One and a Half Problems Solved

          One and a Half Problems Solved

 

 

1.    Liar’s paradox.

 

The Liar’s Paradox is the sentence “This sentence is false”. It can be neither true nor false, and so nothing yielding it can exist in anything governed by binary logic, which denies the existence of paradox. Russell, Cantor, and Goedel constructed such paradoxes; these yield impossibility theorems: incomplete set comprehension, no counting of the continuum, and no complete consistent deductive arithmetic.

I have tried defusing the paradoxes by accepting their existence. My attempt has met with complete success.

Kleenean and diamond logic, which include paradox, support naive set theory, a single infinity, and logistic arithmetical deduction. This is the “Paralogistic Program”: math founded on paradox logic.

          The liar’s paradox also poses a personal problem. It too is fully resolved. I know who I am; I am a fool, but I know that I’m a fool.

 

2.    Voter’s paradox.

 

A voter’s paradox is when three voters support, by shifting majorities, trios of mutually-inconsistent propositions. These “trilemmas” imply Arrow’s Impossibility Theorem: there is no perfect government. All systems are, on some occasions, cruel, weak, foolish, or perverse. Monarchy is cruel, consensus is weak, majority rule is foolish, and utopianism is perverse.

I have tried defusing the voter’s paradox by accepting its existence. My results so far are partial.

A full result: Trilemmas define deduction. Any trilemma defines a deductive system: from any two clauses of the trilemma, deduce the negation of the third. For instance, the some-all-none trilemma yields all of Aristotelian logic.

But that does not resolve the political problem. It just sharpens it.

          A partial result: loop logic. It has three elements; Rock, Scissors, Paper, ordered cyclically as in ro-sham-bo: scissors cuts paper, paper covers rock, and rock breaks scissors. This embodies the “perversity” option in the Impossibility Theorem; a loop where everyone expects a line. Loop logic defines Kleenean logic and Z mod 3 arithmetic, each one three times.

Ro-sham-bo is a children’s choosing game. Ro-sham-bo is fair and decisive, but inwardly chaotic, for every predictable strategy can be bested. Its optimum strategy is random SRP at 1/3 probability each. Ro-sham-bo enforces chaos, and loop logic defines that chaos. I have yet to find loop logic’s defining axioms and normal forms, if they exist.

          The voter’s paradox poses a political problem, which I have yet to fully resolve. My attempt to defuse the Impossibility Theorem yielded both Logic and Chaos. It is as if I were looking for Utopia, but instead I found Aristotle and a children’s game.