Okay, S.M., brace yourself. Apologies in advance for the incredible length, but you guys have been asking for it for a while.
So, IF I exist, the statements “I exist.” and “I do not exist.” have meaning, and IF I does not exist, then they don’t. This is a condition where a statements is not either true or false or meaningless, depending on the condition of reality, but “true or meaningless” and “false or meaningless” respectively. ANY statement of this style would be begging the question, including the meaningful ones (since it can be reasonably said that the speaker indented the sentences as meaningful, and therefore was assuming that “I” exists, even when they claim opposite. This is not acceptable.
You (S.M.) touched on the correct solution; the problem is not that the statements lack meaning, it is that they are being misinterpreted. I claim the blame for this; becuse the above is demonstrative that existence is a first order propert, a position that I’ve been asking we pretend to adopt for simplicity and consistency. Everbody from Descartes on at least occsionally speaks of existence as a property of the “existent” things; all such references are merely shorthand for comments on tehe contents of reality of a whole. Arguing about existence is considerably easier when everybody agrees to pretend that existence is a first order property, as we have been pretty much doing for a long while.
So, in short, I have caught you out in an error, S.M.: both sets of answers should have read like the verbose, more accurate versions. In the interests of someday perhaps being able to make less-than-monstrously verbose arguments, I will now propose an explicit redefinition of phrasings where ‘exists’ is apparently used as a predicate. I’ve pretty much been meaning these very meanings for some time (the whole time) and find them, while more cumbersome to argue with, as intuitive as any other approach. I ask that when I say “I exist” and the like in the future, it be interpreted as stated below.
S.M. may claim credit as inspiration/source for the english phrasing.
[ul][li]“Bs exist” (plural) as “At least one existing thing matches the criteria described by the abstraction B.” Ex.( B(x) )[/li]
[li]“B exists” (singular) as “One specific existing thing matches the criteria described by the abstraction B.” Ex.( B(x) && Ay.( B(y) -> (y=x) ) )[/li]
[li]“B(s) do(es) not exist” (either) as “No existing thing matches the criteria described by the abstraction B.” ~Ex.( B(x) )[/li]
and
[li]“ALL Bs think” (plural) as “Every existing thing that matches the criteria described by the abstraction B has the property of thinking.” Ax.( B(x) -> T(x) )[/li]
[li]“SOME Bs think” (plural) shall NOT be defined since it is particularly problematic, and not overly useful in this argument.[/li]
[li]“B thinks” (singular) as “One specific existing thing matches the criteria described by the abstraction B, AND that one specific existing thing that matches the criteria described by the abstraction B also has the property of thinking.” Ex.( B(x) && T(x) && Ay.( B(y) -> (y=x) ) )[/li]
[li]“B(s) do(es) not think” (either) as “No existing thing both matches the criteria described by the abstraction B AND has the property of thinking.” ~Ax.( (B(x) -> ~T(y)) && (T(x) -> ~B(y)) )[/ul][/li]
As can be seen by these definitions, I CONCEDE that you folks have finally (indirectly while barking up a different tree) demonstrated that, using existence as a forst-order predicate causes errors, we have no choice but to make existence requisite for the statement of other predicates. In other words: I accept that the axiomatic acceptance of the statement “I exist” begs the question, since it includes the axiomatic assumption of existence.
Now that that has been stated, when you all regain consciousness…
Time to reformulate the argument. (You didn’t really think I was done?) At this point, there’s nothing to do but get formal. Very formal.
Symbols:
P(x) = x is a perception. The perception is distinct from that which is perceived (which may or may not exist separately from the perception) and that which perceives/observes the perception.
P(x:y) = x is a perception that seems to its observer to have the content y (y given for information only)
O(x,y) = x observes/perceives y
I(x) = x is I. (Or one of them, if there can be more than one.) As will be made explicit in a premise, that which observes this argument.
IsP5(x) = x is premise P5.
Arg = THIS argument.
Premises
P0: Azc.( P(z:c) -> P(z) )
For the symbolic purists. Of course, it doesn’t work in the other direction.
P1: Az.( P(z) -> Ex.( O(x,z) ) )
That is, that all perceptions are observed (by existant things). This is from the understood meaning of the word “perception”, as being pretty much a nounified verb.
P2: Axyz.( ( O(x,z) && O(y,z) ) -> (x=y) )
That is, that each perception has one observer. Now, the same thing may be percieved my many different observers, but each observer has its own perception. This is from the understood meaning of the word “perception”, as being pretty much a nounified verb. I’m not going to use this yet, but will use it to counter any “something(s)” claims that roll around.
P3: Axz.( ( O(x,z) && P(z:Arg) ) -> I(x) )
P3b: Axz.( I(x) -> ( O(x,z) && P(z:Arg) ) )
That (and only that) which percieves this argument can be called (call itself) I. I can see it now: "But, that means that there are as many different “I"s as there are different perceptions of the argument!” Not necessarily: P3 says one observer per perception, not one perception per observer. But yes, in the absence of other informatin this premise allows that it is possible that there are many different things contemplating the argument (or rather, entertaining the perception that they are contemplating the argument), and that therefore qualify to be called “I”.
P4: Ez.( P(z:Arg) )
And so our story begins: with the perception that the argument is occuring.
P5: ~Ex.( IsP5(x) ) 
JUST KIDDING. After finishing editing I noticed I skipped the number 5. Give me a break, it’s late.
Inferences/Conclusions and maybe one more premise later.
I6: P(parg:Arg)
Instantiation of ‘parg’ for z in P4. Now we have a name for the z in P5.
I7: P(parg:Arg) -> P(parg)
Instantiation of ‘parg’ for z and ‘Arg’ for c in P0. For the purists.
I8: P(parg)
Modus ponens on I6 and I7. Now, for the purists, we have a parg that will fit in most of the other statements.
I9: P(parg) -> Ex.( O(x,parg) )
Instantiation of ‘parg’ for z in P1.
I10: Ex.( O(x,parg) )
Modus ponens on I8 and I9. Note that this says, literally, “There exists some thing x that observes/percieves parg”. This and/or I11 is the point as which we have concluded S.M.'s “something(s)” exist. Except, we’ve only proven a singular existing observer, of course. No (s). This will be discussed further later.
I11: O(begbert2,parg)
Instantiation of ‘begbert2’ for x in I10. We’ve now named the observer “begbert2” (fancy that).
I12: O(begbert2,parg) && P(parg:Arg)
Conjunction of I11 and I6.
I13: ( O(begbert2,parg) && P(parg:Arg) ) -> I(begbert2)
Instantiation of ‘obs’ for x and ‘parg’ for z in P3
I14: I(begbert2)
Modus ponens on I12 and I13. The instantiated object begbert2 (which then does exist) has the propert of being “I”. So, “I exist”.
With the current premises, it cannot be proven that I am the only I that exists. However, it is possible to prove that a given obserer can prove, to itself, that it is the only ‘I’ that exists. This is done by noting that, to a given observer, all perceptions that can be known to exist can be observed by the given observer. So, as far as I(begbert2) can tell, the following premise is true.
P15: Axz.( O(x,z) -> (x=begbert2) )
Do you guys mind if I skip some of the intermediate substeps? I can do them on request if you really want me to.
I16: Axz.( ~(x=begbert2) -> ~O(x,z) )
The contrapositive of P15
I17: Axz.( ~(x=begbert2) -> (~O(x,z) || ~P(z:Arg) )
An disjunction-creation substitued into I16
I18: Axz.( ~( O(x,z) && P(z:Arg) ) -> ~I(x) )
The contrapositive of P3b
I19: Axz.( ( ~O(x,z) || ~P(z:Arg) ) -> ~I(x) )
A demorganized ~( O(x,z) && P(z:Arg) substituted into of I18
I20: Axz.( ~(x=begbert2) → ~I(x) )
Transitivity of implication substitution across I17 and I19
…proving that if it ain’t begbert2, then it ain’t I; and given that by I14 we know that the thing known as begbert2 is I. So, when basing your argument only on perceptions of which one can be aware, exactly one single entity, I, can be proven to exist.
Heavens, that’s a lot of typing there. Have at.