Re: Another way of expressing the difference between first order and second order languages?


andrewspencers@xxxxxxxxx wrote:
> It seems to me that

WHAT IS WRONG with you??
WHY are you always talking about
how things SEEM TO YOU? All of
these concepts HAVE DEFINITIONS!
Why don't you just LOOK THEM UP??

> in a language with logical constants (including
> quantifiers and sentential connectives),
> variables, and nonlogical constants,
> the language is first order if

There is a very clear foundational definition
of what a first-order language is. More to the
point, langauges are not the only things that
can be first-order. LOGIC can ALSO be first-order
(or higher-order).

> it allows the quantifiers to

You have already made a major category mistake here.
A language is a set of strings. A LANGUAGE
*cannot even CARE* what the quantifiers are
"allowed to range over". It does not acknowledge
any ranging of anything. Every last string you can
dream up is either IN the language or OUT of it.
THAT IS ALL. A "language-definition-framework"
(such as the one being invoked in the definition
of "a first-order language") can have knobs that you
can turn, holes that you can put pegs in, or various
other input parameters (every combination of which
produces a first-order language as an output) but it
does NOT have ranging quantifiers.

> apply only to variables or to nonlogical constants
> but not to both,

Uh, NO.

> and the language is second order if it allows
> the quantifiers to apply to both variables and
> to nonlogical constants.

It is possible that some traditional definitions can be
made equivalent to this, but you would be far better
off LEARNING THE TRADITIONAL DEFINITIONS *before* coming
up with alternative hypotheses.

> Is this a correct analysis?

No. It is not an analysis at all. It is a fantasy.


> I've seen the analysis that applying
> quantifiers only to variables
> results in a first order language,

That was your misfortune. Quantifers DO NOT
"apply to" variables in this context. And in any
case, it is a property OF THE LOGIC and NOT of the
LANGUAGE, that determines what the quantifiers "apply"
to (i.e., that determines the things to which they may
legitimately be *instantiated*, what constants can
REPLACE the VARIABLE being quantified WITH).

> and that applying quantifiers to both variables
> and nonlogical constants results in a higher order
> language,

The order of the language is not the place to start.
The logic is the place to start.

> but I've not seen any analysis of a language

Shut Up, he explained.
What you NEED to see is a basic first-order
logic textbook with a basic definition of what
a first-order language is. Immediately after
that, you will learn what first-order logic is.
Higher-order logic can then be defined by CONTRAST
with first-. THEN you will learn that there is no
need to go higher than 2nd, because the higher
orders are reducible. THEN you can have a framework
from which you can ask this question intelligently
(if you still have a question). UNTIL then, you are
JUST babbling.

.

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