Semantically equivalent formulas with different variables I have a small issue regarding the formalism with respect to propositional logic variables when comparing formulas with one another.
Specifically: Are the following two formulas logically equivalent? I.e. is $F_1 \equiv F_2$?
$$
F_1: A \vee B
$$
and
$$
F_2: (A \vee B) \wedge (C \vee \neg C)
$$
I'm pretty sure they are semantically equivalent since every assignment yields the same result for both formulas.
My doubt comes from the fact that $C$ does not appear in $F_1$. Or is the question incomplete, since I am not stating what all possible variables are?
On that same regard, how do we compare:
$A \vee B$ to $C \vee D$?
Intuitively I'd say they are not semantically equivalent, since I would be considering assignments over $A, B, C$ and $D$. Is this correct? Is some further statement about which variables one considers needed or is it save to assume that simply all occurring variables in one or the other formula should be considered?
 A: Your question is a good one, but the answer depends on convention.
In one convention, there is a (usually infinite) supply of variables in play at all times, and all formulas use variables from this supply. An assignment has to assign a truth value to all variables in the supply, even those that do not appear in any formula under consideration. By this convention, $F_1$ and $F_2$ are equivalent.
In another convention, a formula always comes in a context (there are different words for this notion), which tells us what the collection of free variables in play is. For the semantics of a formula, an assignment only assigns truth values to the variables in the context. If we view $F_1$ as a formula in context $\{A,B\}$ and $F_2$ as a formula in context $\{A,B,C,D\}$, then it doesn't make sense to ask whether $F_1$ and $F_2$ are equivalent, since they are in different contexts (so their semantics depend on different assignments). However, we can also view both $F_1$ and $F_2$ as formulas in context $\{A,B,C,D\}$ (the variables in the context don't need to explicitly appear in the formulas). If we do this, then $F_1$ and $F_2$ are equivalent.
Finally, I'm not aware of any convention in which $A\lor B$ and $C\lor D$ are viewed as semantically equivalent. What is true is that these two formulas are related by a substitution of variables (substituting $C$ for $A$ and $D$ for $B$).
