Philose Koshy

Maher [2004. “Bayesianism and Irrelevant Conjunction.” Philosophy of Science, 71:515–520] attempts to resolve the paradox of irrelevant conjunction by arguing that the explicandum underlying the paradox is different from that of Bayesian confirmation theory. However, his solution remains inconclusive as he fails to specify the explicandum that underlies the paradox. In this article, we argue that pursuit worthy confirmation (pw-confirmation) is the explicandum in the case of the paradox. Given background information K, evidence E pw-confirms hypothesis H, meaning H is confirmed as being worthy of pursuit. When many find the confirmation of irrelevant conjunction unintuitive, their notion of confirmation is neither incremental confirmation (i-confirmation) nor absolute confirmation (a-confirmation), but rather pw-confirmation. Following Šešelja, Kosolosky, and Straßer [2012. “Rationality of Scientific Reasoning in the Context of Pursuit: Drawing Appropriate Distinctions.” Philosophica 86:51–82], the article distinguishes between two phases of pursuit worthiness and proposes a Bayesian definition for both phases. We demonstrate that irrelevant conjunctions are not eligible to be considered as candidates for initial pursuit. Often in practice, hypothesis testing can be conceived only as a part of a pursuit. Paradox is a conflation of two explicanda: i-confirmation and pw-confirmation. The standard Bayesian approach fails to tackle the paradox as it does not recognise pw-confirmation as a distinct category of confirmation. While developing a theory of pw-confirmation, we argue that a Bayesian definition of pw-confirmation can address some of the concerns Kuhn raises about scientific rationality.In addition to critically analyzing the standard Bayesian solution to the paradox, we also examine the limitations of Schurz [2022. “Tacking by Conjunction, Genuine Confirmation and Convergence to Certainty.” European Journal for Philosophy of Science 12:1–18], Bandyopadhyay, Brittan, and Taper [2016. Belief, Evidence, and Uncertainty: Problems of Epistemic Inference. Springer], and Chandler [2007. “Solving the tacking problem with contrast classes.” The British Journal for the Philosophy of Science 58:489–502] solutions.

This paper discusses an aspect of the problem of old evidence which I call here the general problem of old evidence. The probability of old evidence is one or close to one, because background information K entails the evidence E or K consists of propositions which make E probable. In the literature, K is considered as a proposition relevant to E. Based on examples, I argue that K does not support the truth of E; instead, K supports the evidential status of E. I define background information as a set of propositions necessary and sufficient to consider E as the evidence of hypothesis H. Background information is relevant to the bearing of E on H, but not to the truth of E itself. My definition of background information implies that background information of E is probabilistically independent of E; that is, in the case of old evidence, neither P = 1, nor P ≈ 1.

In this paper, I critically analyse two strands of Bayesian solution to the paradox: the standard Bayesian solution and the attempts to refute Nicod’s criterion. I argue that the standard Bayesian solution evades the exact challenge of the paradox. I hold that though the NC or instance confirmation is imprecisely formulated, it cannot be ruled out as an invalid form of confirmation. I formulate three conditions of instance confirmation which sufficiently captures our intuitive notion of instance confirmation. Finally on the basis of the conditions of instance confirmation, I show that paradoxical contrapositive instances like white shoe are not contrapositive instances of the raven hypothesis.