A.1.1 Definition. For any nonempty set D, the first-order alphabet obtained by importing the members of D as new constants is denoted by V(D). That is, V(D) = V U D. We will use the terminology D-formulae and D-terms for formulae and terms of the original language L, where some free variables have been replaced by D values.
It is an easy exercise to establish that this is tantamount to saying:
A.1.2 Definition. Given a nonempty set D, a D-formula and D-term are a formula and term over the alphabet V(D), respectively. The set of all D-formulae and D-terms will be denoted by WFF(D) and Term(D) respectively. The augmented language is L(D) = V(D), Term(D), WFF(D).
A.1.3 Exercise. Fix a language L and a nonempty set D. Then Term ⊆Term(D) and WFF ⊆WFF(D).
A.1.4 Exercise. Prove that F((A V B) √(3) A V (√B) by replacing √ with -V)