## 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.