Re: what follows from denying an axiom

On Sat, 17 Jan 2009 13:09:38 -0800 (PST), george <greeneg@xxxxxxxxxxxxx>
said:
On Jan 16, 8:39 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
So writing something down in a second-order language doesn't
guarantee that it has a fixed meaning; you have to stipulate that it
is to be interpreted by standard second-order model theory.

The choice between alternatives semanticses for 2OL ABSOLUTELY IS NOT
analogous to the choice between alternative models of a first-order
theory.

With respect to claims about *what you mean* by a certain sentence, I
believe the situations are rather closely analogous.

The higher-semantics choice is a much bigger, broader, over-arching or
fundamental decision, as well as one among a MUCH SMALLER number of
legitimate alternatives. In FOL, you DON'T specify the intended model
because IT DOESN'T MATTER:

I just have no idea what you are talking about. It looks like a
transparently false empirical claim: I and many other people *do* in
fact specify the intended model of, say, the language of PA or ZF all
the time. As for it not mattering, well, that really depends on one's
purposes, doesn't it? It's hard to understand, for example, work on
nonstandard models of PA if one doesn't specify the intended model. So
specifying the intended model in the context appears to matter.

everything that is valid or contradictory, i.e., determined by the
LOGIC at ALL, is going to be THE SAME in all models.

That is true.

The model only MATTERS for NON-theorems.

Also true.

In the 2OL case, by contrast, what COUNTS AS a theorem in the FIRST
place is going to be affected/determined by the choice. So that is
clearly a deeper choice.

It may in some sense be a deeper choice. But again I was only talking
about the narrow issue of "what you mean" by a given sentence.

One could also insist that you are grotesquely abusing the term "model
theory" if you are talking about alternatives for higher-order
semantics.

One could, but one would thereby be insisting on something that is very
silly.

The classes of things we are talking about THERE are basically too
big to be sets and therefore too big to be models.

What?

FOL in any case CANNOT HAVE a model theory

This is, well, nuts. What is Chang & Keisler, or Hodges, about?

BECAUSE first-order consequence both 1) is model-INdependent --
results derived in a logically valid manner are true IN ALL models,
and things that are provably logically contradictions have NO models
-- and 2) is characterizable PURELY syntactically withOUT ANY appeal
to semantics AT ALL.

I know you know that first-order consequence is a purely model theoretic
relation. So your argument here seems to be self-refuting.

In FOL you have to invoke model theory to prove that things are NOT
consequences.

And that they are, at least to the extent that one has to appeal to the
completeness theorem.

That is sort of an inherently "meta"- AND THEREFORE HIGHER-order
consideration IN ANY case, which is why you have to use set theory,
and as I am not the first to notice, doing set theory in FIRST-order
logic absolutely IS higher- order logic masquerading as something
lower.

Perhaps others have noticed, but I myself don't even know what it is
they noticed. I do not know of any sense in which first-order set
theory is higher-order, and I don't know of anyone who's made the claim.
You aren't thinking of Quine's claim that second-order logic is set
theory in sheep's clothing, are you? That is a very different claim
than the one you make above. Quine was simply arguing against the idea
that second-order logic is *logic*.

Worse, though, In the first-order case, you have a much deeper prior
problem: you have characterized a first-order LANGUAGE as being
finitary (well, it's infinite, but every formula and term IN it is
finite). Managing to convey that is NOT fundamentally POSSIBLE,
DESPITE the fact that it is factually happening.

Looks to me like you very successfully conveyed it.

The point being that finitude is NOT first-order definable and you
therefore CAN'T EVEN SAY WHAT YOU MEAN in your a-priori
characterization of the language.

Well, again, this revolves around what one thinks it means to "say what
you mean" in a first-order language. When I use the language of
first-order set theory, I'm talking about the sets in the set theoretic
hierarchy (perhaps with uruelements). And the definition of "infinite"
in set theory picks out exactly the infinite sets therein. The fact
that there are nonstandard models of set theory -- though deeply
interesting -- just seems to me entirely irrelevant to the fact that
when I use set theory to talk about infinite sets, I'm doing precisely
that; I'm saying precisely what I mean. You seem to think that the fact
of nonstandard models undercuts my ability to do so. I just don't see
why -- I've never found your arguments for the claim in the least bit
convincing.

The short version of the competing problems is this: FOL requires you
to know what "N" means. 2OL, by contrast, only requires you to know
what "all" means, because the 2nd-order quantifiers are supposed to
range over "all" subclasses of the 1st- order universal class.

I won't dispute that.

The two problems are simply not analogous

I certainly agree they are not the *same*, but analogies can be pretty
weak and still be analogies.

because Everybody ALREADY KNOWS what they think "all" means.
Stipulating that "all" will have a standard meaning is INHERENTLY
SIMPLER than stipulating that you will mean "N" when you CAN'T
CHARACTERIZE N *even* with help from all of FOL. "All", UNLIKE "the
infinite set of natural numbers", IS A SIMPLE enough concept that
EVERYbody can understand it. And you see the same retreat in the
first-order case when the quantifiers range over "all" elements OF THE
DOMAIN, EVEN WHEN WE DON'T KNOW what the domain IS: we STILL know what
"all" means.

Well, that's an interesting argument, but all it shows is that
understanding the numbers requires taking a few conceptual steps beyond
understanding "all". Once taken, however, I don't see any reason for
thinking that one can't know what one means by "the natural numbers" as
well as what one means by "all".

.

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