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"A mathematically proven statement would be absolutely correct if all the axioms and inference rules used in the proof are first accepted as absolutely correct. That is the whole purpose of creating formal systems, so that anyone who accepts all the axioms and inference rules will have no choice but to accept all the statements proven within the formal system." I read very similar statements about a logical consequence in most of the logic textbooks and articles on the internet. However I wonder: If I say to you every rule for how to derive a logical consequence and I even tell you, that if you accept those rules and accept premises, than the logical consequence is neccesary.. Shouldn't I also add that the logical consequence is necesary only for me, since my understanding of the words is so that it couldn't be otherwise? And shouldn't I also add that the logical consequence is necessary to you and you have to accept it if and only if you have the same meaning of the words as I do? If so, does it mean that to say that something is a logical consequence of some accepted premises and inference rules, it is always just a subjective statement dependent on the meaning I have in my mind? It could be true for me but I cannot be sure it is true for other person since I cannot be sure other person has the same meaning connected to those words as I do.. Therefore it is just a subjective statement right?

(note: I don't have a solid philosophy background but I would really appreciate to have some knowledge on this topic)

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    There is no way to refute skeptical arguments... – Mauro ALLEGRANZA May 14 at 12:33
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    "Saying" rules is pointless unless one learns to act on them, see the rule-following paradox and What the Tortoise Said to Achilles. So no, you should not bother saying all those things. Deriving logical consequence is not a statement, it is an act, and while the act is performed individually its outcome is not subjective if competent community members agree on it. – Conifold May 14 at 13:08
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    You can see the debate about Wittgenstein on Rules and Private Language: how can we be "certain" that we understand each other ? How can we be certain that we apply the rules of logic and language in the same way ? No certainty at all... but IMO the skeptical attitude is not what Wittgenstein adopted. – Mauro ALLEGRANZA May 14 at 13:27
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    But on of the central theme of W's PI is that language is social, is aimed to communicate much more than "to represent" (contrary to W's Tractatus). Arguments are part of language; logic is part of language and there is no "another language" on which to ground our language. Thus, misunderstanding is possible (it occurs everyday) but there is no "reasonable" ground to assert that we cannot understand each other with language. – Mauro ALLEGRANZA May 14 at 13:30
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    The only thing that makes logical statements necessarily true is that and when they are tautological, which means they have nothing to do with reality. But the question asked about rules, words and consequent presents an important problem for non-mathematical logic. Words are formed in our imaginations Spinoza (TIE). Each person does indeed employ words based on their own unique definitions. This indicates that the way the question is shaped above and its explanatory notes makes the observation that the consequent is subjective, true. Good question, well put. CMS – Charles M Saunders May 17 at 15:17
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Consider:

In math, what is 2x/3y-1 if x=9 and y=2 ?

Some people will say the result is 2, others 11. And in different maths context either can be right.

Even though the pairwise operations like division, multiplication and subtraction can be agreed upon, and multiplication and division take precedence, people can still disagree about the precedence in complex combinations.

Another example: The string "10" can be interpreted as the decimal number 10, or as the decimal number 2 in binary notation. So 10+10 could result in 20 or in 100.

Another example: if we imagine an alien visiting earth, and this alien was taught in alien school that numbers written in black ink have different value than the same numbers written in blue ink, the alien may get different results in maths.

So yes, any 2 people need to agree on a lot of things to make sure that they must reach the same conclusion from the same written premises. But really, it does not matter philosophically, because when any 2 people have any difference, it is just a matter for them to understand their difference. After that, they will necessarily come to the same results, when using approaches without differences.

So logical statements remain objective, not subjective, even if due to differences in unmentioned assumptions, different results could be obtained initially as a misunderstanding. Such misunderstandings do not change an objective statement into a subjective one.

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Short answer: Your confusion stems from not understanding formal systems and their objectivity. Provability in a formal system is almost completely objective, more objective that anything else you can ever hope for. So logical consequence within a formal system is equally objective.

Long answer:

A mathematically proven statement would be absolutely correct if all the axioms and inference rules used in the proof are first accepted as absolutely correct. That is the whole purpose of creating formal systems, so that anyone who accepts all the axioms and inference rules will have no choice but to accept all the statements proven within the formal system.

In the above quote from this post, "accept all the axioms and inference rules" means "accept all the axioms and accept all statements produced by applying any of the inference rules to any accepted statement".

Note that this has almost nothing to do with semantics (i.e. meaning of words). It does not matter what you mean by "accept". It does not matter if you interpret statements differently from me. It only matters that exactly the above is satisfied, namely the statements that you accept include the axioms and are closed under applying inference rules. I say "almost nothing" because we do need to share a very tiny bit of understanding, but that is solely understanding of what each inference rule means, and nothing more. For example, the ∧-intro rule in all natural deduction systems means that if you accept A and accept B then you accept (A∧B). If you want to be 100% precise, in most formal systems it means that if you accept the string A and the string B then you accept the string "("+A+")∧("+B+")", where brackets are used to ensure correct parsing. Some systems may precedence rules, some may not. None of that matters; all that matters is that you understand exactly what each inference rule means, which is why it is so crucial that every formal system can be captured completely by purely syntactic rules, because that guarantees that we can achieve such a shared understanding.

Even if whenever you say "I accept X" you actually mean "I doubt X", it makes no difference at all to the outcome. Suppose you say "I accept the axiom A" and "I accept the axiom B" and "I accept the ∧-intro rule". Then I can by applying the ∧-intro rule guarantee that you have to admit "I accept the statement (A∧B)". I do not need to even know what you mean by "accept", to be able to objectively demonstrate that you must accept every statement that is proven within a formal system whose axioms and rules you accept, as long as I provide the proof (i.e. the sequence of steps each of which states which rule to apply to which statements).

In the past, one may be forgiven for questioning whether even syntactic rules are sufficient to achieve shared understanding of formal systems. But ever since we have had personal computers, that is no longer tenable, because every formal system can be literally implemented by a computer program in a fixed programming language. This even generalizes the notion of syntactic rules in formal systems. An inference rule can be defined as a program that when run on input statements will produce an output statement! Accepting a set A of axioms and a set R of inference rules can then be simply defined as accepting the closure C of A under R. We can very well call C the logical consequences of A under R, and there is essentially no subjectivity concerning whether a statement X is a member of C once I write down explicitly a sequence of steps each specifying which rule in R to run on which statements, because you can completely mechanically check each step by running the specified programs!

The only possible subjectivity in proofs within a formal system is if you doubt your own ability to apply the stated inference rules (i.e. run the given programs) correctly. At that point, it is far more reasonable to doubt that your browser has correctly fetched this webpage for you. (Maybe your browser program had a strange bug/feature and added this sentence which user21820 did not write!)

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This is a question about words and understanding their meaning. An unqualified statement written in the English language assumes that the reader is equally fluent in that language. If the reader struggles with the language, then they have not understood the statement and what they may choose to dicker over is not the statement originally made but, rather, is where their understanding of the language falls short.

The logical game offered is just a vehicle for that confusion.

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