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. Commented 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. Commented 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. Commented 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. Commented Aug 14, 2021 at 7:48
• You can't perform logical calculations between a logical and a non-logical object, in the same way you can't perform arithmetic operations between numbers and water. Commented Jun 10, 2022 at 11:55

Let's follow the usual distinction between syntax and semantics. Logic is concerned with both.

Syntactically, a sentence must be formed in accordance with rules in order to qualify as a well-formed formula. If it is not well-formed, it is just a string of symbols that we are unable to interpret. The string "jdkjfh*J(LPO¬" is not a well-formed formula and might be said to be syntactically meaningless.

Semantically, we can assign interpretations to the terms that appear in a well-formed formula in order to get a handle on the conditions under which a sentence is true. Model theory is a common choice for this purpose. A well-formed sentence may still be semantically meaningless if it is impossible to state its truth conditions. Your example of, "The color of the number 5 smells like cinnamon" is like this. Unless a person has a strange form of synesthesia that allows them to smell numbers, there is no way to state under what conditions this sentence is true or false, or even whether it has a truth value at all.

In such cases, we have a number of options available for handling the logic, and it is largely a matter of convention as to which we adopt.

1. We could stick rigidly with the principle of compositionality and say that if part of a sentence is meaningless then the whole sentence is meaningless, in which case, the conjunction of a false statement with a meaningless one is meaningless.

2. We could treat such sentences as having a third truth value separate from that of truth and falsity, and use a three-valued logic. The truth table for such a logic might accept that the conjunction of false and meaningless is false.

3. We could use the approach of supervaluationism and hold that the conjunction of a false statement with anything else is false, even if the second conjunct is indeterminable or not well defined. This approach is similar to the way some programming languages implement lazy evaluation and permit the programmer to write "IF A AND B THEN DO X" with the effect that in the event A evaluates to false, B is not evaluated.

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.

• "propositions are necessarily either true of false" No, not necessarily. Only if you opt for a two-valued logic. Commented Sep 13, 2021 at 1:08

It can make sense in the Meinongian Jungle where objects are princpally defined by their propertoes which at its most basic level must be taken all at once without logical operators to be talen literally as logic.

Of course it does not make any sense in our existemt world.

My two sikkas for what they're worth.

A meaningless sentence is neither true nor false i.e., relaxing the rules of logic just enough to get my point across, if p is the meaningless sentence, then (~p & ~~p) which is (~p & p), a contradiction which is always false. Any proposition in conjunction with a contradiction is false. In other words, the conjunction of a false proposition with a meaningless proposition is false because ... a meaningless proposition is a contradiction.

This jibes with the fact that contradictions are, if you've ever encountered them, incomprehensible i.e. they're meaningless in that sense.