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if p then q... This seemingly innocent logical proposition can get very confusing when you
look at a its truth table.
|
p
|
q
|
p
then r
|
|
T
|
T
|
T
|
|
F
|
T
|
T
|
|
T
|
F
|
F
|
|
F
|
F
|
T
|
The most difficulty with this truth table comes from the
last row - both p and q are false and the result is TRUE! This can result
in statements like "if the moon is made of green cheese, then I am the
richest person in the world".
|
p
|
q
|
p
then r
|
|
|
T
|
T
|
T
|
p
and q |
|
F
|
T
|
T
|
(not
p) and q |
|
T
|
F
|
F
|
p
and (not q) |
|
F
|
F
|
T
|
(not
p) and (not q) |
You'll notice that the 3rd line (p and (not q)) is not in the Venn diagram -
that's the empty set. Something can't be p and NOT q. q is a sub-set
of P.
A mathematical example will make you more comfortable. Consider the proposition
if
x > 1, then x2 > x.
Note that x > 1 false can result in either x2
> x true or x2
> x false depending on x.
Let's consider the statement:
If you finish a report on time, then the boss will give you
a bonus
The only commitment your boss is making is to give you the
bonus IF you finish the report on time. He's not excluding that he could
give you be bonus no matter what. There are four possible outcomes:
-
Finish on time - get bonus
-
Finish on time - NO bonus
-
Don't finish on time - get bonus
-
Don't' finish on time - NO bonus
#2 is ruled out by the bosses statement so the other three
are possible outcomes. So this implication is false ONLY when you finish
on time and do NOT get the bonus.
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