  For half a century, authors have weakened the rule of necessitation in
various more or less ad hoc ways in order to make inconsistent systems
consistent. More recently, necessitation was weakened in a systematic way, not
for the purpose of resolving paradoxes but rather to salvage the deduction
theorem for modal logic. We show how this systematic weakening can be applied
to the older problem of paradox resolution. Four examples are given: a
predicate symbol S4 consistent with arithmetic; a resolution of the surprise
examination paradox; a resolution of Fitch's paradox; and finally, the
construction of a knowing machine which knows its own code. We discuss a
technique for possibly finding answers to a question of P. \'Egr\'e and J. van
Benthem.
