LESSON 8 -  MORE ABOUT SIGNALS  AND RULES

ASSIGNMENT 8


We are about to make the transition from PLN to standard infix notation, but before we do this is a good time to ask how many signals do we really need as well as how many rules we really need.

We have worked with six signals, actually one more than those used in standard notation, but we could actually have made use of just one.  The trick is to build the idea of negation into the signal itself, and to do this we could use a new signal X that expresses the same relationship as PQAN (as in the gate NAND for computer science) or we could use Y to express the same relationship as PQON.

Using algebraic notation to start with, let's see all sixteen of the possible relationships between any two elements.  (The chart below is for reference only and is not required material.)
 

Algebraic 
PLN
X
Y
Standard
1
PPNO
 PPXPX
PPYPYPPYPYY 
P v ~P
PQ+P+Q
PQO
PPXQQXX 
PQYPQYY
P v Q
PQ+Q+1
QPC
 PQXQX
 PQYQYPQYQYY
Q -> P
P
P
P
P
P
PQ+P+1
PQC
PQXPX
 PQYPYPQYPYY
P -> Q
Q
Q
Q
Q
Q
P+Q+1
PQB
 PQXPPXQQXXX
 PQYPYPQYQYY
P <-> Q
PQ
PQA
PQXPQXX
 PPYQQYY
P & Q 
PQ+1
PQAN
PQX
 PPYQQYYPPYQQYY
 ~(P & Q)
P+Q
PQM
 PQXPPXQQXXXPQXPPXQQXXXX
 PQYPYPQYQYYPQYPYPQYQYYY
 ~(P <-> Q)
Q+1
QN
QQX
 QQY
~Q 
PQ+P
PQCN
PQXPXPQXPXX
 PQYQY
 ~(P -> Q)
P+1
PN
PPX
PPY 
 ~P
PQ+Q
QPCN
 PQXQXPQXQXX
 PQYPY
 ~(Q -> P)
PQ+P+Q+1
PQON
 PPXQQXXPPXQQXXX
 PQY
~(P v Q)
0
PPNA
 PPXPXPPXPXX
 PPYPY
P & ~P

Obviously, using only X or Y as signals would make it extremely difficult to represent all logical relationships.  We could, however, get by with just A, O and N.  This is because we already have the MI equivalence (PQC=PNQO) that lets us convert any conditional to a disjunction or the other way around, and we know that the CN equivalence (PQCN=PQNA) lets us convert a negated conditional to a conjunction.  This, in fact, is what allows us to have what are called the disjunctive normal form and the conjunctive normal form for a wff (this is a topic we are coming back to later).

Could we get by with fewer rules?  As long as we have equivalence rules, we could in principle reduce the number of inference rules needed, but again economy and efficiency are not the same thing.  For instance, we could do away with O elim if we keep C elim, or we could do away with C elim if we keep O elim, but it does appear more efficient to have both available for use.

In this course I have already made certain trade-offs in the rules I propose and the rules I do not use.  Most texts allow the rule called modus tollens (PQC, QN |- PN), while I call for two steps by first using the Contra equivalence then C elim in order to reinforce the idea of working from left to right.  However, I do have the CN equivalence while most texts require a series of steps in order to negate a conditional.  There are other differences as well that I will note as we keep going.

The assignment to go with this lesson is a practice midterm with the answers available followed by a Quizmaster exercise that will be submitted.

Go to assignment 8.