Here is an argument inspired by the Gale-Pruss cosmological argument.
- (Premise) Every first-order truth is knowable.
- (Premise) The conjunction of all basic first-order truths exists and is a first-order truth.
- (Premise) If all the basic first-order truths of a world w1 hold at a world w2, then w2=w1.
- Let p be the conjunction of all basic first-order truths. Let w be a world where there is a being, x, that knows p. (By 1 and 2)
- (Premise) Necessarily, if someone knows p, then p is true.
- p is true at w. (4 and 5)
- w is the actual world. (3, 4 and 6)
- There actually is a being that knows the conjunction of all basic first-order truths. (4 and 7)
Note that the restriction to first-order truths makes this not be just the standard knowability paradox.
I am not giving a definition of basicness. The crucial constraint is that the notion be such that both (2) and (3) hold. If one says that every first-order truth counts as basic, then (3) is very plausible, but (2) is less clear because one worries about a conjunction that has itself as a conjunct. Or maybe we could say that the basic first-order truths are truths expressed by literals, namely truths of the form of the proposition that P(a1,...,an) for some predicate P or its negation. This requires a free logic, an existence predicate, and propositions about all possible non-existent entities. Alternately, we might take "basic" to just mean "fundamental".
Note added later: I forgot the word "basic" in (3) in the original argument.