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

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Reason for

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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Reasons Against

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Reason against

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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Key Terms

Bayesianism(as the main position being discussed): A philosophical and mathematical approach to reasoning about uncertainty that uses a specific formula (Bayes' theorem) to update your beliefs when you get new evidence.
Formal apparatus(as the shared system between Bayesian and non-Bayesian approaches): The mathematical tools, symbols, and rules that a system uses to represent and work with ideas.
Non-Bayesian formalisms(as contrasted with Bayesian approaches): Alternative mathematical systems for representing uncertainty and reasoning that don't rely on Bayes' theorem or traditional probability theory.
Posterior-like degrees of support(as another concept being represented): Measures of how much evidence or reasoning supports a conclusion after considering new information (posterior means 'after' or 'later').
Probability functions(describes what the inductive logic uses to measure likelihood): Mathematical tools that calculate how likely something is to be true, assigning numbers between 0 (impossible) and 1 (certain) to different outcomes.
entailment(Conceptualist framework): Understood in terms of truth at a world
likelihoods(Bayesian confirmation theory): The probability of the evidence given a particular hypothesis, used in conjunction with prior probabilities to determine expectedness
logically necessary(Distinguished from metaphysical necessity in Swinburne's argument): That which could not fail to exist or be true; its non-existence or falsehood would be a logical contradiction

Connections

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Truth & Knowledge1 linked Skepticism1 linked

Any inductive logic that employs the same probability functions to represent bot...Formal isomorphism between mathematical structures doesn't determine their philo...

Details

Type: claim
Perspectives: 2 (1 for, 1 against)
Edits: 1 edit