16.2. If-Then Statements http://www.ck12.org
Example B
Evaluate the truth value of the following conditional statement using a truth table.
If when I go somewhere I always run, then when I run I always go somewhere.
Solution: This statement actually has two layers of conditional statements because both the hypothesis and the
conclusion are conditional statements themselves.
- LetRbe the statement “I run”.
- LetSbe the statement “I go somewhere”.
The original sentence can be rewritten in symbols as:(S→R)→(R→S). To make the truth table for the sentence,
start with all possible truth combinations ofSandR(both true,Strue/Rfalse,Sfalse/Rtrue, both false). Then, find
the truth values for each piece working up to the full sentence.
TABLE16.6:
S R S→R R→S (S→R)→(R→S)
T T T T T
T F F T T
F T T F F
F F T T TThe statement is always true unless you run but do not go anywhere.
Example C
One day a student went to her logic professor and said, “I don’t believe that when the hypothesis of a conditional
statement is true, then the whole statement is trivially true”.
The professor replied, “Alright then. Give me any false hypothesis you can think of and I will mathematically prove
that you are a goat.”
The student thought about it and said, “Okay, prove the statement: ’If 1=2 then I am a goat’.”
How would you prove this ridiculous (but true!) statement?
Solution:If 1=2, then any group of 2 things must also be a group of just 1 thing. Therefore in the two element
set containing the student and the goat, there must be only one element. Thus, the goat and the student must be the
same thing.∴
Concept Problem Revisited
Use a diagram to represent the conditional statement.
If it rains today then you will stay at home and read a book.
- P= it rains today
- Q=you will stay at home and read a book.
You stayed at home and read a book. This implies thatQis true. According to the diagram, if an object is insideQ
it may or may not be insideP. Thus, you can conclude nothing about the rain. Many people will want to incorrectly
conclude that it must have rained, but conditional statements only flow in one direction.