**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.*

we derive this rule triad:

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.*