For
example, the 'closed system' is as described above. But the 'open
system' changes the paradox: on the front side of the card it is
written, 'The statement on the backside of this card is true.' When the
card is turned over, the back side says, 'The surface of the Earth is
composed mostly of water.' Now there is a true statement, which results
in eliminating the paradox.
But now, the solution has very little resemblance to the original
paradox.
Venn Diagrams
What is a Venn diagram?
"A diagram representing mathematical or logical sets pictorially
as circles or closed curves within an enclosing rectangle (the universal set),
common elements of the sets being represented by the areas of overlap among the
circles."^{9}
What is the purpose of a Venn diagram?
A Venn diagram clarifies the relationships of sets through
comparison and contrast of the properties or of the elements of the sets.
How are Venn diagrams constructed?
Venn diagrams are a hierarchy extending from a
universal statement downward to sets of elements with specific properties. It's
similar to deductive reasoning analyzing from the general to the specific. And
if you analyze in the opposite direction—from the specific to the general, then
you will have inductive reasoning. (But you will never find this definition in
any descriptions of Venn diagrams; at least, it has never been my experience to
read one.)
"A Venn diagram or set diagram is a diagram that shows all possible
logical relations between a finite collection of different sets."^{10}
