Re: Torkel Franzen on truth



MoeBlee wrote:
herbzet wrote:
MoeBlee wrote:
herbzet wrote:

MoeBlee's assertion that it is a theorem that every structure is a
model of some theory is a new thought to me.

What is difficult about it?

(1) By definition, M is model of a set of sentences G iff M is a
structure for the language of G and every member of G is true in M. So
every model is a structure, by definition.

Right.

(By the way, in another post I think I might have allowed G to be a
set of formulas. But I think G should be a set of sentences.)

OK.

(2) Every structure M is a model of the set of valid sentences in the
language that M is a structure for.

Yup.

So every structure is a model.

Of any validity, yes. Can we say further that every structure is a
model of a theory (other than the null theory containing only validities)?

I'm not familiar with the expression 'null theory' to refer to the set
of validities, but I'll go along with it.

I probably just made it up, although it seems right. It's the theory
with no non-logical axioms (ie only validities as theorems). It seems
to me that this is a degenerate case of the word "theory": usually
I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic
and to have some non-logical axioms/theorems.

As to the question, I'd have to think about it.

Bingo. That's what's "a new thought to me": that an arbitrary structure
is a model of some non-null theory. I don't happen to know whether
that's true.

Shall we agree as to what sets of sentences constitute "a theory" first?

For simplicity, let's confine to classical first order.

Authors such as Enderton take a theory to be any set of sentences
closed under entailment (which, thanks to the completeness theorem, is
a set of sentences closed under provability). Authors such as Chang &
Keisler take a theory to be any set of sentences. And some authors
take a theory to be a pair <S E> where S is a set of sentences and E
is the set of sentences entailed by S (which, thanks to the
completeness theorem, is the set of theorems of S).

I adopt Enderton's definition.

OK.

I usually think of "a theory" as having a recursive (or at least r.e.)
set of axioms.

That is a recursively axiomatized theory. There are theories that are
not recursively axiomatized.

Yes, they seem a little imaginary to me, but I'm willing to go along
with it.

And as being consistent (i.e. having a model)!

Those are consistent theories. There are theories that are not
consistent.

It would seem that there is but one such theory over any signature:
the theory that consists of all sentences in the language. This
seems like another degenerate case of "theory". I'm willing to
accept the informal locution "inconsistent theory"; I'm willing
to accept an "inconsistent theory" as existing under the Enderton
definition of "theory".

But the extremal cases of "theories" that contain only validities,
or that contain every sentence, do seem a little silly to me. I
guess it's the price you pay for definitional elegance.

Even if it is so,
it does seem to me to be ill-advised to use "structure" and "model"
interchangably (although I'm as slack as anyone else on using
"structure" and "interpretation" more or less interchangeably).

In such contexts as I mentioned, what is the harm of using 'model' and
'structure' interchangably, especially the context I mentioned?:

If I recall correctly, this started when you said to Nam:

Okay, in a technical sense, '2=1+1' is true relative to models because
it's true in some models but not in others.

To which George objected:

No, true in some interpretations but not in others.

which might seem like a rather pedantic distinction (or, as you seem
to think, a false distinction), but I think that in this area with
Nam you have to be very precise, and the distinction is merited.

Anyone may state definitions and explicate a discussion on the basis
of those definitions. Meanwhile, what I said is precisely correct
given ordinary defintions:

'2=1+1' is true in some models and not true in other models.

That is precisely correct.

What is cloudy is the terminology 'true relative to models', which is
Nam's terminology. I don't use that terminology. All I said (or meant
to convey) in the passage you mentioned is that I could see a sense in
which that terminology could be understood.

So, again, to be clear:

My terminology is 'true in a model' and it is precisely correct that
'2=1+1' is true in some models and not true in other models. Nam's
terminology is 'true relative to a model', and though I do not in any
way claim to arbitrate what HE means by that, my point was just to say
that IF he means 'true in a model', then yes, of course, '2=1+1' is
true in some models and not true in other models. Then, there were the
rest of my remarks.

Yuh, I guess the argument turns on whether you want to affirm

a) 2 = 1 + 1 is false in no model (of PA)

or

b) 2 = 1 + 1 is false in some models (of the language of PA).

It seems to be a simple ambiguity to be resolved. I personally would
prefer (a) and would assume that you are using "model" in the sense
of (a) in the absence of an explicit definition otherwise. I concede
that it is possible that Nam uses the word "model" in some sense
other than (a).

I acccept that the literature allows the usage of "model" in the
sense of both (a) and (b). But, as I said before, this seems ill-
advised, in that it allows this ambiguity to arise.

(1) M is a model for a language

(2) M is model of a set of sentences

(3) M is structure for a language.

(4) M is structure of a set of sentences.

Especially, when the precise definitions are given.

(2) and (3) are more common than (1) and (4), though (1) is found in,
for example, Chang & Keisler, though, (4) is admittedly rather odd
sounding and therefore I don't use it, but, as long as my definitions
have been clearly stipulated, it wouldn't be harmful if I did use (4)
even though I don't prefer it.

Sure, as long as you clearly stipulate your definitions, no problem.
I think the default usages are, as you point out, (2) and (3).

More common, not necessarily default.

Well, time will tell, perhaps.

It amazes me that some people can just tune into my
wavelength (if they feel like it) while others must insist
that I'm just evil. [posted by george]

Like anyone else, when I'm in an argument I'm inclined to reject
EVERYTHING my opponent says, no matter how innocuously and obviously
true some of it may be. This is so obviously a form of ad hominem
(If this jerk says X, then X must be false) that it's particularly
embarassing, as a psuedo-logician, to fall prey to it. It's rhetorically
bad, too, to be caught denying what's plainly true.

Except we don't have an example of anyone disagreeing with George
simply because he is otherwise a royal jerk.

On the contrary, I think that people in this forum tend to be argumentative
when they are contradicted. They often will not take a brusque correction
with equanimity. They will put uncharitable and even unreasonable
constructions on what has been said to them. They will move heaven
and earth to show that they were not, in fact, wrong. Do you want
documentation? I'd prefer not to name names. Also, that would be
a very tedious chore.

I'm not interested in such tedium. But to be convinced I would have to
know what examples you have in mind specifically regarding George,
since I don't know of an instance of someone disagreeing with George
merely for his being a jerk.

Such instances would, of course, require an inference as to someone's
motives, since no one's going to assert that they are disagreeing just
to be contrary. That will remain, irredeemably, a matter of opinion.
I'd rather just state my opinions and leave it at that.

I'd like to take this opportunity to say that George has never been
a jerk to me. Of course I attribute this to his acute perception of
my sterling character, but it's more probably that I'm not edjicated
enough to merit abuse. I hope one day to be smart enough to rate
an "oh, SHUT UP" from George. In general I find his explanations
of things to be quite patient and lucid.

I find his explanations usually to be bizarre.

Not usually, but sometimes. Usually there is a core of something
insightful even in the seemingly bizarre assertions.

Of course, he is occasionally wrong. So what?

I think he's more than occasionally wrong, and worse, he's a jerk
while being wrong.

Actually, I find it even more annoying when someone is being a jerk
while being right!

--
hz
.

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