Once again I am overtaken by writing, and must blog more
chapters of “On Troikas and Trilemmas”. Over the next four days I shall blog:

*Inevitability of Trilemmas*

*Finite Boundless Lines*

*Anti-Sorites*

*Disinduction and the Heap*
*******

**Inevitability of Trilemmas**

Given a trilemma A;B;C, it’s easy to
devise a troika that will yield it. Let Moe vote for B and C but not A; let
Larry vote for C and A but not B; let Curly vote for A and B but not C; then
2/3 majorities support each of A, B and C; but their conjunction fails
unanimously. Therefore there is a troika for every trilemma.

But is the reverse true? Does every
election of three voters produce a voter’s paradox? Yes, inevitably! For if
Moe, Larry and Curly truly have three different agendas, then those agendas
must differ by at least two bits; for one bit can distinguish only between two.

The three voters evaluate at least
two propositions differently. Call those
propositions A and B; the voters can evaluate them four different ways:

Moe votes for: A, B

Larry votes for: A, not B

Curly votes for: not A, B

Moe votes for: not A, B

Larry votes for: A, not B

Curly votes for: not A, not B

Moe votes for: A, B

Larry votes for: A, not B

Curly votes for: not A, not B

Moe votes for: A, B

Larry votes for: not A, B

Curly votes for: not A, not B

Let a = not A, and b = not B, then the
first troika supports the trilemma:

A
passes; B passes; not(A and B) passes:
the “Weak And glitch”.

(not a) passes; (not b)
passes; (a or b) passes: the “Strong Or
glitch”.

A passes; B passes; (A
does not equal B) passes:
“Disequivalence Glitch”.

The second troika supports these
trilemmas:

not
A passes; not B passes; (A or B)
passes: Strong Or

a passes; b passes;
not(a and b) passes: Weak And

a passes; b passes; (a
does not equal b) passes: Disequivalence

The third troika supports the
trilemma:

A passes; not B passes;
(A equals B) passes: Disequivalence

A passes; b passes; not(A
and b) passes: Weak And

(not a) passes; (not B)
passes; (a or B) passes: Strong Or

The fourth troika supports the
trilemma:

a passes; not b passes;
(a equals b) passes: Disequivalence

a passes; B passes; not(a
and B) passes: Weak And

(not A) passes; (not b)
passes; (A or b) passes: Strong Or

So any triple of voters yields a
weak-and, a strong-or, and a disequivalence glitch.

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