Several years ago I was teaching a logic course, and we were learning about formal proofs of validity. I enjoy proofs, and to keep myself sharp I was working through a practice quiz in David Kelley’s The Art of Reasoning, when I came across this argument:
D ⊃ (E ⊃ F)
D ⊃ (F ⊃ G)
∴ D ⊃ (E ⊃ G)
I was in a quiet library with plenty of time, but despite all my efforts I could not solve this (without using the Conditional Proof). The next day in class some students were finishing their assignment early, so I challenged them with this proof, thinking to myself, “That ought to keep them busy,” but not really expecting anyone to succeed. Before the end of class, Caroline Jones came forward and said, “I solved it, Mr. Nance.” I scoffed inwardly at first, only to be pleasantly surprised by her correct solution.
Since that time I have called this “The Caroline Jones” proof, and have challenged my logic students to solve it using only the regular rules of inference and replacement. The most elegant proof I have seen requires twelve total steps.
Anyone up to the challenge?