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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?

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    Depends on what counts as "logical statement". You can set up some uninterpreted formal system with rules for generating and proving "statements" that have no meaning attached to them (pure syntax). If this counts then mathematical "statements" can also be meaningless. However, what is usually studied in logic and mathematics comes with interpretation(s), so actual logical and mathematical statements are not meaningless. It is unclear what sorts of findings you are looking for though.
    – Conifold
    Apr 30 at 5:15
  • "meaningless" is used in different contexts: Existentialism: the life has no meaning. Is this your concern? what is the "ultimate aim" of logical statements? Apr 30 at 6:58
  • "If a statement is logical, then it can be proven mathematically" Can you try to be more specific? Some example may be useful. Apr 30 at 6:59
  • A different use of "meaning" is: a statement has meaning when we (humans) can understand it. We understand statements of logic: someone can understand full books concerning logic. Apr 30 at 7:00
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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.

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  • Are you speaking only of Mathematical logic? Meaninglessness is typically discussed by philosophy. As such a meaningless statement does not Express communication that is true or false. In this way all statements are not propositions & vice versa. Most people in math typically confuse statements with propositions & statements with sentences & finally they think propositions are sentences that are true or false. Proper use of terminology is not identical between Philosophy & Math. In math by definition a statement automatically has meaning. Thus you need more information about what is meant.
    – Logikal
    Apr 30 at 12:46
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To get rid of numerous logical and semantic (meaningless) paradoxes such as Liar, modern logical positivism proposed its famous verification principle under which:

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...

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  • The type of sense verifiable statements you refer to is distinct and defined in all sciences as you post. Philosophy would NOT define all statements as sense verifiable as a requirement the way science requires. In this way a statement can be true or false regardless of if there is sense verification. This is typically regarded as objective knowledge. So a statement is not a proposition and proposition is not necessarily a statement by definition. Many people outside philosophy think propositions & statements are the same thing essentially which happens to be false.
    – Logikal
    Apr 30 at 12:35
  • @Logikal thx for your comment. Of course proposition is a better fitting terminology for those meaningful statements with a truth value which I doubt PO is aware. Logical positivism movement in philosophy last century did want to get rid of all those meaningless propositions/statements/arguments (such as cosmological argument) which cannot be verified with a truth value and want to turn philosophy to a pure science. After it went out of favor, meaningless statements came back to philosophy legitimately, and under Popper's fasification it's just non-scientific but still hugely useful perhaps... May 1 at 1:01

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