If UP is analytically embedded in our concept of rational belief-formation, Hume's demand for a demonstrative argument misidentifies the logical status of the principle it seeks to justify.

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Analytically embedded(describing the relationship between UP and rational belief-formation): Built into the very definition or meaning of a concept, so that you couldn't understand the concept without it. For example, 'unmarried' is analytically embedded in 'bachelor'—you can't have a bachelor who is married.
Hume(as the main philosopher discussed in this statement): David Hume was an 18th-century Scottish philosopher who argued that human knowledge comes from experience and observation rather than pure reasoning alone.
Logical status(what the principle's actual nature or foundation is): What kind of truth or proof a statement has—whether it's something that's true by definition, provable through logic, or just based on experience.
Rational belief-formation(the concept that UP is supposedly built into): The process of developing beliefs in a logical, reasonable way based on good evidence rather than just guessing or assuming things.
UP (Uniformity Principle)(a key assumption often thought to underlie all inductive reasoning): The assumption that the future will work like the past—that nature operates by consistent rules and patterns that won't suddenly change.
demonstrative argument(Contrasted with dialectical, rhetorical, and sophistic arguments in Avicenna's taxonomy): An argument in which certainty is transferred from premises to conclusion.

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