The axioms of reason (such as that one thing which follows from another must be true) cannot be false without there being a logical reason, whatever it would be, involving each axiom and the others still being true. These supreme truths are still overlooked or entirely rejected by many, either because non-rationalists are so busy fixating on matters of mere practicality and preference or because they mistake some other truth or idea (God's existence, the universe, subjective perceptions, etc.) for the real uttermost basis of reality. As intrinsic truths, logical axioms depend on literally nothing else, but all else hangs on them.
While everyone relies on various logical facts even if they deny them or never realize this, there are formal ways to articulate certain conclusions which follow from other ideas, called premises, serving as foundations to conclusions. The wording of these proposed syllogisms, though, is by nature able to be revised, and human words themselves are inventions, constructs of individual minds or societies. One would have no need for language if not to communicate, or attempt to communicate, precise, abstract concepts and multifaceted experiences to other people, something that cannot be accomplished by the likes of physical gestures alone. But while words that can convey logical facts are constructs, logic itself is not, for the reason already stated here and elaborated upon extensively in other posts.
The terms, phrases, and sentences that form written syllogisms (two premise statements and a conclusion stating what is supposed to follow from the premises) are constructed as expressions of that which is already objectively true or false. A linguistic syllogism can be created or reworded, but the underlying logical truths behind a correct linguistic syllogism are true entirely independent of thought, belief, and language. Someone can construct a syllogism only in this very narrow sense. I myself used this sort of language when speaking of syllogisms in the past, but I certainly did not mean that logic is some malleable construct of my thoughts or anyone else's. Then, I increasingly saw how very irrationalistic people use similar language and began both naturally and intentionally distancing myself from them.
Logical axioms, true in themselves, cannot possibly have been invented or be altered, not even by God. Neither can the secondary necessary truths that spring from these axioms. But, again, linguistic syllogisms can be contrived or rephrased. Someone could craft sentences to form a syllogism or arrange their thoughts to recognize a logical fact and assign words to that truth, but every logical fact from genuine axioms onward is true because it could not be any other way (moreso with axioms in that even other strictly logical truths are rooted in them, but the latter also cannot have truly differed or been untrue). Written syllogisms can be constructed and reconstructed because wording can be amended; logic is immutable because of its inherent truth.
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