Trilemmas Yield Syllogistics: 6 of 11

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, after ‘untangling’ it by row swaps and modal identities. For instance, take this sorites:

          Some results are known;
          Only systematic things are proven;

Anything under control is guaranteed;

Only unsystematic things are out of control;

Anything unproven is unknown.

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

Some results are known;
          Anything known is proven;
          Anything proven is systematic;

Anything systematic is under control;
          Anything under control is guaranteed.

 

The logical conclusion to this sorites is
                   Some results are guaranteed.
          - since some results are known, proven, systematic, under control, hence guaranteed. To make an anti-sorites, replace that conclusion with “no results are guaranteed”,  and get:

 

          Some results are known;
          Anything known is proven;
          Anything proven is systematic;

Anything systematic is under control;
          Anything under control is guaranteed;

No results are guaranteed.

 

This is a SAAAAN anti-sorites. The inner “all” sequence collapses to “Anything known is guaranteed”, 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.