Up A Level
"And"
"And" of An "Or"
Contrapositive
"For All"
"If and Only If"
"If..., Then..."
"Not"
"Not" of An "And"
"Not" of An "If...Then"
"Not" of An "Or"
"Or"
"Or" of An "And"
Short Tautologies
"There exists"

# "NOT" APPLIED TO AN "IF...THEN" SENTENCE

## Let P and Q be sentences which are true or false, but neither of them is both. "Not(if P, then Q)" means the same thing as "P and (not(Q))".

"Not" Applied To An "If...Then" Sentence
P Q if P, then Q not(if P, then Q) not(Q) P and (not(Q))
T T T F F F
T F F T T T
F T T F F F
F F T F T F

Some people understand this principle as follows. They know that "if P, then Q" is false only when the promise is broken---that is, when P is true and Q is false. Q being false makes "not(Q)" true. So, both P and "not(Q)" are true. This is what is meant by "P and (not(Q))".

The truth table to the right is a different approach to the same principle. Note that columns 4 and 6 have the same truth values.

 Go to an overview of logic. Go to the home page for Tom Ramsey Go to the home page for the UHM Department of Mathematics Your comments and questions are welcome. Please email them ramsey@math.hawaii.edu.