Yes.

> The free distributive lattice on three generators has 18
> elements–

But THAT is, as usual, irrelevant.
Defining what a free distributive lattice is, and what
its generators are, is something that FOL *is* useful
for.  Most people who wanted to do that in fact WOULD USE
FOL to do it.

> that is enough for sixteen basic boolean functions
> and two quantification symbols.

In order to even describe what a lattice is in the first place,
you would need to have some prior structure.  You might
need sentences with truth values.  In other words, you might
ALREADY NEED 16 boolean functions.

As for quantification, the whole notion that you might "need"
something "prior", LIKE " a free distributive lattice on three
generators ",
is just preposterous.  First-order quantification is merely about
extending
something that is already well-defined for the finite case (the
application
of an associative operator to a finite list) to the denumerable case.
The mention of lattices and genreators is, I repeat, IRrelevant
complexity
that PREsupposes a degree of logical machinery JUST to get itself
stated.  It should come as no surprise to anyone that you can
re-formulate
first-order logic in terms of any MORE complicated construct that you
NEEDED first-order logic TO articulate in the first place.  Two
truth-values
and 0th-order logic and the natural numbers, in most people’s opinion,
ARE
LESS of a foundation than "the free distributive  lattice on three
generators".
I repeat, IF you wanted to explain THAT to anybody, YOU would need to
USE first-order logic to do it.

> The problem is that you have to be able to see the ternary structure of FOL.

No, YOURproblem is that YOU have to see that the USUAL explication of
FOL is MORE elementary than "the free distributive lattice on three
generators".

> Two of the generators correspond to "0000" and "####"
> (using the symbols from "logic, triple systems and designs")
> that you can interpret as "FFFF" and "TTTT".  The third
> generator corresponds to "#" that you can interpret as
> FOR ALL.

Given that all three of these concepts (the constant true 2-ary boolean
function,
the constant false 2-ary boolean function, and the conjunctive  w-ary
boolean
function, usually denoted by an unbound variable), ALL ALREADY EXISTED
in the standard treatment, it is hard to see how you could meet a
burden of
proof that they ought to be thought of as generators of a lattice as
opposed to
boolean functions.  Moreover, even if you want to harmonize the
treatments
and say "these 3 boolean functions ARE DEPLOYABLE as generators olf a
free distributive lattice", you still bear a very heavy burden of proof
as to why it
might be desirable to deploy them THAT way as OPPOSED to in the way in
which the standard treatment deploys them.

> Each generator in that lattice is a two-connected node.

If you want to write something called "The Geometry of First-Order
Logic" then NOBODY is going to object.  They ESPECIALLY are not
going to object if you are offering a geometric treatment OF THE SAME
STANDARD CLASSICAL FOL that everybody else is already used to.
But if you begin by claiming that the standard is bullshit, and that
you
have discovered some superior alternative, AND THEN it turns out that
you are just RE-articulating the standard in a different dialect, well,
THEN,
you are going to look stupid.

If I already know that something is a boolean function then I am not
likely to
care that it can also be thought of as "a two-connected node".
That is not going to produce any new results or insights.

> The four four-connected nodes delineated by "0000" and
> "####" correspond to the DeMorgan invariants mentioned
> in the original post.

Here you introduce a new undefined (to the new audience) adjective:
delineated.  You have GOT to DEFINE ALL of YOUR terms.

> I have never restated the standard, George.

You have so, too, and you are so incompetent that you
didn’t even NOTICE that you were restating it WHILE you
were restating it.

>  I respect it.

Liar.  Nobody who "respects it" talks about "Frege’s delusions"
or about things being based on "the arrogance of philosophical
logicians".

> I just don’t believe that one can derive an epistemology for
> mathematics from it.

I just don’t know that anybody CARES about that.
When you know  a first-order proof of a theorem from
some first-order axioms, you CLEARLY know SOMEthing.
Whether the something you know is or isn’t "for mathematics"
is a question that mathematicians in particular would be MOST
likely to consider UNimportant!  And it is NOT like you are going to
get AWAY with caring MORE about "epistemology for mathematics"
THAN *mathematicians* do!  Finally, talk of "an" epistemology for
mathematics — as though ONE size COULD fit all — is just silly.
This is not something that ANYbody BUT you is looking for.

>  And, the more I look into the history of
> it, the more I see it

Bzzt.  Antecedent failure.
What is "it", here?   You last referred to "it" as "the standard".
I had discussed standard classical first-order logic.  You also
referred to that
as the "received paradigm".  Grammar check: it is NOT POSSIBLE,
linguistically, for a pardigm-for-logic TO BE "an example of poor
academic
discipline".  Academic discipline has to do with standards of peer
review and
clarity.  Nobody can attack this paradigm as having been insufficiently
clearly
defined or having been contaminated by incompetent peer review of
various
presentations of it.   The existence of this paradigm, and its general
acceptance
as well, are simply orthogonal to ANY question of "academic
discipline".

> as an example of poor academic discipline
> within the mathematical community.

Another dismissive insult that you cannot even define, just like
"logicism".
You are suffering from TERMINAL hubris if you mistake yourself for
competent to judge "academic discipline" of "the" mathematical
community.   You sound like James Harris.  There Is No Such Thing
as THE mathematical community.  This world’s mathematicians are,
if one is fool enough to try to unify them into members of ONE class,
like the elephant that the blind men were feeling.  Because an elephant
is so much bigger than a man, each of the blind men was feeling a
different
part of it and perceived it as a different KIND of entity.

> > This matters (SUBjectively, INternally) when you had to leave school
> > unsuccessfully.  And it needs NOT to be tolerated.  He needs to grow
> > up,
> > dammit.

> Ah, the usual personal insult…

Don’t be ridiculous.  You’re the one who started with "village idiots".
My replies to you before you pulled THAT stunt DON’T include personal
insults.  I CARE about the math so I will NORMALLY be distracted into
addressing IT.

> When I left school, I had professors enter "A’s" for classes
> in which I had requested "W’s" because of the quality of my
> work.

So what?

> There is a difference between illness and lack of success.

THat you had a lack of success was obvious.
How it interacted with some other entity’s approach to the status
of your student loans is unfortunate.  And since you have never had
integrity enough to explain what your illness was and how it affected
any of this, your bringing this up at all, in this context, is simply
inappropriate.

> What George means to say is that individuals who lack credentials
> should not venture opinions.

Liar.  I do not have any relevant credentials MYSELF.
I started doing this with nothing more than a B.A. in philosophy,
and the M.S. in Computer Science that I got 12 years later wasn’t
directly related to the kinds of foundational issues being discussed
here.
And I still don’t have a Ph.D.  So for you to accuse ME, of ALL people,
of credentialism, is just insane.  My actual position would be more
along
the lines of "fact is, inherently, contemptuous of opinion, in
general".
My point being that given that this is sci.logic, NObody should be
venturing
opinions and EVERYbody should be venturing THEOREMS.  If you haven’t
been able to work out a proof of a theorem yet then you should be
venturing
CONJECTURES, not OPINIONS.  Opinions are inherently just not worth
wasting ANYbody’s time on.  IN my OPINION.

> > Learning how to explain things to people is a survival skill.

> Absolutely.

Oh, shut up.  You don’t actually believe that.

> Learning how to explain things to pompous asses, however, is  not.

Commenting on the lack of academic discipline in the mathematical
community at large, or insisting that you have to understand that
it’s REALLY a free distributive lattice on three generators, is A HECK
OF A LOT MORE pomposity than I personally know HOW to have.

>>Insulting half of it
>>as
>>village idiots or flamers is deserving what you get.

> Actually, George, I was insulting you.  Very few
> people on sci.logic engage in your antics.

"History became legend, legend became myth … And some of the
things that should not have been forgotten were lost."
                             LoTR: "The Fellowship of the Ring"

Well, in this case some of the things that *should have not been
remembered* were rekindled! And it’s not any ring: it’s the phrase
"village idiot"! Imho, you should not have come back after a long
absence with that phrase in the opening post, *unprovoked*!

[Note: I'm not defending GG's foul-language usage, where it occurred
  in the ng.]

–
—————————————————-
Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
                                               Ryokan
—————————————————-

>Torkel Franzen wrote:
>>   Why do you think the manifestations of your disability are irrelevant?

>I DO NOT HAVE a disability regarding conversations of this type.
>YOU DO.

I’ve always felt that when Torkel claimed you just couldn’t
help it he was being remarkably generous. Maybe you should
think about not looking gift horses in the mouth?

(To save time and server space: yes, I’m a FUCKING ASSHOLE;
this is well known, no need to go into it again. Yes, what
I said above is just further PROOF that I’m a FUCKING ASSHOLE.
As is THIS paragraph.)

************************

David C. Ullrich

I DO NOT HAVE a disability regarding conversations of this type.
YOU DO.

> Do you consider the added matter irrelevant if somebody
> comes up to you and says COULD YOU FUCKING
> TELL ME WHAT TIME IT IS, > YOU FUCKING DIPSHIT?

Of course not.

>  Most people wouldn’t.

Nor do I.  It is the fact that you think (and I’m using "think"
VERY loosely here, since, OBVIOUSLY, you are a BETTER
thinker than this) that this situation is analogous to that one
that proves YOUR disability in this matter.

Somebody who walks up to you to ask the time has NOT
had PRIOR relevant interaction with you; you brought this up
as an example of a question that comes out of the blue.
All my cursing at mitch (in this cycle) was in reaction to
his having called ME a village idiot BEFORE  I had had a
chance to say ANYTHING about his latest opus.

In other words, the situations are NOT even REMOTELY
analogous.

But because YOUR disability, namely, BEING A TOTAL FUCKING
ASSHOLE, prevented YOU from seeing THAT, YOU have blessed
us with THIS turd.

When the smoke clears all I will be able to do is wish it MATTERED
more,
ANY of it

>            the elements of X are terms;

>            if p, q are terms, then p\/q,
>            p/\q, ~p are terms.

This is just basic standard propositional 0th-order
logic.  Nothing new is being alleged. Why you
feel the need to reformulate something this basic,
IN A STYLE EQUIVALENT to the original, is
mystifying.  To call this re-inventing the wheel would
be overpraising it.  It is also relevant that it violates
Occam’s razor; if you are going to have Vand ~ then
you don’t NEED /\ — IT IS *DEFINABLE*.

>    Our semantics is the class of orthomodular
> lattices OM.  A valuation v is a homomorphism
> from T(X) in some L of OM, i.e

>            v(p\/q)=v(p)\/v(q),

The V on the right is NOT the same as the V on
the left.  The V on the right is a lattice operator defined
by lattice axioms.  The V on the left is a pure syntactic
functor neither having nor needing any definition whatever.
But there is, obviously, a REASON why you spell them
with the same symbol.   Anybody normal would’ve insisted
that the logic and its truth-values were a lattice  TO BEGIN
with.   But you have gone through all this rigamarole about overloading
the symbol in a semantic AND a syntactic context, and (initially)
limiting the lattice to the semantics, onlyto REimpose lattice
structure on the syntactic side via the forthcoming axioms.

>            v(p/\q)=v(p)/\v(q),

>            v(~p)=v(p)’

But this is silly; you didn’t feel obligated to spell
the semantic version of /\ or \/ differently from the
syntactic one, so why do you NOW feel obligated to
re-spell the syntactic ~ as the semantic ‘  ?
If the first two are going to be clear from context then
the third might as well be as well.

>    The element p of T(X) is a consequence of
> a subset S of T(X), written as (S |= p) if, for
> all valuations v satisfying v(s)=1 for every
> s in S, holds v(p)=1.

One imagines that some sort of  completeness
theorem associating S |= p  with S |- p  is forthcoming,
but the sad part about all of this is that so far, apart
from insisting that valuations be "orthomodular lattices",
THIS MATCHES THE STANDARD.  IFone were going to be
THIS much in AGREEMENT with the received paradigm then
it might have helped TO JUST SAY SO.

>    Our syntax consists of a set of axioms and a
> rule of modus ponens.

No, it doesn’t.
Axioms are not related to syntax unless they
are defining new functors or predicates.
Inference rules are not related to syntax unless
one of the pre-existing functors in the language is
some sort of alias of consequence, or something strongly
correlated with it.

> We write pRq for
> (p/\q)\/(~p/\~q).  The axioms A are:

> A-01   xRx

This is just bullshit.
Propositional logic does NOT need 16 Axioms.

There is an outside chance that you MIGHT know what you are
doing here, but it is, as usual, I repeat like a broken record,
OBSCURED BY IRRELEVANT COMPLEXITY.

Re-spelling ~ as’  is irrelevant complexity.
Invoking 16 axioms WHEN ONE WILL DO
is irrelevant complexity.

 "george" <gree…@cs.unc.edu> wrote:
> Gerry Myerson wrote:

> > This is unpleasant. Do you suppose you could take it somewhere else?

> I am not the one who put "sci.math"  in the to-groups
> list. I  am just replying.

This is a very curious way to apologize for posting filth
to a newsgroup, but I accept your apology, and trust that
this episode will not be repeated.

–
Gerry Myerson (ge…@maths.mq.edi.ai) (i -> u for email)

  Why do you think the manifestations of your disability are
irrelevant? Do you consider the added matter irrelevant if somebody
comes up to you and says COULD YOU FUCKING TELL ME WHAT TIME IT IS,
YOU FUCKING DIPSHIT? Most people wouldn’t.

- Hide quoted text — Show quoted text -

 "george" <gree…@cs.unc.edu> wrote:
> EXACTLY NONE, dumbass.
> I DID NOT, asshole.
> The question, rather, is, if you know what the fuck you
> are talking about,
> Bullshit.
> As usual, QUOTE ME OR SHUT THE FUCK UP.
> I HAVEN’T SAID SHIT about the ability of FOL to subsume that
> question.  
> Dipshit: I AM STARTING FROM THE STANDARD context.
> NO, DIPSHIT:  THIS IS YOUR framework and YOUR paradigm
> Shit.  HOW THE FUCK would the likes of YOU know what MIGHT
> constitute a foundation for mathematics??  Who the FUCK do you
> think YOU are?????  
> You have not begun this way because you CANNOT, because you are
> WAY TOO FUCKING STPUPID to.
> I personally don’t give a fuck,

This is unpleasant. Do you suppose you could take it somewhere else?

–
Gerry Myerson (ge…@maths.mq.edi.ai) (i -> u for email)

- Hide quoted text — Show quoted text -

Gerry Myerson wrote:
> In article <1134333549.544180.44…@o13g2000cwo.googlegroups.com>,
>  "george" <gree…@cs.unc.edu> wrote:

> > EXACTLY NONE, dumbass.
> > I DID NOT, asshole.
> > The question, rather, is, if you know what the fuck you
> > are talking about,
> > Bullshit.
> > As usual, QUOTE ME OR SHUT THE FUCK UP.
> > I HAVEN’T SAID SHIT about the ability of FOL to subsume that
> > question.
> > Dipshit: I AM STARTING FROM THE STANDARD context.
> > NO, DIPSHIT:  THIS IS YOUR framework and YOUR paradigm
> > Shit.  HOW THE FUCK would the likes of YOU know what MIGHT
> > constitute a foundation for mathematics??  Who the FUCK do you
> > think YOU are?????
> > You have not begun this way because you CANNOT, because you are
> > WAY TOO FUCKING STPUPID to.
> > I personally don’t give a fuck,

> This is unpleasant. Do you suppose you could take it somewhere else?

I am not the one who put "sci.math"  in the to-groups
list. I  am just replying. But just for the record, quoting
a whole bunch of things out of context is slanderous.
What is supposed to distinguish the people who care about
math from the people who don’t is that the people who care about
math are more likely to focus on the math than on all the irrelevant
crap that YOU just quoted.

mitch wrote:
> How many times did you say that I needed
> to start with a logic?

EXACTLY NONE, dumbass.
I said you needed to start WITH SOME AXIOMS.
That the right logic was standard/classical/first-order
was OBVIOUS.

> You got one.

I DID NOT, asshole.
YOU DON’T KNOW the DEFINITION of what "a logic" is.

> As for indefensible, you would be the star
> of any debate team.  You have a philosophical
> fallacy for any understandable statement.  Why
> should anyone venture anything beyond
> a bare minimum.

Hardly.
The question, rather, is, if you know what the fuck you
are talking about, why don’t you JUST VENTURE the
bare minimum, NAMELY, SOME AXIOMS.

> Never the less, I have a  long memory.

Bullshit.

>  It took
> only a matter of a few posts by Torkel Franzen
> for you to concede the only substantive position
> you have consistently taken.

Liar.

As usual, QUOTE ME OR SHUT THE FUCK UP.
The comical thing about this is that I always quote you.
I always want to remind

> > There is chronically all manner of IRrelevant complexity.

> Irrelevant complexity.

> Well, why don’t you go back to what you want to
> claim about FOL.

I have certainly never claimed anything irrelevantly complex
about it.  Moreover, since it is the standard treatment, standard
treatments OF it CANNOT be irrelevantly complex.  THAT
much complexity is NORMAL.

>  See what it actually would take
> take to convince Torkel that your opinion is the
> correct–I  mean metaphysically truthful–opinion.

I couldn’t care less.  More to the point, you haven’t even STATED
the opinion in question.  I could always say that I abandoned it
long ago, once you do.

> From what I take from the "standard" to which
> you refer, "syntax" may be compared to a dog
> pissing on a tree.

No, YOUR EXISTENCE in this context could be compared
to a dog pissing on a tree. Syntax just is what it is, not
that you personally are competent to know.

> > The standard classical FOL paradigm ALREADY SUBSUMES
> > anything anybody MIGHT need to say about "a term algebra", in this
> > context, in the notion of  a first-order language.

> The dog has pissed.

Then it’s your dog and your piss, since YOU, by invoking
a term algebra, invoked essentially THE SAME thing
that first-order language invokes.

> For the record, I have always stated that my interests
> have to do with the foundations of mathematics and
> had been motivated by investigation of the continuum
> hypothesis.  What you say here about the ability of
> FOL to subsume that question is incorrect.

I HAVEN’T SAID SHIT about the ability of FOL to subsume that
question.  What I HAVE said is that the usual notion of  a first-
order language looks a lot like a term algebra over the same signature.
YOU YOURSELF ALSO decided to begin with a term algebra.
So any limitations that THAT imposes about investigating the
continuum hypothesis are ones that you also are going to have
to deal with.

> Instead, you defining your own context.

Dipshit: I AM STARTING FROM THE STANDARD context.
The burden of defining a new personal "own" context IS ON YOU.
THAT IS WHAT YOU are doing when you start ranting about
Kalmbach and term algebra.  If you are going to do THAT, you have
to MOTIVATE it.  You have to give people A REASON TO BOTHER
learning it.  You have to offer some clues as to how your context
DIFFERS, in a GOOD way, from the standard.  IF you are going to
start with "a term algebra" and "valuations" then you are going to
wind up looking substantively LIKE THE STANDARD paradigm with
a first-order language and INTERPRETATIONS thereunder.  And, far
worse, LOOKING STUPID because you don’t SEE that you HAVEN’T
said ANYTHING NEW, but have instead just rehashed THE SAME OLD.

>  I suppose
> that is what dogs do when they convert uric acid.

Comparing me to a dog is not a mathematical refutation of anything
I or anyone else has ever said.

> >Both here AND
> > there,
> > THE RELEVANT piece of defining info is A SIGNATURE.

> Yes.  This is the logicist claim that mathematical
> knowledge is grammatical knowledge.

NO, DIPSHIT:  THIS IS YOUR framework and YOUR paradigm
because YOU stressed the importance OF A TERM ALGEBRA.
Term algebras can be defined from signatures.  INCLUDING YOURS.
AS YOU presented it.  So IF there is a false logicist claim in doing
it this way, well, YOU JUST DID IT that way.
But that is not the point.
The point is that "logicism" simply does not exist.
You use it as a dismissive epithet but you have not understood
any of the various definitions it has had over the years in the
various philosophical contexts in which it could’ve had one.
Here and now today is NOT even such a context.

> Apparently, Brentano concluded that logic constituted
> a means of writing textbooks.

What Brentano personally concluded simply has nothing to do with
the way things usually are or what "logic" ACTUALLY means —
a question whose answer you personally are willfully ignorant of
in any case.

>  And, indeed, one needs
> to follow certain grammatical forms when writing textbooks.

Hardly.   But one IS following certain grammatical forms in doing
logic, since logic as WE know it is syntactic.

> That hardly constitutes a foundation for mathematics.

Shit.  HOW THE FUCK would the likes of YOU know what MIGHT
constitute a foundation for mathematics??  Who the FUCK do you
think YOU are?????  The lesson of the day was that ANYthing, almost,
"can constitute" a foundation for mathematics.  Set theory can,
category
theory can,  strings of 0′s and 1′s can.  There are GREAT MANY
frameworks
that are both "comprehensive" and "neutral".

> "look I really do understand this complicated stuff"

> Let’s try it this way.  Frege could not have come up with
> his delusions

No.  We cannot try it that way.
Nobody in the room but you believes that Frege had any delusions
about the important stuff.  Frege made a sort of minor error that
has since been corrected.  If you want to call the whole framework
deluded then you BEGIN with "I think Frege was deluded because x".
You have not begun this way because you CANNOT, because you are
WAY TOO FUCKING STPUPID to.

>  if there was not some sort of invariant associated
> with mathematical topics.

As you so crudely put it, "the dog has pissed".  What dog-piss
REALLY means in this context is irrelevant excessively complex
hypotheses, like the ether and phlogiston.  Mathematics is ABOUT,
among other things, INVARIANTS GENERALLY.  That fact itself is
NOT some SINGLE sort of invariant "associated with mathematical
topics".  Thinking there might be some ONE such thing is just stupid.

>  In the last six months of his life, he
> retracted his logicism and asserted that he had come to the
> conclusion that geometry was probably the foundational discipline
> of mathematics.

I personally don’t give a fuck, and if you can’t come up with a better
arguing style than you have thus far, you will never convince anybody
else to either.  You allege that "Frege recanted his logicism".  To
modern ears, that doesn’t even parse.  Read my lips: LOGICISM DOES
NOT EXIST.  If Frege had any delusions then thinking that logicism
was "retractable" was certainly one of them.  One of yours is that
it is relevant.

> Now, you want an opinion on FOL that isn’t based on the
> arrogance of philosophical logicians?

No, I don’t.  Nobody’s opinions about FOL are even relevant.
It just is what it is.  Anything important that anybody might want
to say about it IS A FACT, NOT an opinion, or IS A THEOREM,
NOT an opinion.