Re: what follows from denying an axiom

Prof. Menzel COULD have said this HIMSELF.
Or he could have referred me to the article.
At any rate, this would help (from the Stanford Encyclopedia of
Philosophy).
The irony of my ACTUALLY HAVING a philosophy degree from Stanford
and not having known this term ("general semantics") is probably not
lost
on anyone, but I would plead 1) Enderton is not from Stanford, and 2)
"general"
is EXACTLY THE OPPOSITE of a CORRECT term for doing THIS:

A key feature of the “standard semantics” discussed in the previous section is that,
for a one-place predicate variable X, the quantifier ∀X ranges over the entire power
set of the universe of discourse. We have seen that this feature gives second-order
languages a high degree of expressive strength.

Here, even HBE is implying that this is a feature OF THE LANGUAGES.


But do we really want the quantifier ∀X to range over the actual power set?

That's not really the question. The actual question is, IF IT'S
ANYTHING LESS,
then ARE YOU STILL even DOING *second*-order LOGIC??

The following is what Prof.Menzel wrongly accused me of being mistaken
or
confused about:
The concept of general semantics for second-order logic avoids
any pretense that the power-set operation is a fixed well-understood resource.

That's not the point. It is SIMULTANEOUSLY BOTH well and badly
understood.
All means ALL! WHAT PART of "all" don't you understand??
The implications are NOT well-understood, though; in particular, we
don't know
whether having absolutely all the subsets you can have is or is not
sufficient
to make sure you have one with a cardinality in between aleph_0 and
that of the
whole powerclass. THE POINT is that the NATURAL liftings of SOME
SIMPLE FIRST-
order axiom-sets BECOME CATEGORICAL if you know you have "all" the
subsets.
THAT IS WHAT MATTERS.

Instead, the range of the quantifier ∀X must be directly specified.

Specified HOW??
What are the legitimate languages are techniques that CAN be used for
that
specification??
More to the point, why is the requirement that the SUBstandard
SUBstitute for the
POWERclass be "directly specified" ANY more or less silly than a
requirement
that you can't do FIRST-order logic without having a DOMAIN that has
been
"directly specified"?

My eventual point is that the whole goal here simply has nothing
whatsoever
to do with SECOND-order ANYthing, LANGUAGE OR OTHERwise. The goal
is just to make things more tractable BY DESCENDING TO FIRST-order.

Here is what they tried:
By a general pre-structure for a second-order language

This here devolves to what Prof. Menzel was insisting was defensible
because
it was common; by direct analogy with the first-order case, we just
start interpreting
symbols to structures.

we mean a structure in the usual sense (a universe of discourse
plus interpretations for the non-logical symbols) together with the additional sets:

If these sets were specified in first-order ZFC then the whole
enterprise is going to
REMAIN subject to all the usual first-order ambiguity, but I imagine
the PARTICULAR
undecided statements will be a lot more obscure and complex and
therefore less
relevant; in particular, SOME things that were undecidable before the
addition of this
machinery may now get decided, EVEN if the appropriate encoding of
"this our whole
system is consistent" continues NOT to be one of them.

* The n-place relation universe for each positive integer n.
This must be a collection of n-ary relations on the universe of
discourse.
In particular, the 1-place relation universe must be some
collection of
subsets of the universe. Thus it is part (perhaps all, perhaps
not) of the
power set of the universe.
* The n-place function universe for each positive integer n.
This must be a collection of n-place functions on the universe
of discourse.

Wow. Plus omega*2 NEW FUMBLING SUB-aliases of "powerset".
This is THE OPPOSITE of "general". You are INTENTIONALLY SPECIALIZING
to the SPECIFIC w+w powersets that YOU HAD TO SPECIFY here! More to
the
point, how does any human ever specify this many powersets anyway??
And even more to the point THAN THAT, IF you are free to SPECIFY any
powerset you know how to, what's to stop you from specifying A FINITE
one
and winding up back down AT ZEROth-order logic, DE FACTO, REGARDLESS
of what "order" you CLAIM your "language" is?? As if anticipating
that, they try
But for second-order logic, we do not really want the 1-place relation universe
to be an arbitrary collection of subsets of the universe. We might not know
everything about the power-set operation, but we know some things about it.

When the smoke clears, "general" is going to equate to "closed under
definability";
in other words, at a bare minimum, the powerset has to include all the
"definable"
subsets, and all those that are implied by a certain class of
comprehension axioms.
Those comprehensions are about ways of interpreting the symbols into
the structures.
One result will be that
In effect, using general pre-structures amounts to treating a
second-order language as a many-sorted first-order language.
which is the first reason why I was so vitriolic vs. CM in dismissing
this as not
relevant to 2nd-order meaning.
The second reason is even more relevant.
Still quoting from the article:
We obtain the general semantics (also called the Henkin semantics)
for a second-order language by considering all general structures.
That is, for a sentence σ to be valid in the general semantics, it must
be true in all general structures.
This is a stronger requirement than saying that σ is valid in the standard semantics.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
THIS COMPLETELY UNDERCUTS
anything CM might have had to say about "fixity of meaning".
EVERYthing that comes up valid under the Henkin semantics WAS ALREADY
A CONSEQUENCE under the standard semantics. FEWER things are
consequences
under the Henkin semantics (but, fortunately, more than would've been
under a first-
order treatment with formula-schemata and NO predicate variables).

SO IF the problem is that when faced with a second-order axiom-set
(and ITS
associated language), we allegedly don't know what a term means
because it
might have DIFFERENT meanings under different semanticses, THAT
PROBLEM
CANNOT ARISE when the different semanticses under consideration are
Henkin and Full, because EVERYthing that is a consequence under the
Henkin
semantics IS GUARANTEED TO MEAN THE SAME thing under the full
semantics --
the Full semantics IS ONE of the structures over which the "general"
semantics
has to generally apply.

So CM was wrong on "fixity of meaning", but the actual point of
contention was
about whether this was or wasn't second-order logic AT ALL. His point
was that
it WAS over "a second-order language". MY point is that the fact that
you CAN do this
OVER "a second-order language" MEANS that the language ITSELF is NOT
sufficient
TO the title "second-order", IF something like THIS is even AVAILABLE
as an alternative.
Because if you use THIS, well, then, THIS happens:

The main feature of the general semantics is a result of the
“nothing but” type: Second-order logic with the general semantics
is nothing but first-order logic (many-sorted) together with the
comprehension axioms. Thus a sentence is valid in the general
semantics iff it is logically implied (in first-order logic) by the set of
comprehension axioms.

This reduction to first-order logic yields at once the following results:

* (Enumerability) In a second-order language with finitely many
non-logical symbols, the set of sentences that are valid in the
general semantics is computably enumerable.
This holds because the set of comprehension axioms is a
computable set (i.e., a recursive set).
* (Compactness) A set of sentences has a general model if every
finite subset has a general model.
* (Löwenheim–Skolem) If a set of sentences has a general model,
then it has a countable general model.

In each of these three cases, there is a sharp contrast to the situation of standard
semantics considered in Section 2. Moreover, a deductive calculus can be given
for second-order logic (adapted from first-order logic and augmented by the
comprehension axioms) that will be complete for the general semantics.

This is NOT an improvement over Godelian incompleteness; this is
complete
in the SAME sense as Godelian completeness (the calculus covers all
the
consequences, but the consequences are still r.e. as opposed to
totally recursive).

My point is simply that if this is the semantics you choose, then the
fact that
predicates are occurring as all three of bound variables, arguments to
functions,
and arguments to other predicates -- which is SUPPOSED to be the
DEFINITION of
what makes this language second-order -- IS NO LONGER SUFFICIENT to
make
ANY of this second-order. And if Chris Menzel doesn't want to take MY
word for it
then he can damn well take Herb Enderton's.
.

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