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.