# The Square of Eli

One of my greatest delights as a teacher is learning something new from my students. That happened today in my logic class, when Eli, Lily and I wandered down a rabbit trail of immediate inferences. We had used two immediate inferences to write a statement equivalent to some other statement, and realized that we could have gotten to that same answer using two different immediate inferences.  Eli asked if there was some “triangle of immediate inferences” that we could go through and end up where we started. After some playing around, we discovered, not a triangle, but what I will call The Square of Eli:

CONVERSE      ⇒     OBVERSE

⇑                                ⇓

OBVERSE   ⇐   CONTRAPOSITIVE

We found that if you start with an E or I statement in the upper left corner, and run through the immediate inferences in the order given, you end up with the original statement in that same place, with several equivalent statements in the process. For example:

1. No Christians are Muslims.   (Original statement)
2. All Christians are non-Muslims.  (OBVERSE)
3. All Muslims are non-Christians.  (CONTRAPOSITIVE)
4. No Muslims are Christians.   (OBVERSE)
5. No Christians are Muslims.  (CONVERSE, and the original statement)

We realized that you cannot start with an A or O statement there, because you cannot take the contrapositive of the resulting obverse. But you can start in the upper right corner with an A or O, and it works. For example:

1. All students are people.   (Original statement)
2. All non-people are non-students.  (CONTRAPOSITIVE)
3. No non-people are students.   (OBVERSE)
4. No students are non-people.   (CONVERSE)
5. All students are people.   (OBVERSE, and the original statement)

There is more to be discovered here, but my students and I found this a fascinating tangent. Sometimes the rabbit trail is the lesson!