|
Any mathematicians (logicians) out there?
|
View this Thread in Original format
| Lira |
I'm studying Fitch's Paradox of Knowability, and I'm stuck with one little problem: the notation.
What does ◊ mean in the following context?
| quote: | Suppose the knowability principle (KP), more carefully, all truths are knowable by somebody at some time:
(KP) ∀p(p → ◊Kp).
And suppose that we are non-omniscient, that there is an unknown truth:
(NonO) ∃p(p ∧ ¬Kp).
If this existential is true, then so is an instance of it:
(1) p ∧ ¬Kp.
Now consider the instance of KP substituting line 1 for the variable p in KP:
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp)
It follows trivially that it is possible to know the conjunction expressed at line 1:
(3) ◊K(p ∧ ¬Kp)
The problem is that it can be shown, independently, that it is impossible to know this conjunction. Line 3 is false.
The independent result presupposes two very modest epistemic inferences: first, a conjunction is known, only if the conjuncts are known. Second, a statement is known, only if it true. Respectively,
(A) K(p ∧ q) Kp ∧ Kq
(B) Kp ⊢ p
Also presupposed is the validity of two modest modal inferences: first, all theorems are necessarily true (the rule of necessitation). Second, if it is necessary that ¬ p, then it is impossible that p (the defition of □). Respectively,
(C) If ⊢ p, then □p.
(D) ¬p ⊢ ¬◊p.
Consider the independent result:
(4) K(p ∧ ¬Kp) Assumption [for reductio]
(5) Kp ∧ K¬Kp from 4, by (A)
(6) Kp ∧ ¬Kp from 5, applying (B) to the right conjunct
(7) ¬K(p ∧ ¬Kp) from 4-6, by reductio, discharging assumption 4
(8) ¬K(p ∧ ¬Kp) from 7, by (C)
(9) ¬◊K(p ∧ ¬Kp) from 8, by (D)
Line 9 contradicts line 3. So a contradiction follows from KP and NonO. The advocate of the view that all truths are knowable must deny that we are non-omniscient:
(10) ¬∃p(p ∧ ¬Kp).
And it follows from that that all truths are actually known:
(11) ∀p(p → Kp).
The ally of the view that all truths are knowable by somebody is forced absurdly to admit that every truth is known by somebody. |
I just want the mathematical notation to make sense :p |
|
|
| Events@Spec |
| Wow, that's another language. I doubt any of the c0r's will knwo anything about that. Maybe some people that ust poke around. |
|
|
| Yan |
| quote: | Originally posted by Events@Spec
Wow, that's another language. I doubt any of the c0r's will knwo anything about that. Maybe some people that ust poke around. |
You'd be surprised how many well educated people we have around the COR. Lira wouldn't just post here not expecting some sort of smart reply. |
|
|
| Lilith |
| quote: | Originally posted by Lira
I just want the mathematical notation to make sense :p |
Perhaps a music forum isnt really the place to ask coming up to new years eve when we're working on getting bombed out of our skulls... not that the date makes much of a difference :haha: |
|
|
| kadomony |
| quote: | Originally posted by Lira
I'm studying Fitch's Paradox of Knowability, and I'm stuck with one little problem: the notation.
What does ◊ mean in the following context?
I just want the mathematical notation to make sense :p |
You sure thats a sound statment? |
|
|
| dallastar |
I dated one!
he was boring, he's still out there!~:happy2: |
|
|
| Lira |
+1 @ what Yan said. We're often surprised by the people that visit the CORe :)
| quote: | Originally posted by kadomony
You sure thats a sound statment? |
Absolutely. If it looks weird, it might be due to bad encoding, so here's the source:
http://www.seop.leeds.ac.uk/entries/fitch-paradox/ |
|
|
| dallastar |
 ;) |
|
|
| sensorium |
According to your source: 'it is possible that' |
|
|
| Lira |
:stongue: :stongue: :stongue:
Heck, I completely missed that line! See, sometimes you don't need an expert to point these things out :D
Cheers, Sensorium :p |
|
|
|
|
