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One can create axioms that make statements like "all bachelors are married" true. What is wrong with calling these truths analytic as a shorthand for the type of truth it is based on the type of axiom it is derived from, much in the way we use the adjectives arithmetic, set-theoretic, or logical to denote those types of formal truths? I feel like one could decide whether a truth is analytic by seeing which (kinds of) axioms need to involved in making it true.

There is nothing stopping you from defining an analytic theorem of a formal system to be one whose derivation requires appeal to at least one member of a designated subset of axioms. But on what basis are you deciding to single out that particular subset of axioms? If you say you're being guided by the fact that those particular axioms express truths about meanings, whereas other axioms express substantive truths about the world, then you owe an explanation of what that distinction amounts to -- and arguably, that will be no easier to give than an outright analysis of "analytic". (You might also look at W.V. Quine's discussion of Semantic Postulates in his paper "Two Dogmas of Empiricism.")

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