Re: what follows from denying an axiom

Chris Menzel wrote:
On Sun, 01 Feb 2009 01:02:20 -0700, Nam Nguyen <namducnguyen@xxxxxxx> said:
Chris Menzel wrote:
On Sat, 24 Jan 2009 13:57:07 -0700, Nam Nguyen <namducnguyen@xxxxxxx>
said:
george wrote:
...
Chris Menzel is the first person I have ever heard say "second-order
language".
Would he then be the one who wrote the content of

http://plato.stanford.edu/entries/logic-higher-order/
I wish!
I'd like to thank you and George for clarifications on this link.

I did write this one, however:

<shameless plug>

http://plato.stanford.edu/entries/actualism

</shameless plug>
Thanks also for this . [For what it's worth, I've been pursuing
what I'd call as "mFOL" (meta FOL) and these 2 links provide some
useful information, though I'm still "dwelling" on both of them,
being slow!].

Glad you found them helpful.

As in your response to Daryl, I think it's _syntactically_ possible to
tell if the reasoning is in fol or sol.

Reasoning was never an issue in my response to Daryl. My claim was only
that it is possible rigorously to define the notion of a second-order
language purely syntactically.

As for different "semantics" of "quantifying over..." individuals of
different sorts in SOL, I think we've opened a can of worm here.

As implied by Hilbert (or even Shoenfield), _formal_ system is sheer
syntactical and is *self-sustained* in the sense that reasoning with
it can be carried out *only with its axioms and the inference rules*:
one doesn't even have to know of any model at all, to infer a theorem.

That is of course quite correct.

Then syntactical reasoning is in effect just a mind game (i.e. in the
meta level) of syntax/symbol manipulation; just like when we take a
syntax-pattern-recognition test, from say the NSA. So, to paraphrase
what Boromir says of Gondor ("The Lord of the Rings"), we could safely
state:

Formal system has no semantics. Formal needs no semantics.
(Or no interpretation for that matter).

Whether or not the formal system is of 1st or 2nd order!

Again, that is quite true. The difference between the two, of course,
is that no purported system of second-order logic can be semantically
complete; for any such system, there will be second-order validities (in
standard second-order semantics) that are not theorems of the system.

Not only we could choose between Standard and Henkin semantics for
quantification of "individuals" of different sorts in SOL, we could
also come up with different semantics for the FOL symbols 'A' and 'E':
semantics that *has nothing to do with quantification* at all!

Well, yeah, sure, we can assign any semantics we want to the symbols of
a language.

All of which means semantics or interpretation is subjective

Now that doesn't follow *at all*.

It actually does, for 2 fundamental reasons.

If I announce that I am assuming
Henkin's semantics to interpret a second-order language, my semantics is
from that point forward entirely objective in the given context.

The 1st reason is that it conforms with the definition of relativity (hence
subjectivity). In any relativity/subjectivity there's a fixed/absolute
component that's invariant to all the concerned individuals; and there's
a component that would vary from one individual's perspective to that
of another. The combined effect of the 2 type of components would rise
to the resultant perception-phenomenon known as relativity, of course.

In this context, the syntax of the quantifier is the invariant component
and the semantic of that syntax is relativistic one. By the time we
choose a specific semantic, Henkin's for example, it's already too late:
a relativity of context or frame has already been subjectively chosen.
(Similarly, even if we say that relatively to the train the speed of
a walking man is absolutely 3 km/hr that still doesn't make the statement
"the walking man's speed is 3 km/h" absolute: simply because all speeds
with the exception of that of light are relativistic).

What
*is* the case is that a semantics doesn't "attach" to a given language
by magic; some such announcement is required. But once it is made in a
given context, the semantics is, within that context, as objective as
can be.

Again, the fact we require such announcement, imho, is a proof that
semantic is relativistic; we've subjectively specified a framework/context
where "true" statements of the language would be made; they'd be "false"
if different "announcement" is committed - by the individuals in question.


and hence is a weaker notion than that of syntactical provability.
Weaker in the sense that if a formula is a syntactical theorem, it's
always a theorem, while if a formula is true under some semantics or
interpretation, it could still be false under others.

I'm not sure I see the contrast. One has to define a proof theory for a
language L no less than one has to define a semantics. The same sentence
of L might be provable in one system and not provable in another.
Provability is thus relative to a proof theory, just as validity is
relative to a semantics.

I'm not sure why one "has to" define a proof theory but let me explain
the contrast. For a fixed axiom-set T, a formula F is either provable or
not provable, and there's no subjectivity no dependency on any individual
at all. On the other hand given a structure-set and in relation to a T,
one would have a freedom to choose to interpret F as true or false.
(For example, all one has to do would be interchanging the semantics of the
quantifiers, and the validity of F, in what's supposed to be a model of T,
would be altered!).

And this sheer freedom to choose different semantic of syntactical formulas
and symbols is the 2nd kind of reason why semantic and interpretation is a
weaker notion than that of provability. In other words, there's no absolute
truth that's invariant in all (subjective) interpretations of syntactical
symbols and/or relations of individual objects.


I'd let go SOL (or any higher order) reasoning all together. And in
its place we could introduce mFOL (meta FOL). The idea of mFOL is that
in the meta level, an n-ary non-logical symbol could in fact be a
variable that might range over more than one constant-symbols. And in
the context of mFOL we could introduce:

(1) The concept of "global" theorems or provability.
(2) The concept of unknown-ability to mathematical reasoning,
making our reasoning a bit more humble than it has been
(since GIT).

Imho, both of these concepts would reflect more of a reality of difficulties
in mathematics that we've been facing. FOL, SOL and the like only mask the
reality.

I don't see the problems with FOL and SOL that you seem to, but perhaps
your approach could prove fruitful as well.

Imho, the 2 areas where FOL is weak and mFOL (meta FOL) would help are:

(a) Description-ability of AI.

mFOL language could serve as language for AI (as you seem to have
alluded to below).

Basically, 1st order language is designed for describing relationships
among inanimate objects (e.g. sets, space-time points, ...) which are
devoid of any transcendental qualities possessed by a reasoning being.
But (AI) transcendental attributes require *transcendental individuals*
to recognize them (the attributes) as transcendental.

By disallowing fixed constant symbols in FOL language and replacing them
with variables, mFOL would force the semantic, truth, and even provability
of formulas to be in meta level. As such, description of objects would
be at a more *transcendental* level than the 1st-order one; and this
would be more conducive to our ability to describe AI attributes.

(b) Description-ability of Relativity.

Our inability to know the truth of every (say, arithmetic) statements
could serve as a technique of mathematically describing relativity
akin to SR. That's to say if in the meta level we accept certain
principles governing these inabilities (or unknown-abilities).

One example is the formal system rN (Relativistic Natural Number)
where the language would comprise of *non-finite* n-tuples of non-logical
symbols (0,S,+,*,<)'s and where the symbols '0', 'S', '+', '*', '<'
are actually symbol variables. [Note also my emphasizing "non-finite".
Here we'd demote the importance of the so called the naturals, or the
standard model of, say, PA. We'd need only to understand "finite" as
a priori and what the nagation "not" means in meta level. Generic infinity
could be understood and defined as being non-finite.]

The motivation here is that so long as FOL doesn't prevent the existence
of an "absolute undecidable" F over the infinite number of sub-systems
N's, then by definition we'd see in meta level that the following statement
is true:

"If F is provable in one subsystem N, it would be unprovable in another
subsystem N'"

Consequently, from the interpretation/model perspective, over the very
*same universe of individual numbers*:

"If there's a model M in which F is true, there's a model M' in which F
is false".

And this is the very spirit that we'd see in SR. Note though the
emphasizing that these opposing models here are of the same universe
of individual numbers; this would make a distinction between a Godel's
undecidability and an absolute undecidability.

Note also that at this moment GC still seems to be a candidate of being
an "absolute undecidability"

You might google "metalevel reasoning" for some work in AI that
you might find relevant.

I take this as a good advise. Thanks.

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")



.

Relevant Pages

  • Re: what follows from denying an axiom
    ... language purely syntactically. ... Formal system has no semantics. ... to the resultant perception-phenomenon known as relativity, ... Provability is thus relative to a proof theory, ...
    (sci.logic)
  • Re: what follows from denying an axiom
    ... to do with SECOND-order ANYthing, ... is just to make things more tractable BY DESCENDING TO FIRST-order. ... second-order language as a many-sorted first-order language. ... That is, for a sentence σ to be valid in the general semantics, it must ...
    (sci.logic)
  • Re: what follows from denying an axiom
    ... Chris Menzel is the first person I have ever heard say "second-order ... Reasoning was never an issue in my response to Daryl. ... language purely syntactically. ... Formal system has no semantics. ...
    (sci.logic)
  • Re: what follows from denying an axiom
    ... have almost exactly *my* characterization of Henkin's semantics! ... "second-order language" can be defined completely syntactically. ... meaning than the use of a first-order language, ...
    (sci.logic)
  • Re: Second-order and Higher-order Logic
    ... admits that it is basically talking about the Henkin semantics. ... by first translating second-order language into a typed first-order language. ... So there are two distinct logics here: ...
    (sci.logic)