****

**Trilemma Deduction**

A trilemma is a triple of
propositions, any two of which can be true but not all three. Any trilemma
implies this deductive principle; from any two, derive the negation of the
third. For instance, this trilemma:

A;

A implies B;

Not
B.

implies these deductive rules:

From: A; A implies B;
deduce: B

From: A implies B; Not
B; deduce: not A

From: not B; A; deduce:
A does not imply B

So a single trilemma implies modus ponens, modus tollens, and
“anti-implication”. All three rules are enfolded into one trilemma. A trilemma
is an encapsuled near-absurdity; and near-absurdity is the engine of deduction.
Every deduction is a narrow escape, one trilemma wide.

Inspection of the trilemmas of the
“Spoofing Classical Logic” chapter shows that all are variants of the following
three trilemmas:

*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.

Both 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 allow existing
things such as Komodo dragons to be called dragons, then one of these three
propositions fails.

Classical syllogism 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. With that as a stated assumption, then the All-All-None
trilemma and the All-None-All trilemma both reduce to the Some-All-None
trilemma. The Some-All-None trilemma is perilously close to the absurd: some
A is B; all B is C; no A is C; choose
two! Yet this peril 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.*

we derive three deduction rules:

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.

As a bonus, if we let each Stooge
agree with the premises and conclusion of each of the three deduction rules,
then we get a troika, 2/3rds affirming each term of the trilemma, but rejecting
their conjunction!

Consider 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. To these rules, add one more:

**Trilemma Deduction:**From any two of;

Some A are B

All B are C

No A are C

deduce the negation of
the third.

Modal identities plus trilemma
yields the core of classical syllogism theory. Adding existential import to
‘all’ 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)

Barbarism,
considered as a troika, is in defiance of classical logic; but considered as a
trilemma, it encodes 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.
Exercise
for the student:

find
deduction rules, and Stooge election, for these trilemmas:

*Some art is beautiful;*

*all beauty is conventional; no art is conventional.*

*Some days are bliss; all bliss is perfect; no days are perfect.*

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