Algeraic notation allows us to see more into the interior


of propositional symbolic expressions, regardless of the notation in which they are framed.  So far in the course we have seen the conventional infix system (p -> q) as well as PLN or postfix literal notation (PQC), and we have also seen how we could reduce all connectives to just one (p|q or PQX).  What we will see towards the end of this course is how we can expand to a multivalue logic (one with three or more "states" and not just our traditional "true" and "false").

Binding all these systems together is the notion of a truth table, and we can easily develop computerized techniques for generating truth tables (see the OneList files for DOS programs that do this with PLN).  Algebraic notation in effect recaptures this programming.  For instance, we can generate the truth table for p&q by multiplying the truth values of the variables (1 or 0) in a mod2 system.  We can also generate the truth table for ~(p&q)--the NAND function which is the basis for the Sheffer stroke (p|q)--by adding 1.

Let's look at all the possibilities with just two variables in our standard two-value system (expressing standard truth-table values horizontally instead of vertically).
 
1111 p v ~ p PPNO 1
1110 p v q PQO pq+p+q
1101 q -> p QPC pq+q+1
1100 p P p
1011 p -> q PQC pq+p+1
1010 q Q q
1001 p <-> q PQB p+q+1
1000 p & q PQA pq
0111 ~(p & q)
p | q		 PQAN, PQX pq+1
0110 ~(p <-> q) PQM p + q
0101 ~q QN q+1
0100 ~(p -> q) PQCN pq+p
0011 ~p PN p+1
0010 ~(q -> p) QPCN pq+q
0001 ~(p v q) PQON,
PQY		 pq+p+q+1
0000 p & ~p PPNA  0

Still another way of looking at algebraic notation concentrates on whether the coefficient of each of the four possible positions is 1 or 0.  The standard form for any two-variable expression is Apq+Bp+Cq+D, and another way of writing any contingent expression is to write just the coefficients when they are positive.   ABCD would be the representation of ~(pvq) and AD would be the representation of ~p.
 

A very optional exercise (for math majors in particular)

Show how you could develop a test for validity using algebraic notation.  Show how you could work the same test with the coefficient representations.

Hint:  if we think of any valid argument form as a tautology, you might want to begin by rewriting the form entirely as a conditional.
P->Q, P |- Q becomes ((P -> Q) & P) -> Q and this in algebraic notation is (pq+p+1)(p)(q)+(pq+p+1)p+1.  Work this out with these ideas in mind: in mod2 pp=p and p+p=0.