Trilemmas Define Syllogistics: 3 of 11

 Inevitability of Trilemmas

 

 

          Given a trilemma A;B;C, it’s easy to devise a troika that supports 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? Is there a voter’s paradox for every election of three voters? 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 differently two propositions. Call them 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 these trilemmas:

 A passes; B passes; not(A and B) passes:  Weak And Glitch         

(not a) passes; (not b) passes; (a or b) passes:  Strong Or Glitch

A passes; B passes; (A does not equal B) passes: Equivalence 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:  Equivalence

                                     

          The third troika supports the trilemma:

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

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:   Equivalence

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 glitch, a strong-or glitch, and an equivalence glitch.