Re: truth/falsity of sentences in first-order logic

Nam Nguyen wrote:
H. J. Sander Bruggink wrote:

Models (or rather, the interpretation of symbols of the
language to objects and relations of the model) *do* provide a completely formal way to seperate language and meaning.

What are you really talking about here? a) I only said formulae,
*not* languages, having 1st-order meanings; b) a language L itself
has no meaning at all (1st-order or otherwise): therefore there is
no need to separate a language _from_ (its non-existing) meaning.

Are you serious? A language is a set of formulas, and
these formulas can have meaning (e.g. by using models).


But formulas an sich don't have first-order meanings. They are
created according to grammar rules.

Consider the following language:

* the letters a and b are formulas;
* if F is a formula, and G is a formula, then (F#G), (F.G) are
formulas.

Now, what is the first-order meaning of the formula (a#(b.a))?

Now, that doesn't look like a wff: what parts of the "following
language" above are the symbols '#' and '.'?

Of course it's a wff, given the grammar above. And the
hole point is of course, that the grammar doesn't say
whay '#' and '.' mean.


That aside, suppose
that is a wff, then its 1st order meaning is: a#(b.c).

Wow! The meaning of a formula is just the formula itself.
How insightful...


[Hint: I never
say said we could always nicely translate a 1st order meaning into
a higher order one!]

I have no idea what you're talking about. What is a
higher-order meaning? (Hint: I have never seen the words
"first-order" and "higher-order" w.r.t. meanings, but
everywhere where I encountered these words, "higher-order"
*includes* "first-order", and so the translation of
"first-order" into "higher-order" is trivial.)



What assertion does the formula (a#(b.a)) make?

Again, it asserts: a#(b.c).

Illuminating...


In other words,
being a formula is synonymous with being well-formed,

Yes.

and all together
is synonymous with having a 1st-order (object) meaning.

Not at all.

Why "Not at all"?

Because "being well-formed" is synonymous to "being part
of the language". No meaning involved.


And this meaning
has nothing to do with any model-interpretation truth or falsehood
about the formula.

The meaning does not exist if you only have a grammar. Period.

Again, why?

Have you ever *looked* at a grammar?


Indeed. Syntax and semantics must not be confused. But that
*is* what you're doing.

That's not what I'm doing:

Yes, that is what you're doing.

I'm *not* talking about natural "semantics"
of formulae, just so you've read my post correctly.

If you *were* talking about natural "semantics", then
at least you were possibly making some sense. But this
is a vague notion of semantics, which has no place in
formal logic.


[snip paragraph about human knowledge being finite, and
theories proving infinitely many theorems]

Formal logic doesn't concern itself with human limits. One

That was what people before Godel said.

Indeed, and they still do.

They believed that there
were not limit in reasoning and given time *any* mathematical
sentence would be provable!

Note how I wrote "*human* limits". Godel proved nothing
about *human* limits.


One specific point: "Formal logic
doesn't concern itself..." is personifying an object (formal logic
in this case). Personification is OK to use (I myself has done
that too), but one must remember in this case that formal logic
is a product of our mind to use: as such we ought to concern with
its limit - that's our limit, for crying out loud!

Not at all! I can envision things that surpass my own
limits. For example, altough my brain is finite, I know
what "infinity" is. It's the same with logic: perhaps
the *rules* are a product of our own mind, but this does
not prevent that in principle things can be proved that
we will never practically be able to prove.


(Do you think
we're limitless in term of knowing all mathematical truths?
Would you happen to know all the 1st digits of *all* the prime
numbers? I don't, so I "personifyingly" warn formal logic that
it ought to concern with my limits.

Just because we have "limits", doesn't mean we have to
incorporate those in the things we think of.

There are *countably* infinitely many truths and *countably*
infinitely many theorems.

Pi is transcendental; there exists one truth then. "e" is
transcendental; there exist one truth then. How many transcendental
would you say there exist?

Formulas are finite, and thus there are only countably
infinitely many of them. The true and provable formulas
are subsets of the formulas. Qed.

groente
-- Sander
.

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