Plausibly, when we adopt a probabilistic standpoint any measure Cb(h, e) of the degree to which evidence e confirms (or is evidentially relevant to) hypothesis (or theory) h relative to background knowledge b should meet these five desiderata:(1) Cb(h,e) > 0 when P(h/eb) > P(h/b; Cb(h,e) < 0 when P(h/eb) < P(h/b); Cb(h,e) = 0 when P(h/eb) = P(h/b).(2) Cb(h,e) is some function of the values P(·/b) and P(·/·b) assume on the at most sixteen truth-functional combinations of e and h.(3) If P(e/hb) < P(f/hb) and P(e/b) = P(f/b) then Cb(h,e) ≤ Cb(h,f); if P(e/hb) = P(f/hb) and P(e/b) < P(f/b) then Cb(h,e) ≥ Cb(h,f).(4) Cb(h,ef) – Cb(h,eg) is fully determined by Cb(h,e) and Cbe(h,f) – Cbe(h,g); if Cb(h,ef) = 0 then Cb(h,e) + Cbe(h,f) = 0.(5) If P(e/hb) = P(e/tb) then Cb(h,e) = Cb(t,e).

This article begins by outlining some of the history-beginning with brief remarks of Quine's-of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it.

In making assertions one takes on commitments to the consistency of what one asserts and to the logical consequences of what one asserts. Although there is no quick link between belief and assertion, the dialectical requirements on assertion feed back into normative constraints on those beliefs that constitute one's evidence. But if we are not certain of many of our beliefs and that uncertainty is modelled in terms of probabilities, then there is at least prima facie incoherence between the normative constraints on belief and the probability‐like structure of degrees of belief. I suggest that the norm‐governed practice relating to degrees of belief is the evaluation of betting odds.

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