# Does the logical conjunction of a false statement and a statement that doesn't make sense result in a false statement?

(This question was originally asked here, but I've decided to ask it in philosophy since it might be a better fit.)

I kind of feel like this is a silly question, but does taking the logical conjunction of two statements make any sense when one of the statements doesn't make sense?

For example, suppose we have the following two statements:

• Statement X: "47 is an even number."
• Statement Y: "The color of the number 5 smells like cinnamon."

The first statement is obviously false, but the second statement doesn't make much sense because 5 doesn't seem to have an inherent color (and colors don't seem to have inherent smells), so I'm not sure if it has a truth value. Is it possible that the second statement has no truth value, or do we force ourselves to assign a truth value to it?

If the second statement has no truth value, then consider statement Z: "The color of the number 5 smells like cinnamon and 47 is an even number." Since X is false, does that mean Z is false, or does Z have no truth value because Y has no truth value?

I'm asking because I'm not sure if it is considered valid to take the logical conjunction of two statements if one of the statements has no truth value. I think the only thing we can be certain of is that Z is not true.

Sorry, not well-versed in logic! Any guidance would be great!

• Well there is an old school of thought in Epistemology that there are some statements that are neither true or false. The other alternative is to categorize MEANINGLESS statements. In your example we clearly see that numbers can't possible posses the qualities assigned to them. In Mathematics the domain of discourse may be different. I say this because math typically uses strict definitions that may not apply outside the math class as you see in your example. The reason is because in math you are told logic is not about CONTENT of the premises themselves. In the real world we know better. Aug 13, 2021 at 5:24
• What do you mean with "a statement that does not make sense"? In a formalized language a statement that... is an ill-formed expressions, i.e. an expression that does not satisfy the syntactical specifications of the language. Thus, every expression including it is not we--formed. Aug 13, 2021 at 9:02
• On the usual convention, if even part of a sentence is malformed then the whole sentence is malformed and is not assigned any truth value. However, one may want to allow some sentences that "do not make sense", treating them not as malformed but as without definite truth value. And there is an approach called supervaluationism that assigns truth values to some sentences even when their parts do not have it. It happens when assigning any truth value to a part produces the same result. So your conjunction will be supervaluated as false. Aug 13, 2021 at 9:31
• Statement Y isn't a proposition it's more like an opinion not a fact, i would have label it as a false. Aug 14, 2021 at 7:48