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

Question

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 sense because 5 doesn't have an inherent color (and colors don't have inherent smells), so I don't know 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!

Answer 1

Mathematical logic deliberately and by design does not care about the "sense" or "meaning" of the statements it deals with. Mathematical logic provides a framework for understanding how we build up and work with complex concepts like "$47$ is even" from more primitive concepts like the definition of the constant "$47$" and the definition of the concept "even". This framework does not depend on our underlying intuitions about the most primitive concepts. Your statement $X$ can become false if we change the definition of $47$ (e.g., by choosing to write decimal numbers with the least significant digit on the left). Your statement $Y$ is only "meaningless" when we impose common intuitions about the terms like "colour" and "smell". From the point of view of mathematical logic, statement $Y$ is simply an assertion of the form $A(5) \mathrel{R} B$ (i.e, "the object $A(5)$ is related by the relation $R$ to the object $B$"), where $A(n)$, $R$ and $B$ are abstractions of the notions "colour of $n$", "smells like" and "cinnamon" respectively. (So mathematical logic is quite happy to help synesthetes with their reasoning.)

Answer 2

It's always said math is the language of science, however, by no means it's the language of other natural languages. When you want to translate arbitrary English sentences to mathematical logic (suppose here you're only talking about the most used classic logic) and apply it correctly, first thing you have to make sure every translated atomic formula is a well formed formula (wff).

Formulas themselves are syntactic objects. They are given meanings by interpretations. Every predicate is interpreted by a determinate property or relation. A determinate property is a property for which given any object there is a definite fact of the matter whether or not the object has the property. Unlike the English predicates, they are given very precise interpretations, interpretations that are suggested by, but not necessarily identical with, the meanings of the corresponding English phrases. For some intrinsically vague English sentences, H. P. Grice introduced conversational implicature to help arrive at a definite fact, such as either...or... case. For meaningless sentences, no such techniques exist to settle its propositional fact as is obvious.