In a sentence theory of truth bearers, what it means to say that a sentence is cognitively meaningful is that it is true or false. To say that it is not cognitively meaningful (i.e. cognitively meaningless) means that it is not true or false.
In contrast, in a proposition theory of truth bearers, what it means to say that a sentence is cognitively meaningful is that it expresses a proposition. To say that a sentence is cognitively meaningless means that it does not express a proposition.1
I wonder if there is some problem with cognitively meaningfulness, logical implication and a sentence theory of truth bearers. Consider:
P. Colorless green ideas sleep furiously.
Q. The Earth is spherical.
Now consider a sentence that is about a logical implication from (P) to (Q):
S. That colorless green ideas sleep furiously logically implies that the Earth is spherical.
Now, it seems to me that if (P) is cognitively meaningless, then any sentence of which (P) is an antecedent or consequent, is cognitively meaningless too.
But now recall how a logical implication is defined. P implies Q iff there is no possible world in which Q is false and P true. But this is the case above. (P) is not true in any possible world at all2 and so any logical implication in which it is the antecedent is true. (Also where it is a consequent.) Thus, (S) is true. Contradiction. Something is terribly amiss.
Maybe some other definition of logical implication is needed. Suppose we stop defining it in terms of truth and falsity, and use the “is the case” phrase instead. Logical implication can then be defined as this: P logically implies Q iff there is no possible world in which P is the case and Q is not the case. Presumably all cognitive meaningless sentences are not the case. They are not false either because the semantic truth relations only hold for cognitively meaningful sentences. Now given the definition of logical implication all logical implications with a cognitively meaningless sentence as the antecedent are true. Again contradiction.
1By “means” I literally mean “means”. Not to be confused with an implication interpretation. I do not want to imply that some pluralistic proposition theory of truth bearers is false.
2Ignore potential problems with sentences meaning something else in a possible world.