Herbrand Universe for Equality
Recall example of formula :
Replace '=' with 'eq':
call the resulting formula .
Consider Herbrand model for the set . The relation splits into two sets: the set of terms eq with , and the set of terms eq with . The idea is to consider these two partitions as domain of new interpretation, denoted .
Constructing Model for Formulas with Equality
Let be a set of formulas in first-order logic with equality and result of replacing '=' with 'eq' in . Suppose is satisfiable.
Let be Herbrand model for . We construct a new model as Interpretation Quotient Under Congruence of under congruence 'eq'. Denote quoient structure by . By theorem on quotient structures, is true in . Therefore, is satisfiable.
Herbrand-Like Theorem for Equality
Theorem: For every set of formulas with equality the following are equivalent
- has a model;
- has a model (where are Axioms for Equality and is result of replacing '=' with 'eq' in );
- has a model whose domain is the quotient of the set of ground terms under some congruence.