Quite some time before Venn, Euler charted out Aristotle's syllogistic via diagrams (as did Leibniz). Euler diagrams capture hierarchy a bit better than Venn, tho' Venn diagrams can represent certain relations that Euler's charts can't. (Neither can capture predicate logic, however, except with a lot of work.
Euler's charts with basic set relationships: All A's are B's; all C's are A's; thus All C's are B's. Hypothetical syllogism in old logic (say, all Insects are Animals; all Cockaroaches are Insects; all Cockaroaches are Animals. QED)
2 comments:
QED??? Not until you've put it through a real Reductio ad Absurdum:
All A's are B's.
All C's are A's
thus, All C's are B's.
In propositional logic the categorical statements (Aristotle's "A") are done with conditionals (any dissenters??)
A -> B
C -> A
thus C -> B
now, RAA (watch closely KrustyRon):
1. A -> B
2. C -> A
3. thus C -> B
negate conclusion ~(C->B)
rules for denied conditionals
anyone?? something like this ~(~C v B), thus C & ~B (Demorgans, right).
So,
C & ~B
(tree test means that both (or all) statements of conjunctions be subject to test for contradiction, right, Krusty):
4. C
~B
5. ~A v B
X
6. ~C X (from 4) v A X (from 5)
Valid. QED
Bonjour
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Bye everyone!
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