## Wednesday, June 26, 2013

### On Troikas: Anti-Sorites

Anti-Sorites

An anti-sorites is a multilemma; that is, a set of statements, not all of which can be true. Therefore all but one true implies the last is false; a form of sorites reasoning. Any anti-sorites encodes several sorites at once.
You make an anti-sorites by appending the negation of a sorite’s classical conclusion. This you can then ‘untangle’, by row swaps and modal identities. For instance, take this sorites:

Some lunches are nutritious;
Anything nutritious is good for you;
Only valuable things are good for you;
Anything valuable is paid for;
free means paid for.

The logical conclusion to this sorites is
Some lunch is free.
- being nutritious, good for you, valuable, paid for, and hence free. After all, you paid for it.
To make an anti-sorites, replace that conclusion with “no lunch is free”,  and get:

Some lunches are nutritious;
Anything nutritious is good for you;
Only valuable things are good for you;
Anything valuable is paid for;
Free means paid for;
No lunch is free.

This anti-sorites, untangled by row swaps and modal identities, becomes;

Some lunches are nutritious;
Anything nutritious is good for you;
Anything good for you is valuable;
Anything valuable is paid for;
Anything paid for is free;
No lunch is free.

This is a SAAAAN anti-sorites. The inner “all” sequence collapses to “Anything nutritious is free”, resulting in a Some-All-None trilemma. Either the initial “some” statement is false, or the final “none” statement is false, or one of the chain of “all”s fails.
Here’s an anti-sorites, row-swapped and then remodulated:

Butterflies are free;
Not all lunches are bland;
Only butterflies are beautiful;
All unbeautiful things are bland;
There’s no such thing as a free lunch.

Not all lunches are bland;
All unbeautiful things are bland;
Only butterflies are beautiful;
Butterflies are free;
There’s no such thing as a free lunch.

Some lunches are not bland;
All not-bland things are beautiful;
All beautiful things are butterflies;
All butterflies are free;
No lunch is free.

Here are some anti-sorites derived from sorites by Lewis Carroll, then untangled:

No interesting poems are unpopular among people of real taste;
No modern poetry is free of affectation;
All of your poems are on the subject of soap-bubbles;
No affected poetry is popular among people of real taste;
No ancient poem is on the subject of soap-bubbles;
Some of your poems are interesting.

Some interesting poems are yours;
All of your poems are on the subject of soap-bubbles;
All poems on the subject of soap-bubbles are modern;
All modern poetry is affected;
All affected poetry is unpopular among people of real taste;
No interesting poems are unpopular among people of real taste.

No kitten that loves fish is unteachable;
No kitten without a tail will play with a gorilla;
Kittens with whiskers always love fish;
No teachable kitten has green eyes;
No kittens have tails unless they have whiskers;
Some kitten with green eyes will play with a gorilla.

Some kitten with green eyes will play with a gorilla;
Any kitten that will play with a gorilla has a tail;
All kittens with tails have whiskers;
All kittens with whiskers love fish;
All kittens who love fish are teachable;
No kittens with green eyes are teachable.

Things sold on the street are of no great value;
Nothing but rubbish can be had for a song;
Eggs of the Great Auk are very valuable;
It is only what is sold on the street that is really rubbish;
Eggs of the Great Auk can be had for a song.

Eggs of the Great Auk can be had for a song;
Anything that can be had for a song is really rubbish;
Anything that is really rubbish is sold on the street;
Anything sold on the street is not very valuable;
Eggs of the Great Auk are very valuable.

This multilemma has a single object at beginning and end, with a property inverting during the A sequence; so an “xAAA~x” anti-sorites.
We know that one of the statements in the Great Auk Anti-Sorites is false. This is a lot more diffuse than a trilemma. An anti-sorites is like the game of telephone, with an error happening somewhere in the deductive chain; then a ‘some’ vanishes into a ‘none’, or the Great Auk’s eggs turn upside down.
All of these can be supported by troikas. Each Stooge need merely deny one of three different terms of the anti-sorites. Perhaps Moe could deny the Some statement, Larry could deny one of the Alls, and Curly deny the None.