Suppose that (S) is true. Then its negation is known and hence true. However, if its negation is true, then (S) must be false. Therefore (s) is false, or what is the name, the negation of (S) is true.
This paradox and its accompanying reasoning are strongly reminiscent of the Lair Paradox that (in one version) begins by considering a sentence this sentence is false and derives a contradiction. Versions of both arguments using axiomatic formulations of arithmetic and Gödel-numbers to achieve the effect of self-reference yields important meta-theorems about what can be expressed in such systems. Roughly these are to the effect that no predicates definable in the formalized arithmetic can have the properties we demand of truth (Tarskis Theorem) or of knowledge (Montague, 1963).
These meta-theorems still leave us; with the problem that if we suppose that we add of these formalized languages predicates intended to express the concept of knowledge (or truth) and inference - as one mighty does if logic of these concepts is desired. Then the sentence expressing the leading principles of the Knower Paradox will be true.
Explicitly, the assumption about knowledge and inferences are:
(1) If sentences A are known, then a.
(2) (1) is known?
(3) If ‘B’ is correctly inferred from ‘A’, and ‘A’ is known, then ‘B’ is known.
To give an absolutely explicit t derivation of the paradox by applying these principles to (S), we must add (contingent) assumptions to the effect that certain inferences have been done. Still, as we go through the argument of the Knower, these inferences are done. Even if we can somehow restrict such principles and construct a consistent formal logic of knowledge and inference, the paradoxical argument as expressed in the natural language still demands some explanation.