28 Jan '16 20:23

I was looking at the Stanford Philosophy site's article on Epistemic paradoxes [1]. It contains the following argument apparently due to Alonzo Church. Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown. The proof proceeds by assuming the converse:

Let K() be a modal operator meaning that it's argument is known by at least one agent.

P & ¬K(P) - states that there is a true proposition P and it is unknown.

We need the following axioms for the modal operator K():

T) K(P) -> P - If P is known P is true because knowledge is justified true belief.

B) K(P & Q) <-> K(P) & K(Q) - knowledge distributes over conjunction.

K) K(P -> Q) -> K(P) -> K(Q)

1. K(P & ¬K(P)) ⊦K(P) & K(¬K(P)) From B

2. K(¬K(P)) ⊦¬K(P) From T

3. K(P & ¬K(P)) ⊦K(P) & ¬K(P) From 1 and 2

4. ⊦¬K(P&¬K(P)) - 3 is a contradiction so the assumption on line 1 was false.

Because the assumption was dismissed the conclusion is

This left me wondering about omnisciences. Can I justify my agnostic stance on the basis of epistemic logic? Let O be the proposition: "There is an omniscient entity.", if there is an omniscient entity then any given proposition P must be known, in symbols O -> (P -> K(P)). To show that it is unknowable that there is

(P -> K(P)) -> O.

In the case of a single entity I can't, but suppose a collective is good enough. Then since by assumption any given proposition is known by at least one agent there is a smallest set of agents who are collectively omniscient. However they may not be individually infallible so although collectively they know all things, it's no use as not all of them realise which of their beliefs count as knowledge (a confounded omniscience). However it's enough for me to claim they are an omnipotent entity and write:

O <-> (P -> K(P))

This means that

¬O <-> ¬(P -> K(P))

and it's not hard to show that ¬(P -> K(P)) <-> (P & ¬K(P)) which in combination with axiom K gives us: ⊦¬K(¬O)

In other words it is unknowable that there is not an omniscience. If there is an omniscience (even a confounded composite one) then it must know it exists and is omniscient so the statement is just that one cannot know things that aren't true. If the omniscience does not exist then it is impossible to know that it does not exist.

[1] http://plato.stanford.edu/entries/epistemic-paradoxes/#KnowPar

[2] http://plato.stanford.edu/entries/logic-epistemic/

Let K() be a modal operator meaning that it's argument is known by at least one agent.

P & ¬K(P) - states that there is a true proposition P and it is unknown.

We need the following axioms for the modal operator K():

T) K(P) -> P - If P is known P is true because knowledge is justified true belief.

B) K(P & Q) <-> K(P) & K(Q) - knowledge distributes over conjunction.

K) K(P -> Q) -> K(P) -> K(Q)

1. K(P & ¬K(P)) ⊦K(P) & K(¬K(P)) From B

2. K(¬K(P)) ⊦¬K(P) From T

3. K(P & ¬K(P)) ⊦K(P) & ¬K(P) From 1 and 2

4. ⊦¬K(P&¬K(P)) - 3 is a contradiction so the assumption on line 1 was false.

Because the assumption was dismissed the conclusion is

*unconditionally*true. If a proposition is true but unknown then the combined proposition is unknowable. This isn't a great problem for an omniscience as it will know P and therefore ¬K(P) is false and the omniscience isn't required to "know" things that aren't true.This left me wondering about omnisciences. Can I justify my agnostic stance on the basis of epistemic logic? Let O be the proposition: "There is an omniscient entity.", if there is an omniscient entity then any given proposition P must be known, in symbols O -> (P -> K(P)). To show that it is unknowable that there is

*not*an omniscience I need to argue that:(P -> K(P)) -> O.

In the case of a single entity I can't, but suppose a collective is good enough. Then since by assumption any given proposition is known by at least one agent there is a smallest set of agents who are collectively omniscient. However they may not be individually infallible so although collectively they know all things, it's no use as not all of them realise which of their beliefs count as knowledge (a confounded omniscience). However it's enough for me to claim they are an omnipotent entity and write:

O <-> (P -> K(P))

This means that

¬O <-> ¬(P -> K(P))

and it's not hard to show that ¬(P -> K(P)) <-> (P & ¬K(P)) which in combination with axiom K gives us: ⊦¬K(¬O)

In other words it is unknowable that there is not an omniscience. If there is an omniscience (even a confounded composite one) then it must know it exists and is omniscient so the statement is just that one cannot know things that aren't true. If the omniscience does not exist then it is impossible to know that it does not exist.

[1] http://plato.stanford.edu/entries/epistemic-paradoxes/#KnowPar

[2] http://plato.stanford.edu/entries/logic-epistemic/