Consider this definition of omniscience: “God believes all true propositions and does not believe in any of their negations.” One might wonder whether this is logically equivalent to “God knows all true propositions, and does not believe any false propositions” or to “For any proposition P, if P is true then God knows it, and if P is false then God does not believe it.”
Let’s start here: a being X is Omniscient if and only if, for any proposition P, if P is true then X knows P, and if P is false, then X does not believe P.
(∀x)(Ox ≡ (∀y)((Ty ⊃ Kxy) & (Fy ⊃ ∼Bxy)))
Where Ox is “x is Omniscient,” Ty is “y is true,” Kxy is “x knows y,” Fy is “y is false,” and Bxy is “x believes y”. However, for a proposition to be false just means for some proposition to be ‘not’-true. So:
(∀x)(Ox ≡ (∀y)((Ty ⊃ Kxy) & (~Ty ⊃ ∼Bxy)))
However, for God to know something presumably requires only that it be true, since it isn’t possible for God to have unjustified beliefs. So,
(∀x)(Ox ≡ (∀y)((Ty ⊃ Bxy) & (~Ty ⊃ ∼Bxy)))
However, that seems to be logically equivalent to the following:
(∀x)(Ox ≡ (∀y)((Ty ≡ Bxy)))
So, a being is omniscient just in case that being believes all true propositions and doesn’t believe any of their negations.