The Elimination of Direct Self-reference

This paper provides a procedure which, from any Boolean system of sentences, outputs another Boolean system called the 'm-cycle unwinding' of the original Boolean system for any positive integer m. We prove that for all m > 1, this procedure eliminates the direct self-reference in that the m-cycle unwinding of any Boolean system must be indirectly self-referential. More importantly, this procedure can preserve the primary periods of Boolean paradoxes: whenever m is relatively prime to all primary periods of a Boolean paradox, this paradox and its m-cycle unwinding have the same primary periods. In this way, we can produce an indirectly self-referential Boolean paradox with the same periodic characteristics as a known Boolean paradox.