In his work God and Evidence: Problems for Theistic Philosophers, Rob Lovering presents an argument for the impossibility of omniscience.
This particular argument rests on the distinction between propositional knowledge (knowledge that p) and experiential knowledge (knowledge of what it is like to x), a seemingly uncontroversial distinction.
Assuming that the distinction is justified, the argument proceeds as follows. One can differentiate between omniscience (complete, unfettered knowledge) with regard to either type of knowledge: a ‘propositionally omniscient’ entity knows all propositional knowledge, while an ‘experientially omniscient entity’ knows all experiential knowledge. A reasonable thought is that an entity isn’t fully omniscient unless it is both propositionally and experientially omniscient. Yet, no entity can be as such because necessarily, an entity that is propositionally omniscient is not experientially omniscient. This is because an experientially omniscient entity will know what it is like not to know that p (‘p’ being a variable for any proposition we like), while a propositionally omniscient entity will know that p and thus not know what it is like not to know that p.
Formally construed, the argument might look something like this:
8. (x) ~Rx [7, UG]
Where R=robustly omniscient (or, ‘fully’ omniscient), P=propositionally omniscient, E=experientially omniscient, and where g=god (though we could plug in any letter we like; the point is to instantiate the variable, use the rules of propositional logic to deduce some things, then generalize back into quantified form, which is what we do to reach premise (8))
Here are a few ways to address the argument:
A. Deny premise (1). Denying premise (1) amounts to denying the posited definition of full omniscience. “That isn’t what it is to be fully omniscient,” we might say. This is engaging in semantics — not necessarily a bad thing — since we will presumably supplant the definition of omniscience in (1) with a different one more to our liking, i.e., we will define the expression such that at least one entity can be fully omniscient.
B. Deny premise (2). This entails insisting that an entity can be both experientially and propositionally omniscient, which in this context amounts to denying that an entity that knows that p can’t also know what it is like not to know that p This may be done in a variety of ways. One way is to indicate that an entity can acquire omniscience by acquiring the knowledge of what it is like not to know that p for any proposition, then acquiring the knowledge that p, thereby achieving omniscience (since the knowledge of what it is like not to know p will presume remain even once one learns that p).
This particular solution won’t be to most theist’s liking, since their god is typically posited as having been timelessly omniscient, or, as having never not been omniscient. But there are strategies available — one imagines that an entity capable of creating the universe might somehow be able to decipher the knowledge of what it is like not to know that p without thereby ever not knowing that p.
C. Accept the argument’s soundness; shrug shoulders. It’s not clear what the import of the argument is in a theological context, other than to deny the possibility of a maximally omniscient (in the sense defined above) god. A fair question to ask at this point is, so what if a god is propositionally omniscient but not quite experientially omniscient? What is the significance of this for theologically minded persons?