One advantage of using
PLN and working with the concept of a color game is that we can develop
the notion of a logic that works with more than two values. Normally
we think of true and false (1 and 0) as the basis for any logic, but if
we go back to where we started in this text we can develop a quite different
logic.
Let's imagine a game that
works with three colors: red, blue, white. Let's also set up a special
signal F, which operates by this rule:
when the elements are the same color then the signal has that color also,
but when they are different then the signal takes on the third color.
If we let the numbers 1,2 and 0 represent these colors (note we are now
into a mod 3 arithmetic just as in the last section we were working with
a mod 2 arithmetic), we have the following truth table.
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Since there are three values
we can no longer think of negation as the simple reciprocal relationship
we had before. We can, however, redefine the signal "N" so that it
represents a "rotation" of values.
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As we have before, we can use our truth table relationships to determine the value of any WFF in this three-value system. For instance, the value of PQFPNF would be determined by setting up a truth-table as follows
P Q || P Q F
P N F
1 1
1 1 1 1 2 0
1 2
1 2 0 1 2 1
1 0
1 0 2 1 2 2
2 1
2 1 0 2 0 0
2 2
2 2 2 2 0 1
2 0
2 0 1 2 0 2
0 1
0 1 2 0 1 0
0 2
0 2 1 0 1 1
0 0
0 0 0 0 1 2
Note that a truth table with two variables will have nine lines. A full development of this logic would allow us to determine new signals and also note equivalences and inferences just as we have done with our standard two-value logic. However, instead of sixteen basic possibilities for a truth-table series we have nearly 20,000.