[FOM] basic model theory question

[Jim Farrugia]

Chris,

Thanks for your reply (and thanks, too, to Matt for his).

I think I understand what you said. Let me go from there to try to make
my question sharper.

Chris said:

...

> Example: Consider a language L consisting of a binary function symbol +
> and a constant symbol 0. It is not particularly interesting (at least, not
> to most *mathematical* logicians) to study the class of all L-structures
> alone, or the set of sentences true in all L-structures (e.g., 0=0). But
> if we add a few axioms (or further conditions, if you like), then the
> picture changes radically:
> 
> \forall xyz, (x+y)+z=x+(y+z)
> \forall xy, x+y=y+z
> \forall x, x+0=x
> \forall x \exists y, x+y=0
> 
> Models of this theory are abelian groups. Clearly, they are worth
> studying, no matter what symbols we happen to pick to express the group's
> operation and indentity.


If we take away the axiom for commutativity, we are left with the axioms
for groups. I do see that there will be differences in models of the two 
theories, because the sets of axioms that characterize these models are 
different. 
 
But this kind of difference is one thing, focusing on different 
collections of sentences (theories) that are true under any interpretation 
of the nonlogical vocabulary. 

I think this kind of difference could be characterized as: different 
theories have different models, and although not all models of a 
given theory interpret the nonlogical vocabulary in the same way, under 
any such interpretation for a given model, all the sentences in the 
given theory are true.

What I'm wondering, though, is something else (at least I think it is): 

What if anything can be said about how different interpretations of 
nonlogical vocabulary affect the truth values of a collection of sentences?

For example, consider the axioms you gave above. Suppose we have a structure 
whose domain is the integers, but with the symbol "+" being interpreted as 
subtraction on the integers. (Let "0" have its usual interpretion of zero.) 

Then the first two axioms you gave above are false in this structure, but 
the second two axioms are true in this structure. 

On the other hand, if we change the interpretion the "+" symbol to be 
addition on the integers, then all four of the axioms are true.

In this scenario, the different choices of interpretation of "+" lead
to different true/false values for the four sentences you gave.

Does model theory investigate these kinds of differences?

Thanks again for any help you can give.

Jim