[FOM] question about Mendelson's Intro to Math Logic

[Jeff Hirst]

Dear Bob et al:

I see that several solutions to the problem in Mendelson have been outlined.  I have questions and an observation.

(1)  If we let M3 denote the third axiom of Mendelson and A3 denote the (~B->~C)->(C->B) scheme, working with
Mendelson’s first two axioms and modus ponens as the only rule of inference, what is the minimum number of
applications of M3 that are needed to deduce A3?  Similarly, what is the minimum number of applications of A3
that are needed to deduce M3?  

(2)  I used A3 to describe the contrapositive form because it is the third axiom of the system P_2 introduced at the
beginning of Chapter 2 in Church’s “Introduction to mathematical logic.”  I e-mailed Prof. Mendelson one time and asked
about the origins of his axiom system.  I thought that perhaps he called it L in homage to Lukasiewicz.  He told me that
L was for Logic.  (So much for my career as a historian.)  He also mentioned that using M3 rather A3 made some of
the deductions of additional negation properties (double negation elimination, double negation introduction, etc.)
easier for the students.  By easier, I think he meant easier, not necessarily requiring fewer applications of the scheme
as described in (1).

Best,

Jeff

Jeff Hirst,  Professor of Mathematics
Department of Mathematical Sciences
Appalachian State University
Boone, NC 28608
828-262-2861  jlh at math.appstate.edu