Can a logical statement be meaningless? If a statement is logical, then it can be proven mathematically, but if logical statements can be meaningless still doesn't that prove that mathematical proofs are also meaningless? What are the philosophical findings on this?
I know it sounds like I am being a wise guy, but what do you mean by meaningless.
The statement 'If the sky is blue, then the sky is blue,' is a logical statement that does not convey any information, so I would call it meaningless.
Contrast that with the statement 'If the sun rises in the North, then you owe me $100.' Statements like this are described as "true, but only because it doesn't say anything."
There are plenty of other ways you can define meaningless. I cannot help answer the question without you telling me what you are referring to when you say meaningless.
Just a side not, but logical statements can be false. A statement is logical simply if it can be represented by a system of logic. The statement 'If fish can swim, then birds can shoot lazer beams from their eyes' can be represented in first-order logic, but it is also false. Logical statements that are true are called sound. Also, a valid argument is an argument which would be sound if the premises were true. Terms like these can get confusing and easy to mix up. Just keep going at it and you will get the hang of it eventually.
The logical positivists' initial stance was that a statement is "cognitively meaningful" in terms of conveying truth value, information or factual content only if some finite procedure conclusively determines its truth. By this verifiability principle, only statements verifiable either by their analyticity or by empiricism were cognitively meaningful. Metaphysics, ontology, as well as much of ethics failed this criterion, and so were found cognitively meaningless. Moritz Schlick, however, did not view ethical or aesthetic statements as cognitively meaningless. Cognitive meaningfulness was variously defined: having a truth value; corresponding to a possible state of affairs; intelligible or understandable as are scientific statements.
Verification principle is the philosophical doctrine which maintains that only statements that are empirically verifiable (i.e. verifiable through the senses) are cognitively meaningful, or else they are truths of logic (tautologies).
So under this POV, only those logical statements which are either analytic tautologies or can be empirically verified their truth value are meaningful, other arbitrarily constructed logical statements are meaningless. So of course logical statements can be meaningless, and perhaps their cardinality is larger than those meaningful.
As for your concern about math proofs meaninglessness, a math theory under the most popular formalism has no meaning themselves besides its definitions and axioms, it's through their model theories' interpretations it's imparted meaning, possibly many different types of meanings if the theory is noncategorical. If a logical statement expressed in math FOL like format is meaningless, it's not its proof (theory) meaningless but its interpreting model is meaningless since the statement cannot be verified either from its axioms analytically or from sense experiences empirically...