Is "P=M" a well-formed predicate log formula in a domain?

Question

A predicate $P(x)$ "$x$ is possible" and $M(x) $ "$x$ is a mission" and all in the "Possible Mission domain of disclosure."

Is $P=M$ a well-formed predicate logic?

I would say yes because, both the predicates are in the same domain. The question is a bit confusing for me.

Thanks

Answer 1

NO, it is not.

Predicates are used to form statements, while the equality symbol $=$ is a relation between terms : variables and (individual) constants, i.e. "names" of objects.

Thus, we may write e.g. $x=y$ and $\forall x Px$ but not $Px=Mx$.

If we want to formalize the statement : "Every mission is possible" we have to write :

$\forall x (Mx \rightarrow Px)$.