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    It is not the case that If non-Bayesian formalisms can represent both likelihoods and posterior-like degrees of support using the same formal apparatus, the alleged entailment from shared probability functions to Bayesianism is not logically necessary.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Using probability functions to represent both likelihoods and degrees of support practically requires conditional probability calculus, which Bayes' theorem axiomatically governs.
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    • 2.Non-Bayesian systems claiming to use 'posterior-like' support must either lack formal coherence or tacitly presuppose Bayesian updating assumptions they deny.
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    • 3.The claim conflates syntactic equivalence (same notation) with semantic equivalence (same interpretation), which are distinct philosophical properties.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Formal isomorphism between mathematical structures doesn't determine their philosophical interpretation or foundational commitments.
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    • 2.Imprecise probabilities, interval-valued likelihoods, and other non-Bayesian formalisms demonstrably use probability functions without endorsing Bayesian updating rules.
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    • 3.Necessity requires that Bayesianism logically follows from the mathematical apparatus alone; alternative interpretations of shared formalism block this entailment.
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