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(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!

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  • 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.
    – Logikal
    Aug 13 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 at 9:02
  • 2
    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.
    – Conifold
    Aug 13 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.
    – Dasem
    Aug 14 at 7:48
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The first answer one is tempted to give is : logical operators only take propositions as inputs , propositions are necessarily either true of false, and nothing can be either true or false unless is has meaning. So , the first idea that show up is : there is no conjunction involving an expression that does not make sense, or, a conjunction involving such an ewpression has itself no meaning, and therefore, cannot be false.

But, (1) an antology is absurd ( say : X&~X) ) (2) and one can buid a conjunction involving an antilogy ( or even two) (3) and this conjunction is itself an antilogy,hence, a false sentence.

This does not answer the question, but shows the question itself is not absurd.

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  • "propositions are necessarily either true of false" No, not necessarily. Only if you opt for a two-valued logic.
    – lemontree
    Sep 13 at 1:08

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