This is my first post, so if I am doing anything wrong, please notify me.
In predicate logic, can one produce a truth-functional extension of a sentence containing 3 constants for a set containing two constants? For example, is it possible to produce a truth-functional extension of the following sentence : '(∀x)(Px ∨ ¬Px) ∧ ((Pa ∧ Pb) ∧ Pc) for the set for the set {a, b}?
By ''truth-functional extension'' I mean an equivalence without quantifier.
Thanks!
First realize that the $a,b,c$ in the sentence are constant symbols, while the $a $ and $b$ in the domain are objects, and these are different. That is, an interpretation needs to map each of the constants $a,b,c$ to either one of the objects $a$ and $b$ ...which is a little confusing... But can of course be done. And, once you have decided on such an interpretation(say, constant $a$ maps to object $b$, while constants $b$ and $c$ both map to object $a$), then the truth-functional expansion of your sentence with regard to this interpretation becomes:
$(P(a) \lor \neg P(a)) \land (P(b) \lor \neg P(b)) \land ((P(b) \land P(a)) \land P(a))$
In here, I use a 'new' constant symbol $a$ to refer to object $a$, and 'new' constant symbol $b$ to refer to object $b$, just as you would in any truth-functional expansion.