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



Nam Nguyen wrote:



H. J. Sander Bruggink wrote:

Charlie-Boo wrote:

You seem to be stuck in the status quo, unable to think for yourself.



Contrary to you, at least I *can* think.


It seems an irony that, for a shy-math-guy [C-B's email], C-B has
actually talked "quite a bit" about mathematical logic. [Although
a lot of what he said, like "Truth has nothing to do with models."
or "Godel has been dethroned.", seems to point to different
kind of logic than one most of us are used to!]

But that aside, the notion that a formula doesn't have a meaning
until it's interpreted in a model - as you mentioned below - is
*technically incorrect*. Now, sure, I think I understand what you meant;
and I also "sympathize" with a _convnetion_ that we're very familiar
with and have used often. But this is mathematical logic, not just
mathematics, and anything that's technically wrong at the foundation
of reasoning could "sink the ship of reasoning", naturally.


Theorem proving is meaningless playing with symbols. I could
use the following axioms

1 + 1 = 3
(x + y = z) -> (Sx + Sy = Sz)

and still be able to prove theorems with it. Why not? I just
have a set of rules, and nothing to check them with.


First of all, theorem proving is *not* a "meaningless playing with
symbols" or with formulae: theorem proving requires the *1st-order
meanings* of the formulae involved in the proof. Period. In fact,
the (1st-order) meaning of a formula, which is simply the assertion
that the formula makes, is guaranteed by the very grammar of the
formula's being a *wff* (*well-formed-formula*). In other words,
being a formula is synonymous with being well-formed, and all together
is synonymous with having a 1st-order (object) meaning. And this meaning
has nothing to do with any model-interpretation truth or falsehood
about the formula.

The models give meaning to the symbols. And only when they
have meaning, we can say that 1+1=3 as an axiom doesn't make
sense.



Secondly, technicality speaking, regarding to model-interpretation,
models don't yield any meaning at all to a formula: not any high-level
(natural) meaning and certainly not the (1st-order) object meaning.
What models yield is the truth or falsehood of the (interpreted)
formula. If we, any reasoning beings using FOL, see a "natural
meaning" through the truth value of the interpreted formula, that would
be *above and beyond what FOL framework would _sanction_*. [Put it
differently, using a Java terminology, the 1st-order meaning of a
formula is "final" and can't not be extended or "casted" into a natural
meaning of a higher level! So officially, we could not claim that
"models give meaning to the symbols (or formulae)".]

Thirdly, the two observations above point right at the very heart of the
edifice of what is known as First Order Reasoning: when we use a theory
(and its theorems) to reflect natural meanings or concepts, we do so
through models (interpretations); but when we construct a 1st-order
proof, we shall, beside the rules of inference, rely *solely* on the
syntactical formation of the formulae - and not on their natural
meanings at a higher level.

Fourthly, the third observation in the above paragraph hints that
there are a couple of prices (of using FOL reasoning) we've been
reluctant to pay (so far), but we'd probably have to eventually,
especially if we contemplate on the solutions of some outstanding
arithmetic problems such as Goldbach Conjecture (GC). In proving a
theorem, we have to depend on the syntactical formulation of formulae
involved (and not their natural meanings that we might be more familiar).
But formulae are of *finite* length; proofs are *finite* sequences of
formulae; and our knowledge - including mathematical knowledge - is also
*finite*. Now, we all would remember this familiar arithmetic dichotomy:
given m is different from n, then either m > n or n > m. For a
sufficiently "rich" theory such as an incomplete theory, say T,
there are infinite numbers of theorems increasing proof lengths.
Then at some points, there will be a theorem in which either its finite
proof length, or the finite length of a formula involved in the proof,
would exceed our finite knowledge throughout our entire existence
(which is also finite). In this case, we would tumble upon a known
finite formula without ever knowing what its proof is. This is what
I'd call as the passive-relativity problem of FOL reasoning: what
mathematical truth to be known, or mathematical proof to be proven,
would depend - and thus be relative to - the finite knowledge of
a reasoning being. And given there are (uncountably) infinite truths
to be interpreted, and (uncountably) infinite theorems to be proven,
and that we are finite being with mortal knowledge, there will be
truth unknown to us, and theorems unprovable by us. In addition,
contrary to our reluctance to admit, that we're free to choose one
out of (possibly infinite) different model-interpretations w.r.t. to
a given "structure" means truth or falsehood of a formula is a
"subjective" value relative to who is doing the model interpretation.
This is what I'd call as the active-relativity problem of FOL reasoning:
there is no absolute mathematical truth that FOL would sanction; and
all mathematical truths are subject to the beholders of the truths.

[to be continued...]


Fifthly, the "finiteness" foundation of FOL (finite length of
formulae and proofs, finite human-knowledge) means that in formulating
a formula, we have to *know* that formula: symbol-by-symbol.
Consequently, if we couldn't *know* or *state* a formula involved
in a proof, we technically and practically don't know the proof. But
herein lies the Achilles' point of FOL framework: it allows the
use of *existences of formulae* in our reasoning (e.g., generalized
inductive definition of formulae; axiom schema). The Achilles' point is
that if we know (how to spell) a formula then we know its existence, but
not necessarily the other way around! *Knowing the existence of a
formula may _not_ necessarily mean _we_ know what the formula is*;
and if the formula is involved in a proof, then that proof is
unknowable to _us_. [Though the formula might be known to a reasoning
species whose "finite cardinality" of knowledge is greater than ours.]

Let's take an example on the Achilles' point. Let L(0, 1, 2, ... n-1, S)
be a language with 0, 1, 2, ... n-1 be individual symbols, S be a
"successor" function. Let the following be axioms of a theory T on L:

A1) S(0) = 1 /\ S(1) = 2 /\ S(2) = 3 /\ , ..., /\ S(n-2) = n-1
A2) S(n-1) = 0

For abbreviation, let S0 = S(0), SS0 = S(S(0)), ....
Also, let Sk = SSSSSSSSSS....SS0 where there are k numbers of
occurrences of 'S'. Consider the formula:

(1) Sk = 0 (where k is an arithmetic number)

Now unless we claimed that we possess an infinite knowledge, we _must_
admit that there will exist an arithmetic k in which we can *not*
know the proof of (1), or ~(1)! This is a simple example illustrating
knowing only the existence of a formula would contravene the
"finiteness" principle of formulating a proof. As well, this example
also exemplify the passive-relativity problem mentioned above: we don't
know all the 1st-order proofs; and what proofs we could possibly know
would be limited by our finite knowledge, individually, or collectively
as a reasoning species. In addition, with existence of formulae being
admissible in reasoning, existences of theories are also admissible!
But the admissibility of theory-existences would be the dead end for
any absolute reasoning: it's possible that for a given formula, we might
not be able to know which theories, out of infinite existences of
theories (all of which share the same axiom-description), in which the
formula is provable. [In a later thread, we'll show that what is known
as "the" PA theory would, instead, have to be a collection of infinite
"PA" theories.]

Finally, we wouldn't be the first one who would introduce the concept of
"relativity" in mathematical logic. Godel did, albeit rather implicitly.
Before Godel's time, each mathematical sentence was supposed to be
reflecting an absolute that be provable - and *not be "interpretable"*.
But we know the rest of the story: we're free to "relativise" the truth
value for a given formula. But just as the collection of the sentences
G's doesn't include, say, the 5th Postulate, AC, GCH, ...., the sense
of relativity in model interpretation is not "comprehensive" enough.
It misses the fact that our ability to reason is limited to
(thus relative to) the finite knowledge that God has endowed us,
individually or collectively.

---Nam



groente
-- Sander



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