 In probabilistic inductive logic the likelihoods carry the empirical import of hypotheses. A likelihood is a support function probability of form \(P[e \pmid h_i\cdot b\cdot c]\). It expresses how likely it is that outcome \(e\) will occur according to hypothesis \(h_i\) together with the background and auxiliaries \(b\) and the experimental (or observational) conditions \(c\).[5]  If a hypothesis together with auxiliaries and experimental/observation conditions deductively entails an evidence claim, the axioms of probability make the corresponding likelihood objective in the sense that every support function must agree on its values: \(P[e \pmid h_i\cdot b\cdot c] = 1\) if \(h_i\cdot b\cdot c \vDash e\); \(P[e \pmid h_i\cdot b\cdot c] = 0\) if \(h_i\cdot b\cdot c \vDash{\nsim}e\). However, in many cases a hypothesis \(h_i\) will not be deductively related to the evidence, but will only imply it probabilistically. There are several ways this might happen: (1) hypothesis \(h_i\) may itself be an explicitly probabilistic or statistical hypothesis; (2) an auxiliary statistical hypothesis, as part of the background b, may connect hypothesis \(h_i\) to the evidence; (3) the connection between the hypothesis and the evidence may be somewhat loose or imprecise, not mediated by explicit statistical claims, but nevertheless objective enough for the purposes of evidential evaluation. Let’s briefly consider examples of the first two kinds. We’ll treat case (3) in  Section 5,  which addresses the the issue of vague and imprecise likelihoods.