I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable, which is true, and If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable, which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What's more, I argue that these counterpossibles don't just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models.

It is commonly held that the ascription of truth to a sentence is intersubstitutable with that very sentence. However, the simplest subclassical logics available to proponents of this view, namely K3 and LP, are hopelessly weak for many purposes. In this paper, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality o ered by proponents of LP are far less promising than those available to proponents of K3. This undermines the ability of proponents LP to engage in public reasoning in classical domains. IntroductionGottlob Frege famously held that "nothing is added to [a] thought by . . . ascribing to it the property of truth" ( , ). This idea is commonly expressed with the slogan that truth is transparent: φ and T r ( φ ) -the sentence that says that φ is true-are fully intersubstitutable in extensional contexts. Unfortunately, in classical logic, the law of excluded middle, i.e.φ ∨ ¬φ, and the rule of explosion, i.e. φ, ¬φ ∴ ψ, allow us to derive any sentence from the liar sentence if we have transparency. It's tempting to put the blame on transparency here.However, it isn't entirely obvious what to replace transparency with. That is why a number of authors have instead blamed classical logic. Saul Kripke ( ), Robert Martin and Peter ' φ ' is a term for φ in the object language. Note that the corner quotes here are Gödel quotes, not Quine quotes. To be fully precise, I would need to use both kinds of quotes; however, since corner quotes are commonly used for both, I allow myself the usual use-mention sloppiness here and throughout.See McGee ( ) and Halbach ( ) for surveys of some of the options.

Timothy Williamson has argued that the safety condition on knowledge places certain limits on iterations of knowledge. But at the same time, Williamson claims that interpersonal iterations of knowledge aren’t so restricted as to rule out ordinary cases. The present authors show that Williamson’s discussion misconstrues the challenge to iterated interpersonal knowledge. The proper argument against interpersonal iterations is rather what the authors call a third-person argument that does not share the major weaknesses of the argument Williamson considers. The challenge that the safety condition poses to interpersonal iterations of knowledge therefore seems robust, even in ordinary cases. But the authors also identify an underlying assumption that their argument relies on, and they show that Williamson’s original argument as well as his argument against intrapersonal iterations of knowledge rely on analogous assumptions. In assessing the extent of the clash between safety and iterated knowledge, the focus must be on the viability of these assumptions.