﻿ paths of logic

In the article, Quantum Theory Derived from Logic, a logical expression was derived for n parallel paths of m steps each. This is equation [7] shown again below.

(q0 qn) = ni1,i2,i3,i4...im=0  (q0 qi1)(qi1 qi2)(qi2 qi3)(qi3 qi4)...(qim qn).

To gain some confidence that this expression is true, you can cut and paste the text into this truth-table generator. This is an expression for when n=4 and m=3. Altogether there are (n+1)m=53=125 different terms of 4 implications each.

(q0 => q4) <=>

((q0 => q0) /\ (q0 => q0) /\ (q0 => q0) /\ (q0 => q4))
\/ ((q0 => q0) /\ (q0 => q0) /\ (q0 => q1) /\ (q1 => q4))
\/ ((q0 => q0) /\ (q0 => q0) /\ (q0 => q2) /\ (q2 => q4))

\/ ((q0 => q0) /\ (q0 => q0) /\ (q0 => q3) /\ (q3 => q4))
\/ ((q0 => q0) /\ (q0 => q0) /\ (q0 => q4) /\ (q4 => q4))

\/ ((q0 => q0) /\ (q0 => q1) /\ (q1 => q0) /\ (q0 => q4))
\/ ((q0 => q0) /\ (q0 => q1) /\ (q1 => q1) /\ (q1 => q4))
\/ ((q0 => q0) /\ (q0 => q1) /\ (q1 => q2) /\ (q2 => q4))

\/ ((q0 => q0) /\ (q0 => q1) /\ (q1 => q3) /\ (q3 => q4))
\/ ((q0 => q0) /\ (q0 => q1) /\ (q1 => q4) /\ (q4 => q4))

\/ ((q0 => q0) /\ (q0 => q2) /\ (q2 => q0) /\ (q0 => q4))
\/ ((q0 => q0) /\ (q0 => q2) /\ (q2 => q1) /\ (q1 => q4))
\/ ((q0 => q0) /\ (q0 => q2) /\ (q2 => q2) /\ (q2 => q4))

\/ ((q0 => q0) /\ (q0 => q2) /\ (q2 => q3) /\ (q3 => q4))
\/ ((q0 => q0) /\ (q0 => q2) /\ (q2 => q4) /\ (q4 => q4))

\/ ((q0 => q0) /\ (q0 => q3) /\ (q3 => q0) /\ (q0 => q4))
\/ ((q0 => q0) /\ (q0 => q3) /\ (q3 => q1) /\ (q1 => q4))
\/ ((q0 => q0) /\ (q0 => q3) /\ (q3 => q2) /\ (q2 => q4))

\/ ((q0 => q0) /\ (q0 => q3) /\ (q3 => q3) /\ (q3 => q4))
\/ ((q0 => q0) /\ (q0 => q3) /\ (q3 => q4) /\ (q4 => q4))

\/ ((q0 => q0) /\ (q0 => q4) /\ (q4 => q0) /\ (q0 => q4))
\/ ((q0 => q0) /\ (q0 => q4) /\ (q4 => q1) /\ (q1 => q4))
\/ ((q0 => q0) /\ (q0 => q4) /\ (q4 => q2) /\ (q2 => q4))
\/ ((q0 => q0) /\ (q0 => q4) /\ (q4 => q3) /\ (q3 => q4))
\/ ((q0 => q0) /\ (q0 => q4) /\ (q4 => q4) /\ (q4 => q4))

\/ ((q0 => q1) /\ (q1 => q0) /\ (q0 => q0) /\ (q0 => q4))
\/ ((q0 => q1) /\ (q1 => q0) /\ (q0 => q1) /\ (q1 => q4))
\/ ((q0 => q1) /\ (q1 => q0) /\ (q0 => q2) /\ (q2 => q4))
\/ ((q0 => q1) /\ (q1 => q0) /\ (q0 => q3) /\ (q3 => q4))
\/ ((q0 => q1) /\ (q1 => q0) /\ (q0 => q4) /\ (q4 => q4))

\/ ((q0 => q1) /\ (q1 => q1) /\ (q1 => q0) /\ (q0 => q4))
\/ ((q0 => q1) /\ (q1 => q1) /\ (q1 => q1) /\ (q1 => q4))
\/ ((q0 => q1) /\ (q1 => q1) /\ (q1 => q2) /\ (q2 => q4))
\/ ((q0 => q1) /\ (q1 => q1) /\ (q1 => q3) /\ (q3 => q4))
\/ ((q0 => q1) /\ (q1 => q1) /\ (q1 => q4) /\ (q4 => q4))

\/ ((q0 => q1) /\ (q1 => q2) /\ (q2 => q0) /\ (q0 => q4))
\/ ((q0 => q1) /\ (q1 => q2) /\ (q2 => q1) /\ (q1 => q4))
\/ ((q0 => q1) /\ (q1 => q2) /\ (q2 => q2) /\ (q2 => q4))
\/ ((q0 => q1) /\ (q1 => q2) /\ (q2 => q3) /\ (q3 => q4))
\/ ((q0 => q1) /\ (q1 => q2) /\ (q2 => q4) /\ (q4 => q4))

\/ ((q0 => q1) /\ (q1 => q3) /\ (q3 => q0) /\ (q0 => q4))
\/ ((q0 => q1) /\ (q1 => q3) /\ (q3 => q1) /\ (q1 => q4))
\/ ((q0 => q1) /\ (q1 => q3) /\ (q3 => q2) /\ (q2 => q4))
\/ ((q0 => q1) /\ (q1 => q3) /\ (q3 => q3) /\ (q3 => q4))
\/ ((q0 => q1) /\ (q1 => q3) /\ (q3 => q4) /\ (q4 => q4))

\/ ((q0 => q1) /\ (q1 => q4) /\ (q4 => q0) /\ (q0 => q4))
\/ ((q0 => q1) /\ (q1 => q4) /\ (q4 => q1) /\ (q1 => q4))
\/ ((q0 => q1) /\ (q1 => q4) /\ (q4 => q2) /\ (q2 => q4))
\/ ((q0 => q1) /\ (q1 => q4) /\ (q4 => q3) /\ (q3 => q4))
\/ ((q0 => q1) /\ (q1 => q4) /\ (q4 => q4) /\ (q4 => q4))

\/ ((q0 => q2) /\ (q2 => q0) /\ (q0 => q0) /\ (q0 => q4))
\/ ((q0 => q2) /\ (q2 => q0) /\ (q0 => q1) /\ (q1 => q4))
\/ ((q0 => q2) /\ (q2 => q0) /\ (q0 => q2) /\ (q2 => q4))
\/ ((q0 => q2) /\ (q2 => q0) /\ (q0 => q3) /\ (q3 => q4))
\/ ((q0 => q2) /\ (q2 => q0) /\ (q0 => q4) /\ (q4 => q4))

\/ ((q0 => q2) /\ (q2 => q1) /\ (q1 => q0) /\ (q0 => q4))
\/ ((q0 => q2) /\ (q2 => q1) /\ (q1 => q1) /\ (q1 => q4))
\/ ((q0 => q2) /\ (q2 => q1) /\ (q1 => q2) /\ (q2 => q4))
\/ ((q0 => q2) /\ (q2 => q1) /\ (q1 => q3) /\ (q3 => q4))
\/ ((q0 => q2) /\ (q2 => q1) /\ (q1 => q4) /\ (q4 => q4))

\/ ((q0 => q2) /\ (q2 => q2) /\ (q2 => q0) /\ (q0 => q4))
\/ ((q0 => q2) /\ (q2 => q2) /\ (q2 => q1) /\ (q1 => q4))
\/ ((q0 => q2) /\ (q2 => q2) /\ (q2 => q2) /\ (q2 => q4))
\/ ((q0 => q2) /\ (q2 => q2) /\ (q2 => q3) /\ (q3 => q4))
\/ ((q0 => q2) /\ (q2 => q2) /\ (q2 => q4) /\ (q4 => q4))

\/ ((q0 => q2) /\ (q2 => q3) /\ (q3 => q0) /\ (q0 => q4))
\/ ((q0 => q2) /\ (q2 => q3) /\ (q3 => q1) /\ (q1 => q4))
\/ ((q0 => q2) /\ (q2 => q3) /\ (q3 => q2) /\ (q2 => q4))
\/ ((q0 => q2) /\ (q2 => q3) /\ (q3 => q3) /\ (q3 => q4))
\/ ((q0 => q2) /\ (q2 => q3) /\ (q3 => q4) /\ (q4 => q4))

\/ ((q0 => q2) /\ (q2 => q4) /\ (q4 => q0) /\ (q0 => q4))
\/ ((q0 => q2) /\ (q2 => q4) /\ (q4 => q1) /\ (q1 => q4))
\/ ((q0 => q2) /\ (q2 => q4) /\ (q4 => q2) /\ (q2 => q4))
\/ ((q0 => q2) /\ (q2 => q4) /\ (q4 => q3) /\ (q3 => q4))
\/ ((q0 => q2) /\ (q2 => q4) /\ (q4 => q4) /\ (q4 => q4))

\/ ((q0 => q3) /\ (q3 => q0) /\ (q0 => q0) /\ (q0 => q4))
\/ ((q0 => q3) /\ (q3 => q0) /\ (q0 => q1) /\ (q1 => q4))
\/ ((q0 => q3) /\ (q3 => q0) /\ (q0 => q2) /\ (q2 => q4))
\/ ((q0 => q3) /\ (q3 => q0) /\ (q0 => q3) /\ (q3 => q4))
\/ ((q0 => q3) /\ (q3 => q0) /\ (q0 => q4) /\ (q4 => q4))

\/ ((q0 => q3) /\ (q3 => q1) /\ (q1 => q0) /\ (q0 => q4))
\/ ((q0 => q3) /\ (q3 => q1) /\ (q1 => q1) /\ (q1 => q4))
\/ ((q0 => q3) /\ (q3 => q1) /\ (q1 => q2) /\ (q2 => q4))
\/ ((q0 => q3) /\ (q3 => q1) /\ (q1 => q3) /\ (q3 => q4))
\/ ((q0 => q3) /\ (q3 => q1) /\ (q1 => q4) /\ (q4 => q4))

\/ ((q0 => q3) /\ (q3 => q2) /\ (q2 => q0) /\ (q0 => q4))
\/ ((q0 => q3) /\ (q3 => q2) /\ (q2 => q1) /\ (q1 => q4))
\/ ((q0 => q3) /\ (q3 => q2) /\ (q2 => q2) /\ (q2 => q4))
\/ ((q0 => q3) /\ (q3 => q2) /\ (q2 => q3) /\ (q3 => q4))
\/ ((q0 => q3) /\ (q3 => q2) /\ (q2 => q4) /\ (q4 => q4))

\/ ((q0 => q3) /\ (q3 => q3) /\ (q3 => q0) /\ (q0 => q4))
\/ ((q0 => q3) /\ (q3 => q3) /\ (q3 => q1) /\ (q1 => q4))
\/ ((q0 => q3) /\ (q3 => q3) /\ (q3 => q2) /\ (q2 => q4))
\/ ((q0 => q3) /\ (q3 => q3) /\ (q3 => q3) /\ (q3 => q4))
\/ ((q0 => q3) /\ (q3 => q3) /\ (q3 => q4) /\ (q4 => q4))

\/ ((q0 => q3) /\ (q3 => q4) /\ (q4 => q0) /\ (q0 => q4))
\/ ((q0 => q3) /\ (q3 => q4) /\ (q4 => q1) /\ (q1 => q4))
\/ ((q0 => q3) /\ (q3 => q4) /\ (q4 => q2) /\ (q2 => q4))
\/ ((q0 => q3) /\ (q3 => q4) /\ (q4 => q3) /\ (q3 => q4))
\/ ((q0 => q3) /\ (q3 => q4) /\ (q4 => q4) /\ (q4 => q4))

\/ ((q0 => q4) /\ (q4 => q0) /\ (q0 => q0) /\ (q0 => q4))
\/ ((q0 => q4) /\ (q4 => q0) /\ (q0 => q1) /\ (q1 => q4))
\/ ((q0 => q4) /\ (q4 => q0) /\ (q0 => q2) /\ (q2 => q4))
\/ ((q0 => q4) /\ (q4 => q0) /\ (q0 => q3) /\ (q3 => q4))
\/ ((q0 => q4) /\ (q4 => q0) /\ (q0 => q4) /\ (q4 => q4))

\/ ((q0 => q4) /\ (q4 => q1) /\ (q1 => q0) /\ (q0 => q4))
\/ ((q0 => q4) /\ (q4 => q1) /\ (q1 => q1) /\ (q1 => q4))
\/ ((q0 => q4) /\ (q4 => q1) /\ (q1 => q2) /\ (q2 => q4))
\/ ((q0 => q4) /\ (q4 => q1) /\ (q1 => q3) /\ (q3 => q4))
\/ ((q0 => q4) /\ (q4 => q1) /\ (q1 => q4) /\ (q4 => q4))

\/ ((q0 => q4) /\ (q4 => q2) /\ (q2 => q0) /\ (q0 => q4))
\/ ((q0 => q4) /\ (q4 => q2) /\ (q2 => q1) /\ (q1 => q4))
\/ ((q0 => q4) /\ (q4 => q2) /\ (q2 => q2) /\ (q2 => q4))
\/ ((q0 => q4) /\ (q4 => q2) /\ (q2 => q3) /\ (q3 => q4))
\/ ((q0 => q4) /\ (q4 => q2) /\ (q2 => q4) /\ (q4 => q4))

\/ ((q0 => q4) /\ (q4 => q3) /\ (q3 => q0) /\ (q0 => q4))
\/ ((q0 => q4) /\ (q4 => q3) /\ (q3 => q1) /\ (q1 => q4))
\/ ((q0 => q4) /\ (q4 => q3) /\ (q3 => q2) /\ (q2 => q4))
\/ ((q0 => q4) /\ (q4 => q3) /\ (q3 => q3) /\ (q3 => q4))
\/ ((q0 => q4) /\ (q4 => q3) /\ (q3 => q4) /\ (q4 => q4))

\/ ((q0 => q4) /\ (q4 => q4) /\ (q4 => q0) /\ (q0 => q4))
\/ ((q0 => q4) /\ (q4 => q4) /\ (q4 => q1) /\ (q1 => q4))
\/ ((q0 => q4) /\ (q4 => q4) /\ (q4 => q2) /\ (q2 => q4))
\/ ((q0 => q4) /\ (q4 => q4) /\ (q4 => q3) /\ (q3 => q4))
\/ ((q0 => q4) /\ (q4 => q4) /\ (q4 => q4) /\ (q4 => q4))