Omniscience - Theological Representations

Theological Representations

The concepts of omniscience can be defined as follows (using the notation of modal logic):

x is omniscient =_def

In words, for total omniscience:

x is omniscient =_def For all propositions p: if p (is true), then x knows that p (is true)

For inherent omniscience one interprets Kxp in this and the following as x can know that p is true, so for inherent omniscience this proposition reads:

x is omniscient =_def For all propositions p: if p (is true), then x can know that p (is true)

But a critical logical analysis shows that this definition is too naive to be proper, and so it must be qualified as follows:

x is omniscient =_def

In words:

x is omniscient =_def For all propositions p: if p (is true) and p is (logically) knowable, then x knows that p (is true)

The latter definition is necessary, because there are logically true but logically unknowable propositions such as "Nobody knows that this sentence is true":

N = "Nobody knows that N is true"

If N is true, then nobody knows that N is true; and if N is false, then it is not the case that nobody knows that N is true, which means that somebody knows that N is true. And if somebody knows that N is true, then N is true; therefore, N is true in any case. But if N is true in any case, then it is logically true and nobody knows it. What is more, the logically true N is not only not known to be true but also impossibly known to be true, for what is logically true is impossibly false. Sentence N is a logical counter-example to the unqualified definition of "omniscience", but it does not undermine the qualified one.

There are further logical examples that seem to undermine even this restricted definition, such as the following one (called "The Strengthened Divine Liar"):

B = "God does not believe that B is true"

If B is true, then God (or any other person) does not believe that B is true and thus doesn't know that B is true. Therefore, if B is true, then there is a truth (viz. "B is true") which God doesn't know. And if B is not true (= false), then God falsely believes that B is true. But to believe the falsity that B is true is to believe the truth that B is not true. Therefore, if B is not true, then there is a truth (viz. "B is not true") which God doesn't know. So, in any case there is a truth that God does not and cannot know, for knowledge implies true belief.

While sentence N is a non-knower-relative unknowability, B is a knower-relative unknowability, which means that our concept of omniscience apparently needs to be redefined again:

x is omniscient =_def

In words:

x is omniscient =_def For all propositions p: if p (is true) and p is (logically) knowable to x, then x knows that p (is true)'