- An Axiomatic Version of Fitch's Paradox.Samuel Alexander - 2013 - Synthese 190 (12):2015-2020.A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the (...)
about D: K(!x) -> !P(x)
Université paris 1
notation : I use ! for 'not'
Perhaps you can avoid paradox but you have to admit this very strange proposition :
K !K x -> !P K x
If you know that you ignore (x) it's impossible that you know (x)
I don't see how it could be compatible with the knowability principle :
x -> P K x
else you can't have
(x) and (K !K x)
(excuse me if this message is out of place, I ignore the policy of tis forum,
excuse also my probable mistakes in english)
about D: K(!x) -> !P(x)
Reply to Bruno Maret
Ohio State University
Thank you very much for the comment.
Yes, when I wrote the paper I was unaware K(!x)->!P(x) was so controversial. (I was also, obviously, mistaken that it came from Salerno. I intend to write an erratum to correct that.) The axiom was derived from the less controversial rule of inference: "From !x, deduce !P(x)". If knowledge and provability are identified, the rule and the axiom are equivalent. But I am an ignorant man and did not realize that not all philosophers identify knowledge and provability!
While this state of affairs greatly weakens the "paradox" part of the paper, it actually strengthens somewhat the "resolution" part. When a paradox is "resolved", the stronger the original assumptions, the more surprising the resolution. (To be clear, I use "resolve" in Chow's sense, so that a paradox can have multiple resolutions and it's subjective which is the best resolution)
K(!x)->!P(x) says that the knower is extremely cautious: if there is any tiny possibility that there is an elephant in Bertrand Russel's room, this knower refuses to know there's no elephant. The knowability thesis seems a lot less plausible for such cautious knowers. But the paper shows that, in spite of this, knowability is nevertheless consistent, even with such paranoid knowers, provided the knower is not required to know his own factivity.