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Three open questions in the theory of one-symbol Smullyan systems

are symbols in P. A theorem of S is any string which is either an axiom of S or is derivable from the axioms by a finite number of applications of (a) uniform substitution of non-null strings in K* for variables, and (b) modus ponens. A string ~ ~ K* is said to be B-generated by S if Bet is a theorem of S, and a set L is said to be generated by S if there is a B ~ P such that L is the set of all B-generated strings. These definitions are based on the presentation of elementary formal systems in Smullyan 1961.