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  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.