## Monday, May 27, 2013

### On Troikas 11.75: Trilemma Deduction

Once again writing overtakes me, and I must interpolate another chapter. The following naturally follows the "Spoofing Classical Logic" chapter.

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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.