It turns out, despite what mom told us, that two wrongs do make a right, and it’s an obscure rule of logic that is the culprit.
Take a look at the city map pictured here, and we’ll illustrate this concept.
Let’s suppose that A calls you on the phone and asks for directions to X. You say to him, “Go straight for one block, then turn left. Go straight for one more block.”
A few minutes later, you look up from your table at the cafe, and someone walks up. You suddenly realize you have no way of recognizing A, because you’ve never met him. This stranger looks at you and says, “I followed your directions: I went one block–turned left–went one more block, and here I am.”
Is it A or B?
Maybe B is a spy that tapped your phone and heard your directions. But the fact is, B could have followed your directions and got to the same spot.
That example shows us that if we start at the wrong place, we can still get to the right location. Just because a stranger arrives at city center does NOT mean it is person A. In fact there are two other locations that have the exact same directions to city center as well.
We must know where we are starting from.
In statistics and logic, we start from fundamental assumptions. Logic tells us how to evaluate “if-then” statements. It takes three steps.
1. Put “if” phrase into truth computer: output is True or False.
2. Put “then” phrase into truth computer: output is True or False.
3. Then output overall truth value of “if-then” statement based on this formula:
T + T = T
F + T = T
T + F = F
F + F = T
In the last line, you can see that 2 Wrongs = 1 Right. The truth computer (the rules of logic) tell us that our “if-then” statement is True overall.
The point is this: if our assumptions (the “if” part) are wrong then we can logically conclude something using correct reasoning and come to a ludicrous conclusion. Don’t let it happen to you.
Have you heard of confirmation bias–do you think that a similar concept?
How do we find our way then, when we are exploring the unknown, blazing a trail into uncharted territory? How do we apply elementary statistical principles to transform uncertainty into decisive action? What is to prevent us from making a preposterous application of ATOMs when we deal with very complex situations, those in which our intuition fails?
These questions are not much different from those faced by Chuck Yeager before he ever broke the sound barrier or Neil Armstrong as he took that first step on the moon. Neither of these men, nor anyone around them–with hundreds or thousands of highly educated, very scientific people on these teams–knew what to expect. Or did they…?
ATOMs is a monthly column that introduces analytical tools of mathematics and statistics and illustrates their application. To read more about ATOMs, you can read Where Do We Go From Here, or view the online workbook here.