A propositional function is an expression that contains at least one occurrence of a free variable; whereas a proposition is an expression that does not contain any free occurrence of any variable.

Now, (1) contains two occurrences of free variables (namely two occurrences of "x"). So, (1)^* is NOT a proposition, but a propositional function. Since (1) is a propositional function, it does not have a truth-value. On the other hand, (2)** contains no free occurrence of any variable; the two occurrences of "x" in (2) [I read (2) as "For all values of x, if x is human, then x is mortal"] are bounded by the quantifier 'for all values of x'; and hence, they are not free variables, they are bound variables. Since, there is no free variable in (2), it is a proposition, NOT a propositional function. That is why, (2) has a truth-value.

Note that, a proposition may contain bound variables, but cannot contain any free variable. And, a propositional function may contain both- bound and free variables, and must contain at least one free variable.

Does it help, Jason?

-MNM-

* (1) If x is human, then x is mortal.
** (2) For all values of x, (1) is true.