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 T F F T F F T T F T F F T 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 T F F T T F F T F F T F F T T F T T F T F F T F F T 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: If I put too much air in my bike tires, then they ll pop. If a figure is a square, then it is a rectangle. If I choose the wine in front of me, then I am a great fool. Contrapositive of original statement: If my bike tires didn t pop, then I must not have put too much air in them. If a figure is not rectangle, then it is not a square. 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 q If p, then q? p q ~p ~q If ~p, then ~q T T T T T F F T F F T F F T F T T F T T F F F T F F T T 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: If I put too much air in my bike tires, then they ll pop. Non equivalent inverse of original statement: 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 an instructor cares, he ll send emails to his students regularly. 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 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. B. C. First Place: 58 minutes First Place: 55 minutes First Place: 55 minutes Second Place: 58.5 minutes Second Place: 58.5 minutes Second Place: 56 minutes Third Place: 59 minutes Third Place: 59 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). Conditional Converse p q If p, then q? p q If q, then p? T T T T T T F F T F F T T F T F F T F 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!