Download Slides - people.vcu.edu

GEB
class notes jan 18
A Logic Puzzle
Classification Problems,
Equivalence Relations,
and Disjoint Unions
Logic and Set Theory
Basic Question
What hope is there that
a certain “portion of reality
can be imitated in its
behavior by a set of
meaningless symbols
governed by formal rules”?


I offer two prizes --- Prize 1 and Prize 2. You
are to make a statement. If the statement is
true, then I will give you one of the prizes (I’m
not saying which one). If your statement is
false, then you get no prize. Obviously you can
be sure of winning a prize by saying, e.g.,
“Two plus two is four,” but suppose you have
your heart set on Prize 1; what statement
could you make that would guarantee that you
will get Prize 1?
HINT: Try a statement of the form “You will not
give me …”
Classification Problems



Statements can be self-referential,
well-formed, true, provable,
meaningful …
The Game of 20 Questions
The “… three of these things are
kind of the same” Game
Set Complements
The Universe
A
not A
A Partition of the Universe
animal
vegetable
other
mineral
A Partition by Overlapping Sets
A
B

How many disjoint subsets in this partition?
What does “kind of the same” mean?
3 essential properties



For every A, A “is kind of the same as” A.
If A “is kind of the same as” B,
then B “is kind of the same as” A.
If A “is kind of the same as” B
and B “is kind of the same as” C,
then A “is kind of the same as” C
RESULT:


If a relation is symmetric, reflexive,
and transitive,
Then the collection of sets of things
that are “kind of the same” forms a
partition of the Universe into a
disjoint union.
DeMorgan’s Laws


Sets: not(A U B) = not A I not B
In Logic,
Not(A OR B) =(Not A) AND (Not B)
Summary: at a very basic level,




reality seems to be imitated by sets,
set operations seem to correspond
to logical operations,
logical operations seem to
correspond to thought processes,
thought processes seem to be wellexpressed through the language of
AND, OR, and NOT.
A Set Operation for
If … Then …
Which Box Contains the Gold?




Two boxes are labeled "A" and "B".
A sign on box A says "The sign on box
B is true and the gold is in box A".
A sign on box B says "The sign on box
A is false and the gold is in box A".
Assuming there is gold in one of the
boxes, which box contains the gold?
What’s Going On?


Your reasoning was sound. If each
sign is either true or false, then the
only interpretation that is logically
consistent with those signs is “the
gold is in Box B.”
Experimentally, we saw that the
logically consistent interpretation
was not consistent with reality --who said it had to be?