Re: alternative 1st-order quantifier semantics

On Dec 15, 11:27 am, Jan Burse <janbu...@xxxxxxxxxxx> wrote:
This is common practice. '

No, it isn't.

It runs under the names term model,
henkin model, herbrand model, etc..

All those are INDIVIDUAL models under THE REGULAR semantics.
I am talking about an alternative interpretation of the quantifier
that admits ONLY term models and therefore winds up with
different consequences. It is a whole different semantics
and you cannot characterize it in terms of one model that also
occurs under the standard semantics.


The mathematical reason for being able to do that is the
following: Every algebra is a quotient algebra of a
free algebra.

It translates basically as follows, instead of having
objects x and y from your domain, and x=y indicates
identity of your objects, you have objects equating to
terms,

Well, yes, it IS, indeed, that, but I completely fail to see
how that is a TRANSLATION of anything. It's just DIFFERENT
from the other way.

and x=y is an equivalence relation and congruence over terms.

No, it isn't. It is syntactic equality of terms.
And you don't need equality for this in any case.
Any theory rich enough to be interesting is going to be
capable of DEFINING equality in any case. Though, of
course, if it is going to define a relation and spell it = ,
it had dangwell better be an equivalence relation.

Such that:

     t = s   :<=> evaluated t = evaluated s

"evaluated" is not even part of this.

.

Relevant Pages

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