I have removed the original text in favor of showing the specific passages in the book that I am stuck on. Sorry about the inconvenience.
I am wondering whether $p ∧ q$ can be a proposition when $p$ and $q$ are both propositional variables. It seems to me that my book in one case says that a proposition can't have a variable truth value and then states that $p ∧ q$ is a proposition, even though it surely also has a variable truth value.
Consider the following sentences.
- What time is it?
- Read this carefully.
- $x + 1 = 2$
- $x + y = z$
Sentences 1 and 2 are not propositions because they are not declarative sentences. Sentences 3 and 4 are not propositions because they are neither true nor false. Note that each of sentences 3 and 4 can be turned into a proposition if we assign values to the variables.
Note that we will use the term "compound proposition" to refer to an expression formed from propositional variables using logical operators, such as $p ∧ q$.
Rosen, K. H. (2019). Discrete Mathematics and Its Applications Eighth Edition. New York, NY: McGraw-Hill Education.
Couldn't we have stated the following?
"$p ∧ q$ is not a proposition because it is neither true nor false. Note that it could be turned into a proposition if we assign values to the variables $p$ and $q$."