Project: Some Applications of Logic (and how to avoid some logical mistakes!)
Part 5: Logical Equivalency
For this next bit, we’ll look at a pretty important component of logical reasoning – how to tell if two logical
statements are equivalent. Logical equivalency implies that two statements, for all given values of their hypotheses (p)
and conclusions (q), have identical truth tables. Remember this example from last time?
“If you do the dishes, then I’ll take out the trash.”
Lemme see if I can show you what I mean by making up an
example that’s logically equivalent to that!
If I don’t take out the trash, then you must not have done the
dishes.”
But are you sure that the statement I just made up is logically
equivalent to the first? And how can you be so sure?
Remember – opinion doesn’t mean anything in logic! You might look at that second statement and say, “yeah –
it looks good.” Or, “Nope! I don’t buy it.” But, if you stop at that, you’re basing your choice on opinion! Unless you’re
practicing to be a politician, I wouldn’t go there. Let’s make Mona Lisa happy, and attack this logically see how that
one’s truth table would look…let’s start by deconstructing the statements!
“If you do the dishes, then I’ll take out the trash.” This one’s in “If p then q” form!
You do the dishes  p
I’ll take out the trash  q
If I don’t take out the trash, then you must not have done the dishes.”
I don’t take out the trash  ~q
You don’t do the dishes  ~p
That one’s also in “if then form” – but the two statements that make it up have been switched around and
negated! It’s in “if not q, then not p” form! Let’s build a truth table for it! Wahoo!
p
q
~p
~q
If ~q, then ~p
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
To understand the far right column (the one containing “If ~q, then ~p”), first note that the hypothesis is “~q”
and the conclusion is “~p”. Therefore, that conditional will only be false if ~q is true and ~p is false – which happens in
row 2.
Now – let’s compare the truth table we just created (If ~q, then ~p) has the same right – hand column as the
truth tables we created for the conditional (If p, then q)? Don’t trust me? OK! Here they are again!
p
q
If p, then q?
p
q
~p
~q
If ~q, then ~p
T
T
T
T
F
F
T
F
T
F
F
T
F
T
F
T
T
F
F
F
T
F
T
T
F
F
T
T
T
F
T
T
That means that the statements “If you do the dishes, then I’ll take out the trash” and “If I don’t take out the
trash, then you must not have done the dishes” are logically equivalent. Sweet!
When you take any conditional statement and rewrite it as we did above, you have formed the
contrapositive of the conditional statement.
Contrapositives are logically equivalent to their corresponding
conditionals! Here are some examples!
Original conditional statement:
Contrapositive of original statement:
If I put too much air in my bike tires, then they’ll
pop.
If my bike tires didn’t pop, then I must not have
put too much air in them.
If a figure is a square, then it is a rectangle.
If a figure is not rectangle, then it is not a square.
If I choose the wine in front of me, then I am a
great fool.
If I am not a great fool, then I will not choose the
wine in front of me.1
Now you try some!
1. (2 points each) Form the contrapositive of each of the following statements:
a. Form the contrapositive of the statement “If there’s a hole in my gas tank, then my car won’t start.”
b. If a number is greater than 10, then it is greater than 5.
1
Paraphrased.
Part 6: Logical Non-Equivalency
I. Inverse Error
Unfortunately, sometimes people can fall into logical traps. For example, suppose that we reuse the conditional
“If you do the dishes, then I’ll take out the trash”. Some people might think this is logically equivalent to the statement,
“If you don’t the dishes, then I won’t take out the trash.” That is, folks might erroneously think that “If p, then q” is
logically equivalent to “if not p, then not q”.
2. (4 points…1 point for each cell) Complete the truth table below (the grayed cells) to show that these two
statements are not logically equivalent:
p
T
T
F
F
q
T
F
T
F
If p, then q?
T
F
T
T
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
~q
F
T
F
T
If ~p, then ~q
See how the far right columns of the two truth tables are different? That means the statements are not logically
equivalent.
The statement “if not p, then not q” is called the inverse of the statement “If p, then q”. If someone
mistakenly switches one for the other, they have made what is called a “logical inverse error”. Here are some
examples!
Original conditional statement:
Non equivalent inverse of original statement:
If I put too much air in my bike tires, then they’ll
pop.
If I don’t put too much air in my tires, then they
won’t pop. (they sure might! From some other
factor besides air pressure)
If my car is out of gas, then it won’t run.
If my car is not out of gas, then it will run. (Not
necessarily true! It could, for example, have a
dead battery)
If an instructor cares, he’ll send emails to his
students regularly.
If an instrcutor doesn’t care, he won’t send
emails to his stuidents regularly. (c’mon, now!
There are other ways to show you care!)
What I gave you in parentheses up there in each example are called counterexamples – remember them from
the last project?
Here’s one I heard up at the Mount Bachelor Nordic center: a friend’s son had just finished sixth in a XC ski race.
She was trying to figure out if he finished with a good enough time to move onto the next race in the series. Here’s the
rule:
“If a skier’s time is no more than 7% greater than the average of the top 3 skier’s times, then they move onto
the next round.”
As we were crunching the math, a gentlemen (let’s call him Fred) came up and said, “Just check to make sure
he’s within 7% of the top finisher. If he is, he qualifies.”
Fred’s statement can be succinctly wrapped up as follows:
“If a skier is within 7% of the fastest skier’s time, then he is within 7% of the
average of the top three skier’s times.”
(convince yourself this is true before moving on)
I then asked him, “But what if he isn’t?” To which I (predictably) got a weird look, and then Fred skated away.
Let’s analyze!
3. (2 points) Form the contrapositive of Fred’s statement. This is, logically, true.
4. (2 points) Form the inverse of Fred’s statement.
That last one is not logically equivalent to the original – it’s possible that someone could have taken more than
7% of the time it took the first place finisher to race, but yet still be within 7% of the average of the top three finishers!
5. (3 points) Suppose that a skier finished his race in 1 hour (60 minutes). Which of the following “Top three”
arrangements would show that the inverse is not logically equivalent to the original rule (in other words, a
counterexample)? Don’t round – use exact values for times (they do in races)!
A.
First Place: 58 minutes
Second Place: 58.5 minutes
Third Place: 59 minutes
B.
First Place: 55 minutes
Second Place: 58.5 minutes
Third Place: 59 minutes
C.
First Place: 55 minutes
Second Place: 56 minutes
Third Place: 57 minutes
This type of error is also seen in advertising. Often. Here’s an example I saw on a billboard near our home:
In case it’s a little hard to see, it’s a huge bottle of
Budweiser next to a burger with the catchline “Not A Fad”. I
imagine that’s a dig at the types of beers that Budweiser feels
are fads (if you remember the Superbowl commercial they ran,
they expressed this sentiment very heavy – handedly:
http://for.tn/16a0r1z). At any rate, let’s analyze why it’s a
logically faulty statement!
Budweiser wants you to believe that “If it’s a Bud, then
it’s not a fad.” OK! I’ll allow that as a true statement…bud’s
been around for about 150 years now; seems fair to not deem
it a “fad”.
6. (2 points) Form the inverse of that statement. This is the statement Budweiser marketers want you to take
away from that billboard…and it’s not logically true!
Inverse error happens more often than you’d care to believe. And it’s easy to do! Human nature makes it
easy to “just change all the signs” in a pseudo – mathematical application. Heck, my son even gets upset
because of this: if I tell his friend, “Hey, Max’s friend! Nice pass!”, Max will sometimes get upset and say “What
about me?!?” And I tell him, “Buddy! Just because I think so-and-so’s pass was nice doesn’t mean yours isn’t!”
Here’s the skinny – once you negate the hypothesis of a conditional, all logical bets are off. 
II. Converse Error
We’ll wrap up with one more, far less prevalent error. Converse Error happens when someone takes a
conditional, switches the hypothesis and conclusion, and then assumes that the resulting converse is equivalent to the
original conditional. For example, they might think that “If you do the dishes, then I’ll take out the trash” is logically
equivalent to “If I take out the trash, then you’ll do the dishes.” Thus, if a conditional statement is in the form “if p,
then q”, its converse would have the form “if q, then p.”
7. (4 points…1 point for each cell) Complete the trith table for the conditional (grayed cells).
p
T
T
F
F
Conditional
q If p, then q?
T
T
F
F
T
T
F
T
p
T
T
F
F
Converse
q If q, then p?
T
F
T
F
Because the end columns are different, the statements must not be equivalent! Here’s a concrete example to
see that difference…
Let’s use the conditional, “If you live in zip code 97701, then you live in Bend, Oregon.” This statement is true!
8. (2 points) Write the converse of that statement.
9. (2 points) Give a counterexample to prove that your statement in 8 is incorrect.
******
This is just a teensy – weensy look into the world of logic. This awesome world spans far more than we could
uncover even if I devoted all of MTH 105 to it (some schools have entire courses devoted to logic). I hope that it’s
intrigued you enough to chase down more resources – becoming more logical can help you in everything from
conducting better web searches to understanding how to diagnose electrical problems in your house. Heck yeah!