Tuesday, August 31, 2021

Trilemmas Yield Syllogistics: 5 of 11

Trilemma Deduction

All of the trilemmas in the “Spoofing Classical Logic” section are variants of these three Anti-Syllogisms:

Some-All-None Trilemma

Some A are B;

All B are C;

No A are C.

For instance; some angels are bats; all bats are cats; no angels are cats.

All-All-None Trilemma

All A are B;

All B are C;

No A are C.

For instance; all angels are bats;  all bats are cats; no angels are cats.

All-None-All Trilemma

All A are B;

No B are C;

All A are C.

For instance; all angels are bats;  no bats are cats; all angels are cats.

The Some-All-None trilemma is perilously close to absurdity. It is a disappearing trick, a kind of sleight of hand. Some A is B; all B is C; no A is C; deny one! Yet the anti-syllogism’s absurdity makes it a deduction engine. From any two of the Some-All-None trilemma, deduce the negation of the third. The trilemma is both a parody of logic, and a succinct summary of it.

For instance, from this trilemma:

Some angels are bats;

All bats are cats;

No angels are cats.

From: some angels are bats; all bats are cats;

Deduce: some angels are cats.

From: all bats are cats; no angels are cats;

Deduce: no angels are bats.

From: no angels are cats; some angels are bats;

Deduce: some bats aren’t cats.

Both the All-All-None and All-None-All trilemmas implicitly assume that A exists; but modern logic does not assume that, and allows vacuous implication. For instance;

All dragons are red;

Nothing red is blue;

All dragons are blue.

All three are true, because there are no dragons! This is simply a refutation of dragons. If you let existing things like Komodo dragons be called dragons, then one of these three propositions fails.

Classical syllogist logic assumed that ‘all’ had existential import; that is,

“All A are B”  =   “All A are B” and “Some A are B”.

Without that assumption, the vacuous case can apply, and both All-All-None and All-None-All trilemmas can refer to nonexistent A. If existential import is a stated assumption, then both All-All-None and All-None-All trilemmas reduce to the Some-All-None trilemma.

Applying the Two Thirds Rule to the Some-All-None Trilemma yields Some-All-None Deduction:

From any two of:

Some A are B

All B are C

No A are C

deduce the negation of the third.

To that add these modal identities:

Swap:

All A are B                            =       All not-B are not-A

No A are B                            =       No B are A

Some A are B                        =       Some B are A

Some A are not B                 =       Some not-B are not not-A

Negation:

Not (all A are B)                   =       Some A are not-B

Not (no A are B)                   =       Some A are B

Not (some A are B)              =       No A are B

Not (some A are not-B)       =       All A are B

Mode Switch:

All A are B                   =       No A are not-B

No A are B                   =       All A are not-B

Some A are B              =       Some A are not not-B

From one side of an equation, deduce the other.

Modal identities plus Some-All-None Deduction yields most of Aristotle’s syllogistics. Adding existential import yields the rest.  For instance, modal identities, substitutions and swap can transform the Some-All-None Trilemma to Barbarism in three steps thus:

Some A are not not-B

All not-C are not-B

All A are not-C                      (by modal identities)

Some X are not-Z

All Y are Z

All X are Y                    (substitute X=A, Y=not-C, Z = not-B)

All X are Y

All Y are Z

Some X are not-Z                  (swap)

An anti-syllogism is not a syllogism itself, but it’s always ready to explode into three conflicting syllogisms. For instance; by 2/3 Rule and modal identities, Barbarism encodes these three classical logic rules:

From: All X are Y; all Y are Z; Deduce: all X are Z.

From: All Y are Z; some X are not-Z; Deduce: some X are not-Y.

From: Some X are not-Z; all X are Y; Deduce: some Y are not-Z.

The last two can be changed, by substitutions and swaps, to:

From: Some X are Z; no Y are Z; Deduce: some X are not-Y.

From: Some X are Z; All X are Y; Deduce: some Y are Z.

Friday, August 27, 2021

Trilemmas Define Syllogistics: 4 of 11

Spoofing Classical Logic

There are nineteen syllogisms of classical logic. Each one has a trilemma that spoofs it. These are the syllogisms and their spoofs:

1:AAA: Every M is P; every S is M; therefore every S is P

Contra 1:AAA

Moe: Some M is not P; every S is M; some S is not P

Larry: Every M is P; some S is not M; some S is not P

Curly: Every M is P; every S is M; every S is P

So by 2/3: every M is P; every S is M; some S is not P.

Exercise for the student: find troika and triad for this trilemma:

All brainiacs wear glasses;

All scientists are brainiacs;

Some scientists don’t wear glasses.

1:EAE: No M is P; every S is M; therefore no S is P

Contra 1:EAE

Moe: Some M is P; every S is M; some S is P

Larry: No M is P; some S is not M; some S is P

Curly: No M is P; every S is M; no S is P

So by 2/3: No M is P; every S is M; some S is P

Exercise for the student: find troika and triad for this trilemma:

No beggar is honest;

All televangelists are beggars;

Some televangelists are honest.

1:AII: Every M is P; some S is M; therefore some S is P

Contra 1:AII

Moe: some M is not P; some S is M; no S is P

Larry: Every M is P; no S is M; no S is P

Curly: Every M is P; some S is M; some S is P

So by 2/3: Every M is P; some S is M; no S is P

Exercise for the student: find troika and triad for this trilemma:

All men are beasts;

Some saints are men;

No saints are beasts.

1:EIO: No M is P; some S is M; therefore some S is not P

Contra 1:EIO

Moe: Some M is P; some S is M; every S is P

Larry: No M is P; no S is M; every S is P

Curly: No M is P; some S is M; some S is not P

So by 2/3: No M is P; some S is M; every S is P

Exercise for the student: find troika and triad for this trilemma:

The only good writer is a dead writer;

Some Americans are good writers;

Every American is alive.

2:EAE: No P is M; every S is M; therefore no S is P

Contra 2:EAE

Moe: Some P is M; every S is M; some S is P

Larry: No P is M; some S is not M; some S is P

Curly: No P is M; every S is M; no S is P

So by 2/3: No P is M; every S is M; some S is P

Exercise for the student: find troika and triad for this trilemma:

Teachers are never enthusiastic;

You are always enthusiastic;

You are sometimes a teacher.

2:AEE: Every P is M; no S is M; therefore no S is P

Contra 2:AEE

Moe: Some P is not M; no S is M; some S is P

Larry: Every P is M; some S is M; some S is P

Curly: Every P is M; no S is M; no S is P

So by 2/3: Every P is M; no S is M; some S is P

Exercise for the student: find troika and triad for this trilemma:

Every dog has his day;

No skinflint has his day;

Some skinflints are dogs.

2:EIO: No P is M; some S is M; therefore some S is not P

Contra 2:EIO

Moe: Some P is M; some S is M; every S is P

Larry: No P is M; no S is M; every S is P

Curly: No P is M; some S is M; some S is not P

So by 2/3: No P is M; some S is M; every S is P

Exercise for the student: find troika and triad for this trilemma:

No president is a moron;

Some illiterates are morons;

Every illiterate is president.

2:AOO: Every P is M; some S isn’t M; therefore some S isn’t P

Contra 2:AOO

Moe: Some P is not M; some S is not M; every S is P

Larry: Every P is M; every S is M; every S is P

Curly: Every P is M; some S is not M; some S is not P

So by 2/3: Every P is M; some S is not M; every S is P

Exercise for the student: find troika and triad for this trilemma:

All classics are good books.

3:AII: Every M is P; some M is S; therefore some S is P

Contra 3:AII

Moe: Some M is not P; some M is S; no S is P

Larry: Every M is P; no M is S; no S is P

Curly: Every M is P; some M is S; some S is P

So by 2/3: Every M is P; some M is S; no S is P

Exercise for the student: find troika and triad for this trilemma:

Every weekend I play golf;

Some weekends I’m with my father;

I never play golf with my father.

3:EIO: No M is P; some M is S; therefore some S is not P

Contra 3:EIO

Moe: Some M is P; some M is S; every S is P

Larry: No M is P; no M is S; every S is P

Curly: No M is P; some M is S; some S is not P

So by 2/3: No M is P; some M is S; every S is P

Exercise for the student: find troika and triad for this trilemma:

No man is an island;

Some men float;

Everything afloat is an island.

3:IAI: Some M is P; every M is S; therefore some S is P

Contra 3:IAI

Moe: No M is P; every M is S; no S is P

Larry: Some M is P; some M is not S; no S is P

Curly: Some M is P; every M is S; some S is P

So by 2/3: Some M is P; every M is S; no S is P

Exercise for the student: find troika and triad for this trilemma:

Some days I’m happy;

Every day I have the blues;

I’m never happy on the days I have the blues.

3:OAO: Some M is not P; every M is S; therefore some S isn’t P

Contra 3:OAO

Moe: Every M is P; every M is S; every S is P

Larry: Some M is not P; some M is not S; every S is P

Curly: Some M is not P; every M is S; some S is not P

So by 2/3: Some M is not P; every M is S; every S is P

Exercise for the student: find troika and triad for this trilemma:

Some women are not pretty;

Every woman is lovely;

Every lovely is pretty.

3:AAI: Every M is P; every M is S; therefore some S is P

Contra 3:AAI

Moe: Some M is not P; every M is S; no S is P

Larry: Every M is P; some M is not S; no S is P

Curly: Every M is P; every M is S; some S is P

So by 2/3: Every M is P; every M is S; no S is P

Exercise for the student: find troika and triad for this trilemma:

Every sex act is nasty;

Every sex act is sacred;

Nothing sacred is nasty.

3:EAO: No M is P; every M is S; therefore some S is not P

Contra 3:EAO

Moe: Some M is P; every M is S; every S is P

Larry: No M is P; some M is not S; every S is P

Curly: No M is P; every M is S; some S is not P

So by 2/3: No M is P; every M is S; every S is P

Exercise for the student: find troika and triad for this trilemma:

No parents love war;

Every parent is a producer;

Every producer loves war.

4:AEE: Every P is M; no M is S; therefore no S is P

Contra 4:AEE

Moe: Some P is not M; no M is S; some S is P

Larry: Every P is M; some M is S; some S is P

Curly: Every P is M; no M is S; no S is P

So by 2/3: Every P is M; no M is S; some S is P

Exercise for the student: find troika and triad for this trilemma:

Everything that he likes is esoteric;

Nothing esoteric is on TV;

There’s something on TV that he likes.

4:EIO: No P is M; some M is S; therefore some S is not P

Contra 4:EIO

Moe: Some P is M; some M is S; every S is P

Larry: No P is M; no M is S; every S is P

Curly: No P is M; some M is S; some S is not P

So by 2/3: No P is M; some M is S; every S is P

Exercise for the student: find troika and triad for this trilemma:

No criminal is kind;

Some kind people are poor;

All poor people are criminals.

4:IAI: Some P is M; every M is S; therefore some S is P

Contra 4:IAI

Moe: No P is M; every M is S; no S is P

Larry: Some P is M; some M is not S; no S is P

Curly: Some P is M; every M is S; some S is P

So by 2/3: Some P is M; every M is S; no S is P

Exercise for the student: find troika and triad for this trilemma:

Some important truths are obvious;

Everything obvious is dull;

Nothing dull is important.

4:AAI: Every P is M; every M is S; therefore some S is P

Contra 4:AAI

Moe: Some P is not M; every M is S; no S is P

Larry: Every P is M; some M is not S; no S is P

Curly: Every P is M; every M is S; some S is P

So by 2/3: Every P is M; every M is S; no S is P

Exercise for the student: find troika and triad for this trilemma:

Every person is an object;

Every object is a pattern in Hilbert space;

No pattern in Hilbert space is a person.

4:EAO: No P is M; every M is S; therefore some S is not P

Contra 4:EAO

Moe: Some P is M; every M is S; every S is P

Larry: No P is M; some M is not S; every S is P

Curly: No P is M; every M is S; some S is not P

So by 2/3: No P is M; every M is S; every S is P

Exercise for the student: find troika and triad for this trilemma:

Nothing I can explain takes long to read;

Anything that takes long to read is profound;

I can explain anything profound.

In each case, Curly repeats the classic syllogism, but he is overruled in the third term!

Many thanks to Rudy Rucker, from whose book “Mind Tools” I cribbed the nineteen Aristotelian syllogisms here spoofed.