Can a proposition contain variables?

Question

I understand that proposition logic can be expressed as 3=3, and predicate logic can be expressed as x+3=5. However I am unsure if proposition logic can contain a statement such as x=3? A simple yes or no answer is all I am searching for. Thanks in advance! :)

Answer 1

Variables are definitely not part of propositional logic.

I suppose we could define our propositional logic language to allow expressions like $Siblings(a,b)$ for readability sake to express atomic statements (instead of your typical $P$'s and $Q$'s) as long as you do not analyze that statement as consisting of those parts. That is, in propositional logic, atomic statements are exactly that: atomic statements without any further composition or structure. Indeed, if you were to use $3=3$ in such a language for propositional logic, it would be considered just like any other atomic statement, i.e. something that could be either true or false. But in predicate logic, the $=$ is a logical symbol, and there $3=3$ is a tautology.

Answer 2

Siblings(a, b), meaning a and b are siblings, is a declarative statement whose truth value is undetermined. Therefore, is not a proposition. 2 > 3 is a proposition.