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.