In "Epistemic Permissiveness", Roger White presents several arguments against Extreme Permissivism, the view that there are possible cases where, given one's total evidence, it would be rational to either believe P, or to believe ~P. In this paper, we carefully reconstruct White's arguments and then argue that they do not succeed.

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams (...) are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the inference steps of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive -rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here. (shrink)

The principle of suspension says that when you disagree with an epistemic peer about p, you should suspend judgment about p. In “Epistemic Abstainers, Epistemic Martyrs, and Epistemic Converts,” Scott F. Aikin, Michael Harbour, Jonathan Neufeld, and Robert B. Talisse argue against the principle of suspension, claiming that it “is deeply at odds with how we view ourselves as cognitive agents.” I argue that their arguments do not succeed.

The concept of virtual machine allows us to combine the dynamical and computational hypotheses in an investigation of cognition. Van Gelder explicitly rejects this approach, but not only does it allow us to use the modelling technique most appropriate to the task, it also opens up a new range of phenomena where these techniques interact.

The principle of suspension says that when you disagree with an epistemic peer about p, you should suspend judgment about p. In “Epistemic Abstainers, Epistemic Martyrs, and Epistemic Converts,” Scott F. Aikin, Michael Harbour, Jonathan Neufeld, and Robert B. Talisse argue against the principle of suspension. In “In Defense of Epistemic Abstemiousness” I presented arguments that their arguments do not succeed, and in “On Epistemic Abstemiousness: A Reply to Bundy” they argue that my arguments are not successful. I here clarify (...) and defend my arguments. (shrink)

This suggests that those of us building artificial reasoning systems should also build what I have called extended theorem provers that evolve and compare their methods of reasoning.".