originally off, and you pressed the
button an odd number of times, the lamp is on, and if you
pressed the button an even number of times the lamp is off.
Suppose now that the lamp is off, and I succeed in pressing the
button an infinite number of times, perhaps making one jab
in one minute, another jab in the next half-minute, and so on,
according to Russell's recipe. After I have completed the whole
infinite sequence of jabs, i.e. at the end of the two minutes, is
the lamp on or off? It seems impossible to answer this question.
It cannot be on, because I did not ever turn it on without at
once turning it off. It cannot be off, because I did in the first
place turn it on, and thereafter I never turned it off without at
once turning it on. But the lamp must be either on or off. This
is a contradiction."

Here's your presentation:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00

From these we can then deduce:

C1. The lamp is either on or off at all tn >= 10:00
C2. The lamp is on at some tn > 10:00 iff the button was pushed at some ti > 10:00 and <= tn to turn it on and not then pushed at some tj > ti and <= tn to turn it off
C3. If the lamp is on at some tn > 10:00 then the lamp is off at some tm > tn iff the button was pushed at some ti > tn and <= tm to turn it off and not then pushed at some tj > ti and <= tm to turn it on

From these we can then deduce:

C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction] — Michael

I want to get back to looking at this more closely, but in the meantime, do you consider your presentation equivalent with Thomson's statement of the problem?

Fairy tale characters are an abstract universal. They are general, and they don't actually exist.

Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.

Fairy tale characters are abstract universals, and Cinderella is an abstract particular.

In your world you don't have any abstraction at all. I think you're taking a point too far. — fishfry

Nicely said.

You first claimed that I was offensive to you. So I pointed out that you don't realize how offensive you often are. So I just gave you that info. I don't sweat being offended in posts. But you carelessly misconstrue what I've posted, and claim I've said things I haven't said, and write back criticism of my remarks by skipping their substance and exact points. And that is what I post my objections to.

Meanwhile, what you say about my posting style is rot. You say it's too long. But you also say it doesn't explain enough. Can't have it both ways. And I do explain a ton. But, again, I can't fully explain without having the prior context back to chapter 1 in a text already common in the discussion. And l explain somewhat technically because being very much less technical threatens being not accurate enough. Meanwhile, your own posts are usually plenty long, so take that tu quoque.

"wut" is a standard Internet location, and though it carries a bit of snarkitude, it's not considered overly aggressive in the scheme of things. Just an expression of puzzlement. — fishfry

wut?

wut? axiom of infinity. what's wrong with you tonight? — fishfry

My response was to 'what's wrong with you tonight?', not so much to 'wut?'.

Convenient for you now to self-justify by highlighting 'wut?' and not 'what's wrong with you tonight?'.

There was nothing wrong with what I posted that night. You just snapped-at as if there were, when actually the problem is that you, as often, reply to your careless mis-impression of what is written rather than to what is actually written.

Hey, I get your whole "Aw shucks, I'm just a scorpion who's gonna do what a scorpion's gonna do. I don't mean nothin' by it" routine. But it doesn't mean jack to me as far as feeling any less right in answering right back.

It is simple indeed:


x is an element iff Ey xey

x is a class iff (x=0 or Ey yex)

x is a proper class iff (x is a class & ~Ey xey)

x is a set iff (x is a class & Ey xey)

x is an urelement iff x is not a class


Classical set theory theorem: Ax x is a set

Classical class theory theorem: Ax x is a class & Ex x is a set & Ex x is a proper class

Set theory with urelements theorem: Ex x is a set & Ex x is an urelement

The crank clown can't understand what the rest of humanity understands:

AN ordering of the children is not the ONLY ordering of the children.

And back to 'The Adventures Of The Crank Radio Hour':

Crank: Hey boss, I put our sales products in this spreadsheet in the order.

Boss: Which order?

Crank: The order.

Boss: Order by revenue or by items sold or by catalog number or what?

Crank: You know, the order.

Boss: Remind me how you got this job.

The crank says, "TPF's head sophist has a sense of humour."

So the sophist crank finally comes close to a true sentence, but still only half true. I'm not a sophist, neither philosophically nor rhetorically.

The sophist crank says, "the principle you stated, the elements of a set are not things".

I never said any such thing. I've said the opposite. The sophist crank again lies about me, as a function of his abysmal confusion.

The sophist crank says, as a paragraph, "Etc.."

More eloquent than a rock, by a word.

As far as S5 is concerned, if F is a formula, then AxF and ExF are both treated as if they are just a propositional letter. Is that not correct?

I do see now that to show that (3) is not the case, we need rely only on pnQ -> nQ and the fact that it is not the case that pQ |- nQ.

But It is difficult to follow you as you jump around among very different formal formulations and among different English formulations and different kinds of examples. I started out trying to sort out your original argument as originally formulated but now you've twice jumped to different, though related, formulations. I'm giving up for now. It would help if you would give one self-contained argument with transparent inferences from start to finish.

You asked readers to consider a formal argument you started. Since that was interesting to me, I considered it in detail as far as I could. The argument involves uniqueness, inferences in S5 and inferences with both quantification and modal operators. I asked questions whose answers might allow me to understand your locutions about the argument and to see that your argument would be completed. But then your answer is to just drop that formal buildup; moreover, to give an English argument that does't come close to the specifics of your previous formal argument. So I don't understand your point in your formalisms if you don't follow through with them; I don't see why I should have spent my time on them if you're just going to ditch them anyway.

But regarding your answer (I'm using 'Q' rather than 'G' or 'U' to steer clear of theological or fictive connotations):

If I understand (I've not read subsequent posts to your answer to me), your argument starts with: Q is consistent and ~Q is consistent, so S5 proves ~pnQ v ~pn~Q.

I can see that argument if these are theorems of S5:

Q -> ~pn~Q

~Q -> ~pnQ

Are they? If not, then what is the argument that "Q is consistent and ~Q is consistent" implies that S5 proves ~pnQ v ~pn~Q?

Then you say, "Therefore, we cannot just assume that because some X is not a contradiction that it is possibly necessary."

I take that to mean: "Q is consistent" does not imply S5 |- pnQ.

doesn't always explain himself, or is just typing stuff in. — fishfry

I explain in detail. And it's a stupid thing to say that I just type stuff. But in post or even a series of them, I can't fit in an explanation all the way back to the basics of the subject, so if one doesn't have the benefit of a context of adequate knowledge, it's not my fault that I can't supply all that needed context in even several posts.

Then you tell [the kids on the playground] to line up by height. Now you have an ordered set of kids. Or you tell them to line up in alphabetical order of their last name. Now you have the same set with a different order.

It's an everyday commonplace fact that we can have a set of things in various orders.

Now maybe you are making the point that everything is in SOME order. The kids in the playground could still be ordered by their geographical locations or whatever.

But sets don't have inherent order. — fishfry

Exactly and well put. I've given the crank that same explanation. He will never understand it, because he wants to not understand it. If he found himself understanding it one day, then he would face the crisis of seeing that he's been confused and in the dark for years and years (decades?).

The crank's rejoinder is that we may state the positions and that that is "an order".

He is exactly right there. It is AN order. He said it himself! It is not "THE" order since there are different orders, each of them AN order.

The crank asked about rocks. But we were not talking about rocks. We have been talking about sets. Sets of rocks, or set of numbers, etc. Sets have orderings, but if a set has more than one members then it has more one ordering. For example, a set with two members:

{0 1} = {1 0}

There are two orderings of that set:

{<0 1>} and {<1 0>}

So there is not "THE" ordering of that set, since there are two of orderings of the set.

But we may indicate the set with regards to a particular ordering. The notation is:

<S R> where S is the set and R is a particular ordering. For example"

<{0 1} {<1 0>}>

is the set {0 1} along with the ordering that is the greater-than relation on the set.

For example, the set whose members are all and only the bandmates in the Beatles has 24 orderings. So there is not "THE" ordering of that set.

But we may indicate that set with regards to a particular ordering. For example, the alphabetical ordering by first name:

{<George Harrison, John Lennon> <George Harrison, Paul McCartney> <George Harrison, Ringo Starr> <John Lennon, Paul McCartney>, <John Lennon, Ringo Starr> <Paul McCartney, Ringo Starr>}

As a sequence: {<1 George Harrison> <2 John Lennon> <3 Paul McCartney> <4 Ringo Starr>}

As a list: George Harrison, John Lennon, Paul McCartney, Ringo Starr.

But, obviously there are many other ways to order the Beatles: by age from youngest to oldest, by age from oldest to youngest, by height from tallest to shortest, by height from shortest to tallest, by wealth number of record sales as an artist after the Beatles, ...

So there is not "THE" ordering of the set whose members are the bandmates in the Beatles.

But what about that rock? If it's the one that is the crank's head, then it is indeed empty and there is only one ordering of the set of its particles, which is the empty ordering.

But what about more complicated, more intelligent rocks? The rock is not a set. However, we may speak of the set of particles of the rock. And in that case, again, there is no "THE" ordering of that set. But the crank mentions structure. Yes, we may describe the rock in terms of a certain structure. But the rock, even as described per a certain structure is not a set; it's a rock. Moreover, we may describe a rock as different isomorphic structures. Your structure is based on rock's pointy tip facing up, and my structure, isomorphic to your structure is based on the rock's pointy tip facing down.

/

The crank says I use definitions out of context. The crank confuses self-description with outward observation.

/

The crank says that he doesn't know what I mean by 'identity theory' even though I've stated and explained the axioms of identity theory at least a few times. (Or if I hadn't done that prior to the crank's post, then nothing was stopping him from asking me to do it.)

/

The crank makes the ridiculous claim that I misunderstand the rules of axiom systems. I understand the formation syntax of the formal languages, the formation syntax of the formulas, the formation syntax of the axioms, the formation syntax of the inference rules - all recursively. And the formation of the semantics for the meaning of the formulas - all by inductive definition. I understand exactly how to check that a purported formal proof is a proof and also I understand exactly how to interpret the meaning of formulas.

The crank doesn't know what he's talking about regarding mathematics or the axiomatic method or regarding me. Then he says that I annoyed him when we met but now I merely amuse him. Ah, the classic arch line, "You merely amuse me". The crank is not only a feeble thinker, he's a lame flamer. And why was he initially annoyed? Because as he was freely spewing confusion, ignorance and disinformation on this forum, I corrected him.

/

The crank repeats his argument that the notion of identity in mathematics is wrong since mathematics regards objects that don't exist. So, yet again, the crank just ignores the responses I've given to that. Just to start: He ignores even the examples I've given of sets of non-abstract objects, such as the set of pencils strewn on my desk, etc.

@fishfry

Now that we got the axiom of extensionality straightened out, it's apropos to get the rest of the dissension worked out.

It starts with these good posts:

https://thephilosophyforum.com/discussion/comment/911857

https://thephilosophyforum.com/discussion/comment/913150

I wasn't clear; I didn't mean a URL link; I meant a reply link. Does the link in this post do what you want?

one or both of these is true:

4. ¬◇∃x□(Fx ∧ Ax)
5. ¬◇∃x□(Fx ∧ ¬Ax) — Michael

I think I'm with you that far. But I'm not sure what the following quotes mean or how they follow from the above quote:

Therefore we cannot assume that ◇∃x□Px is true for any logically consistent Px. — Michael

(What do you mean by 'logically consistent' rather than plain 'consistent'?)

Am I correct that by "we cannot assume pEx(nPx) is true for any logically consistent Px" you mean "For all consistent Px, we have that pEx(nPx) is not logically true"?

(I would think that to say "we cannot assume Q" means "We don't have sufficient basis to assume Q since Q is not logically true".)

or do you mean

"It is not the case that for all consistent Px we have pEx(nPx)"?

I surmise you mean the former, since:

we cannot assume that a necessary unicorn [...] is possible. — Michael

I take it that by a "A necessary unicorn is possible" you mean "It is possible that there is an x such that necessarily x is a unicorn". I.e. pEx(nUx).

Are you saying: If Ux is consistent, then pEx(nUx) is not logically true?

If I'm not mistaken, pEx(nUx) is not logically false:

Let Ux be Dx <-> Dx. So nUx. So Ex(nUx). So pEx(nUx).

If I understand correctly, you're saying that the first part of your argument (up to 5.) shows that if Ux is consistent then pEx(nUx) is not logically true? What is your argument for that?

If I understand correctly, you are saying that

(ExFx -> E!xFx) -> (~pEx(n(Fx & Ax)) v (~pEx(n(Fx & ~Ax))) (which seemed correct to me when I glanced over it)

implies

If Ux is consistent, then pEx(nUx) is not logically true

If that is what you're saying, then what is your argument?

/

P.S. I'm assuming we have "If Q is consistent then Q is not logically false".

Yes, Tarski was very much concerned with both formal and natural languages.

I bet you are fun at parties — Banno

I don't go to parties to talk about modal logic. Have your party hearty fun about the ontological argument. I'm not stopping you. I merely pointed out that the modal theorem you cited is not correctly applied as you did.

No, you are not correctly applying the formulas.

This is correct:

If it is not necessary that Q, then it is not possible that is necessary that Q.

That is not equivalent with your incorrect application:

If it is not necessary that Q, then it is not possible that Q.

Some people believe that Godel-Rosser has implications not confined to mathematics and questions in the philosophy of mathematics. They argue that Godel-Rosser pertains to questions in epistemology, ontology, and even other subjects. But I don't know what you mean by 'applicable to non-formal languages'.

An excellent book that discusses arguments about Godel-Rosser outside of mathematics and philosophy of mathematics is 'Godel's Theorems' by Torkel Franzen.

If, in S5, if god is possible then god is necessary — Banno

S5 does not say that pQ -> nQ.

Or am I missing something in your context?

2. ◇∃x□(Fx ∧ Ax) ∴ ∃x□(Fx ∧ Ax)
3. ◇∃x□(Fx ∧ ¬Ax) ∴ ∃x□(Fx ∧ ¬Ax) — Michael

both 2 and 3 are valid under S5 — Michael

EDITED post:

I think I see how you got :

pEx(nQ) -> Ex(nQ)

(I'm using 'Q' instead of e.g. the more specific 'Fx & Ax'.)

I don't know the deductive system, but I guess this is a validity:

pEx(nQ) -> Ex(pnQ)

And we have:

pnQ <-> nQ

So we have:

pEx(nQ) -> Ex(nQ)

But you say that is in S5. But, as far as I know, S5 is merely a modal propositional logic.