Matthias Jenny

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 article, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality offered 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.

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.