Considerations of accuracy – the epistemic good of having credences close to truth-values – have led to the justification of a host of epistemic norms. These arguments rely on specific ways of measuring accuracy. In particular, the accuracy measure should be strictly proper. However, the main argument for strict propriety supports only weak propriety. But strict propriety follows from weak propriety given strict truth directedness (which is non-negotiable) and additivity (which is both very common and plausible). So no further argument is necessary.

We present a semantics for a language that includes sentences that can talk about their own probabilities. This semantics applies a fixed point construction to possible world style structures. One feature of the construction is that some sentences only have their probability given as a range of values. We develop a corresponding axiomatic theory and show by a canonical model construction that it is complete in the presence of the ω-rule. By considering this semantics we argue that principles such as introspection, which lead to paradoxical contradictions if naively formulated, should be expressed by using a truth predicate to do the job of quotation and disquotation and observe that in the case of introspection the principle is then consistent.

We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap's revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability function. One such property is Leitgeb's Probabilistic Convention T, which says that the probability of ϕ equals the probability that ϕ is true.

It is natural to think that there is something epistemically objectionable about avoiding evidence, at least in ideal cases. We argue that this natural thought is inconsistent with a kind of risk-avoidance that is both widespread and intuitively rational. More specifically, we argue that if the kind of risk-avoidance recently defended by Lara Buchak is rational, avoiding evidence can be epistemically commendable.In the course of our argument we also lay some foundations for studying epistemic value, or accuracy, when considering risk-avoidant agents.Is it ever reasonable not to gather evidence? Sure it is. Gathering and processing evidence is almost never free, costing time and energy if nothing else. Even if it were free, you might know that the evidence doesn't bear on anything important. And even if the evidence were both free and relevant, you might still be worried that you'll misevaluate it.But what about cases in which none of these worries arise? Ideal cases, in which (i) gathering the evidence incurs no cost whatsoever, (ii) the evidence is potentially relevant, and (iii) you're certain to process it rationally? Here, it seems wrong to ignore evidence. Good [1967] shows that classical decision theory agrees: it says that not gathering the evidence is always instrumentally irrational, a bad way of pursuing your goals. However, Wakker [1988] shows that decision theories that allow for risk-avoidance (a form of risk-aversion not permitted by classical decision theory) do not share this verdict. The two thus disagree about:Look-I In ideal cases, one is instrumentally required to gather the evidence.So it looks as though evidence-avoidance can be rational (in ideal cases) if and only if risk-avoidance is.Recently, however, Lara Buchak [2010] has pointed out that this reasoning ignores an important distinction. For, even if avoiding evidence is rational