While compiling a set of notes for high school students about logic and proof, I wrote
Let $A$ and $B$ be statements. Then $A \Rightarrow B$ means ''if $A$ is true then $B$ is true.''
This was my definition of a statement:
A (mathematical) statement is a sentence that is either definitely true or definitely false. (It cannot, therefore represent an opinion, like "apples taste better than bananas" or events that may or may not be true, like "It will rain tomorrow in London".) Statements can therefore be proved/disproved by providing an argument that rigorously demonstrates that they are true/false.
By the above definition, $A$ and $B$ don't have to be statements for $A \Rightarrow B$ to make sense. For instance, for real $x$, "$x>4 \Rightarrow x>2$" makes sense (and is true) and yet it isn't possible to assign a definite true/false value to these sub-statements "$x>4$" and "$x>2$" in isolation.
This leads me to my question: is there a proper name for things like "$x>2$" and "$x>4$", where it makes grammatical sense to call them true or false, but which cannot truly be called either, without being given more information?
I was thinking of calling them boolean phrases, but I would rather not just invent a name for something if a proper name is already in use. Another option is just to redefine what I call a (mathematical) statement to include this kind of thing, but then add another definition like "decidable statement" to describe sentences that are either definitely true or definitely false.
This question also applies to other examples. What name should be given to sentences like the below?
- "This sentence is false" (seems like it can be true or false, but both lead to contradictions)
- "$x$ is even" (We cannot decide if it's true or false without more information about $x$)