Wednesday, May 15, 2013

On Troikas 3: More Glitches



            More Glitches


Here is a Sorites Failure:  (in honor of Lewis Carroll)
Curly believes:   that babies are logical;
that illogical people are despised;
and that despised people cannot manage a crocodile;
and babies can manage a crocodile, being logical and respected.
Larry believes:   that babies are illogical;
that illogical people are not despised;
and that despised people cannot manage a crocodile;
and babies can manage a crocodile, being illogically respected.
Moe believes: that babies are illogical;
that illogical people are despised;
and that despised people can manage a crocodile;
and babies can manage a crocodile, being despised.
Majorities agree: (L&M)  Babies are illogical;       
(M&K)  Illogical people are despised;
(K&L)   Despised people cannot manage a crocodile;
yet they all agree that babies can manage a crocodile!


The sorites failure implies this heap glitch:
Curly says: 100 grains of sand are the smallest heap.
Larry says: 101 grains are the smallest heap.
Moe says:  102 grains are the smallest heap.
Majorities believe:
* 100 grains of sand are not a heap.
* 101 grains are a heap.
* 101 grains are not the smallest heap.

Here is a Set Loop:

                      Moe              Larry            Curly
Apes             {Dr.0}        {Dr.0, #1}          {}
Bozos         {Dr.0, #1}          {}              {Dr.0}
Crooks              {}             {Dr.0}         {Dr.0, #1}

Majorities agree that:
* All apes are bozos.
* All bozos are crooks.
* All crooks are apes.
Yet not the reverse! That is:
* Not all bozos are apes.
* Not all crooks are bozos.
* Not all apes are crooks.
And worst of all, every Stooge agrees:
** These three classes form a BARBARA syllogism:
all X are Y; all Y are Z; therefore all X are Z.
So the Stooges, those bunglers, made a huge mess of BARBARA, in the very act of affirming it! These spinning set loops make mincemeat of classical logic. How barbaric!

Here is a Double Bind:
Moe                 Larry     Curly
Can we choose chocolate?                            yes                   no             no
Can we choose vanilla?                                no                   no            yes
Do we have to choose?                                 yes                   no            yes
Majorities believe:                
* We can’t choose chocolate.
* We can’t choose vanilla.
* We have to choose.
The Stooges are in a collective double-bind, even though no one of them is!

Here is an Orlov Doubt State:
Moe                 Larry               Curly
Can we tell chocolate from vanilla?             yes                   no                    no
Can we choose at random?                           no                   no                    yes
Do we have to choose?                                 yes                   no                    yes
            Majorities believe:    
* We can’t tell chocolate from vanilla.
* We can’t choose at random.
* We have to choose.
The collective is in a quandary, even though no individual Stooge is.


               In this Failed Reductio troika, reductio ad absurdum fails:
Moe: Alice did not fall down a rabbit hole
Larry: Alice fell down a rabbit hole, and I am not a monkey’s uncle.
Curly: Alice fell down a rabbit hole, and I am a monkey’s uncle.
So by 2/3 majorities each:
               MK: If Alice fell down a rabbit hole, then I am a monkey’s uncle.
               LM: If Alice fell down a rabbit hole, then I am not a monkey’s uncle.
               KL: Alice fell down a rabbit hole.


               In general the Failed Reductio troika looks like:
               Moe:  not A
               Larry: A and not B
               Curly: A and B
So by 2/3 majorities each:
               MK: If A then B
               LM: If A then not B
               KL: A

               Here is its dual; the “Failed Dilemma”;
               Moe:  B
               Larry: not B and A
               Curly: not B and not A
So by 2/3 majorities each:
               MK: A implies B
               LM: not A implies B
               KL: not B

               For instance, this “Permission Troika”:
               Moe:  Everything is permitted.
               Larry: Not everything is permitted, and God does not exist.
               Curly: Not everything is permitted, and God does exist.
So by 2/3 majorities each:
               MK: If there is no God, then everything is permitted.
               LM: If there is a God, then everything is permitted.
               KL: Not everything is permitted.

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