Re: Fitch's paradox and OWA

 Jan Hidders says...
>
>On 3 jan, 14:09, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
>> Jan Hidders says...
>>
>> >The purpose of (3) was only to illustrate the translation of formulas
>> >in the original logic to your logic. You are right that by itself it
>> >does not show the paradox. But if this translation exists then all
>> >formulas used in the proof of the paradox will have their equivalents
>> >in your logic.
>>
>> Yes, but my point is that in the more expressive logic, the
>> knowability principle can be expressed as
>>
>> forall p:P, exists w:W, k(w,p)
>
>Er, I think you forgot the part where it requires that p is true.

Right. Thanks.

>But if you fix that, then this is indeed equivalent with the one used in
>the Stanford page. This one will still lead to the conclusion that all
>truths are known.

No, it doesn't. I already went through this. In the Stanford logic, if p is some proposition such that p & ~K(p), then the application of the knowability principle gives (in some world w')

K(p & ~K(p))

which is a contradiction. My rule does *not* lead to that conclusion. Instead, we have, for some world w,

p & ~k(w,p)

If we apply my version of the knowability principle, we get, for some world w'

k(w',(p & ~k(w,p)))

which is *not* a contradiction. Proposition p is known in world w', but not in world w.

For clarification, the propositions in this type theory are *non-modal*.

>> The original knowability principle, when translated into this
>> new logic, would look something like this:
>>
>> forall f:W -> P, forall w:W, f(w) -> exists w':W, f(w') & k(w',f(w'))
>>
>> The "propositions" of modal logic are actually functions on worlds.
>
>Really? This is actually a stronger principle that implies the
>previous one since as a particular case I can take for f the function
>that maps each world to the same predicate p in P.

Right. It's *too* strong, which is why it leads to a contradiction (together with the principle of non-omniscience).

>Also, I don't understand what you mean by "propositions are actually
>functions on world" except that the same proposition can have a
>different semantics in different worlds,

That's exactly what I mean. For each modal proposition p (which varies from world to world) we can associate a function f_p from worlds to nonmodal propositions as follows:

f_p(w) == the nonmodal proposition "p is true in world w"

>and that was already taken into account in the old semantics.

Yes. It's the old *syntax* that was inadequate to express a reasonable knowability principle.

--
Daryl McCullough
Ithaca, NY