Satisfaction
Let 
be a relational system, and let 
be a language which is appropriate for 
. Let 
be a well-formed formula of 
, and let 
be a valuation in 
. Then 
is written provided that one of the following holds:
1. 
is of the form 
, for some variables 
and 
of 
, and 
maps 
and 
to the same element of the structure 
.
2. 
is of the form 
, for some 
-ary predicate symbol 
of the language 
, and some variables 
of 
, and ![<span style=]()
{s(x_1),...,s(x_n)}" src="https://mathworld.wolfram.com/images/equations/Satisfaction/Inline25.svg" style="height:22px; width:134px" /> is a member of 
.
3. 
is of the form 
, for some formulas 
and 
of 
such that 
and 
.
4. 
is of the form 
, and there is an element 
of 
such that 
.
In this case, 
is said to satisfy 
with the valuation 
.
REFERENCES
Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1969.
Enderton, H. E. A Mathematical Introduction to Logic. Boston, MA: Academic Press, 1972.
