Any inductive logic that employs the same probability fun...

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Bayesian inductive logic(Distinguished from the narrower contemporary usage which associates the term exclusively with subjectivist or personalist accounts of probability.): In the broad sense: any inductive logic that gives Bayes' theorem a prominent role in inductive inferences, as opposed to approaches that largely eschew Bayes' theorem.
Evidence claims(the observations or data we're working with in reasoning): Statements about facts or observations that we think we've discovered or measured in the world.
Inductive logic(describes the type of logical system being discussed): A method of reasoning that starts with specific observations or evidence and uses them to draw general conclusions, rather than starting with general rules and deriving specifics from them.
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.
Thomas Bayes(the historical figure the 'Bayesian' approach is named after): An 18th-century English mathematician and minister who developed a formula for updating beliefs based on new evidence, which became foundational to probability theory.
hypothesis(Phase three of Dewey's pattern of inquiry): A construction that imaginatively utilizes both theoretical ideas and perceptual facts to forecast the possible consequences of various operations

An inductive logic using the same probability functions for both directions of i...Bayes' theorem expresses a necessary connection between the probabilities of evi...Bayes' theorem follows directly from the axioms that any probability function mu...Bayesian inductive logic is constitutively defined by the use of prior probabili...

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Halpern and colleagues have shown that probability-like functions satisfying wea...If non-Bayesian formalisms can represent both likelihoods and posterior-like deg...

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Elements of a logicist conception of inductive logic live on today as part of the general approach called Bayesian inductive logic. However, among philosophers and statisticians the term ‘Bayesian’ is now most closely associated with the subjectivist or personalist account of belief and decision. And the term ‘Bayesian inductive logic’ has come to carry the connotation of a logic that involves purely subjective probabilities. This usage is misleading since, for inductive logics, the Bayesian/non

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Any inductive logic that employs the same probability functions to represent both the probabilities of evidence claims due to hypotheses and the probabilities of hypotheses due to evidence claims must be a Bayesian inductive logic.

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