Trilemmas Define Syllogistics, 1 of 11

 Trilemma, Troika, and Triad

 

 

A trilemma is a trio of sentences (a; b; c), any two of which can be true, but not all three. For instance:

Superman can fly;

Clark Kent can’t fly;

Superman is Clark Kent.

 

Any two imply the negation of the third; that is the Two Thirds Rule. Therefore any trilemma yields a triad of deduction rules. In this case:

If Superman can fly and Clark Kent can’t fly,

then Superman is not Clark Kent.

If Clark Kent can’t fly and Superman is Clark Kent,

then Superman can’t fly.

If Superman is Clark Kent, and Superman can fly,

then Clark Kent can fly.

 

These deduction rules are exemplified by a trio of models; a three-voter election, or troika, here represented by the Three Stooges:

Moe:

Superman can fly, Clark Kent can’t fly, Superman is not Clark Kent.

Larry:

Clark Kent can’t fly, Superman is Clark Kent, Superman can’t fly.

Curly:

Superman is Clark Kent, Superman can fly, Clark Kent can fly.

 

The Three Stooge’s votes combine, by majority rule, to affirm each sentence in the trilemma, though none of the Stooges affirm all three. Thus the trilemma is a voter’s paradox.

A trilemma is the voter’s-paradox generated by a majority rule in a troika; the troika exemplifies a triad of deduction rules; the triad of rules is a consequence of the 2/3 rule on the trilemma.

 

So trilemma, troika, and triad are themselves an interdependent triple of, respectively, an absurdity, three possibilities, and three laws.

         

In general the trilemma (A; B; C) is associated with this troika:

 

          Moe: A; B; not-C

          Larry: A; not-B; C

          Curly: not-A; B; C

 

          Here is its associated rule triad:

 

          From A and B, deduce not-C

From C and A, deduce not-B

From B and C, deduce not-A