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.
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