Re: Torkel Franzen on truth

On Jan 4, 2:44 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:

For what it's worth, I actually don't use a different definition of the
word "model" here, in FOL context. The phrase that seems difficult to some
to understand is:

(1) 'true relative to a model'

Now suppose you believe PA is consistent and let M be a model of PA,
then the following meta level statement would be true:

(2) '2=1+1' is true in M.

We don't need 'belive' there. We could just say:

If M is a model of PA, then '2=1+1' is true in M.

The question though, if we change the inference rules and come up with
a different reasoning framework, say (FOL)', then would (2) still be
true?

Yes.

The inference rules don't alter what sentences are true in what
models.

What the inference rules change is what theories we get.

First order PA is DEFINED to be the set of sentences that are entailed
in classical first order logic from the first order PA axioms.

With a set of inference rules that doesn't yield the same theorems as
classical first order logic you get a DIFFERENT theory from PA, you
get HA (if the rules are intuitionistic logic) or whatever you want to
name each DIFFERENT theory depending on a different logic.

The answer is "Not necessarily", depending on what changes of
rules of inference that have taken place of course. Now if we consider
the formal system PA as just a *collection of axioms* then syntactically
PA is the same.

But we DON'T consider PA as just a collection of axioms. We consider
PA to be the set of sentences entailed by that collection of axioms.
(Or some people say it is the pair <A T> where A is the collection of
axioms and T is the set of sentences entailed by A.)

But what we consider as M might or might not be an absolute
model of PA, right?

What does "absolute model" mean?

In that context then, '2=1+1' is being true in M is
relative to the fact M might or might not be a model, depending on who's
doing the reasoning. In that context, then the following statement
would make sense:

I'd grant that it depends on how we're doing the reasoning in the META-
theory, since 'true in the model M' is defined in a meta-theory for
first order PA and our proofs of whether something is true in the
model are either done in the meta-theory or we rely upon the soundness
theorem (which also is proven in the meta-theory) to infer that what
is proven in the object theory is true in every model of the axioms of
that object theory, as well as it is in the meta-theory that we prove
that axioms are true in whatever models we claim the axioms to be true
in.

But that's not what you're talking about. Your present line of
argument is, as usual, confused and ill-premised.

(3) '2=1+1' is true, relative to a model.

which is not different in nature from the statement:

(3') the speed of the train is 100 km/hour, relative to the framework M.

At any rate, '2=1+1' is no more absolute than 'the speed of the train is
100 km/hour'.

Those who believes otherwise would not realize the relativity nature of
human mathematical reasoning (through FOL at least).

Now there is another "more technical" way to demonstrate the model-relativity
connoted in (3), and I did allude to this way a couple of times in the past.
Basically, if we consider a FOL system sPA ("super" PA) whose languages
contains *infinite* symbols:

L(0, S,+,*,<, S',+',*',<', S'',+'',*'',<'', S''',+''',*''',<''', ...)

In other words, besides symbol 0, the rest would be grouped together and
each group, when coupled with 0, would form a language we could use to
formalize a theory we'd name "PA". [This alone would signify the relativity
of "PA" and associated models - and (3). Wouldn't it?]

NO! That's silly! Just CALLING something 'PA' doesn't show any
relativity other than the quite prosaic sense we all already know that
if you say "Sally Fields played the role of James Bond" then that is
true if by 'Sally Fields' you are referring to the person Sean
Connery.

In some given overall mathematical context, such as a particular set
theory to serve as a meta-theory, we DEFINE PA to be a certain exact
mathematical object: a certain exact set of finite strings of symbols.
Once we make that definition, it's just silly to worry about what
happens if you say, "Oh, we get something different if we use 'PA' to
stand for some other thing."

The (infinite number of) axioms of sPA would be the union of those individual
axioms per each ("PA") group mentioned above.

Now let's examine the anatomy of a model M of a general FOL formal system T.
In a nutshell, the major components of M are:

c1: a set S of individuals of which certain n-ary relations would exist.

Okay, a non-empty set.

c2: a collection of n-ary relations, each of which would correspond to an
     n-ary symbol of the language.

Hmm, I suppose that's okay. But I'd rather say, a function that
assigns to each n-ary predicate symbol of the language an n-ary
relation on S.

c3: collection of (subjective) interpretations, each of which would predicate
     a theorem-formula as true.

YOU want c3 for whatever odd reason you do. I have no use for c3.
Through c2 you had a nicely rigorous mathematical definition going,
but then you obliterated it with the UNDEFINED terminology "subjective
interpretations" and "would predicate a theorem-formula as true".

Now MoeBlee suggested above:

   "Anyone may state definitions and explicate a discussion on the basis
    of those definitions."

Sure. Though that was not intended to disallow that we may find
certain definitions and explications to clash so strongly with
ordinary terminology as to be irritating to work with.

So in this context here and now, let's call S a "structure" (and temporarily
forget if some text books reserve this word for something else).

So, a 'structure' is defined by you now to be what we ordinarily call
'the universe' or 'the domain of discourse' of a structure.

What is the advantage of you so confusingly switching these
defintions?

Then it's
not hard to see that the relativity of a model M would come from component c3!

Since c3 is UNDEFINED NONSENSE, it's hard to see what comes from it!

For example, if the formula is "a < b", S = {a',b'}, and the n-ary relation
is {(a',b')}, I still could at my subjective willingness interpret (or predicate)
"a < b" as false while you or any other as true.

So who the L cares about that?

We already have a formal mathematical definition of 'true in the
model', but you'd rather we use some junky UNDEFINED nonsense about
"subjective interpretation" instead. What possible advantage comes
from that?

And so relative to whose
predicating or interpreting, M would be or *not* be a model of say T = {a < b},
or for that matter of T = {~(a < b)}.

Now back to sPA, let S be the structure (i.e. the _set_) of individuals of a
model of the integers (i.e. not the natural numbers). The long and the short
of it is out of S, we know there exist _uncountably many_ "successor" functions
S()'s, hence uncountably many n-aries "addition", "multiplication", and "less-than".
Put if differently, not only S is a structure for sPA, it would be the very same
structure for _uncountably many_ models of each "PA" theory (written in L(sPA)).
But it's not hard to demonstrate that due to the subjective interpretation in c3,
if one interpret S as a model of a "PA", others might disagree and (re)interpret S
as *not* a model this "PA".

Forget about whatever involves c3. If you'd have me adopt your c3,
then I might as well have you adopt: A sentence is true in a model M
iff one of the snails in my garden leaves, on the walkway, a trail
more than 4 inches on the second Wednesday of the next month.

Of course when we talk about "PA" we typically talk about it outside the context
of this orangutan sPA system. But it should not matter! Given *any* L(PA),

No, there is no "any" L(PA). L is an OPERATION. For each theory T,
there is exactly ONE object that is L(T). For each theory, there is
exactly one object that is THE language of that theory.

one could
consider it as part of L(sPA).

No problem with L(sPA) as different from L(PA) if sPA and PA are
different and happen also to have different languages.

And given a model - over a structure S - of *any*
"PA" theory one could interpret this structure S as a non-model of this "PA".

In summary, a model is always (at least) *relative* to:

In summary, apply c3, then check the snail trails each first Wednesday
of the month, then take the Boolen product, then toss a coin for an
arbitrary number of trials, then disregard having done all that, then
declare the "relativity of predicating", then take deep inhalations
from a bottle of model airplane glue and forget about everything
whatsoever.

MoeBlee
.

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