>>"The basic of model-theoretic semantics can be roughly described as the following.
For a formal language L, a model M consists of descriptions about
objects and their factual relations in a domain. The descriptions are written in
another language Lm, which is a meta-language, and can either be a natural
language, like English, or another formal language. An interpretation I maps
the words in L onto the objects and relations in M. According to this theory,
the meaning of a word in L is defined as its image in M under I, and whether
a statement in L is true is determined by whether it is mapped by I onto a
fact in M."
from "Experience-Grounded Semantics:A theory for intelligent systems"
>>According to model-theoretic semantics, for any formal language L, the necessary
and sufficient condition for its terms to have meaning and for its statements to have truth value is the existence of a model. In different models, the meaning of terms and truth value of statements may change; however, these changes are not caused by using the language. A reasoning system R that processes sentences in L does not depend on the semantics of L when the system runs. That means, on the one hand, that R needs no access to the meanings of terms and truth values of statements — it can distinguish terms only by their forms, and derive statements from other statements only according to its (syntactically defined) inference rules, but it puts little constraint on how the language can be interpreted. On the other hand, what knowledge R has and what operations R performs have no influence on the meaning and truth
value of the terms and sentences involved. Such a treatment is desired in pure mathematics.