Three-value logic

THREE-VALUE LOGIC

Three-value logic At the beginning we talked about playing the circle-box game with colors instead of numbers or letters representing true and false. Our ordinary logic is two-value (bipolar), but we can think of what it would be to have a logic of more than two values. Imagine, for instance, that the colors are red, blue, and white (represented by the numerals 1,2,0). A truth-table would then have nine lines, and if we use a signal F to indicate a relationship by which playing the same color leads to the same color but playing different colors leads to the third color (a same-different rule), we have the following:

P Q || PQF
1 1.........1
1 2.........0
1 0.........2
2 1.........0
2 2.........2
2 0.........1
0 1.........2
0 2.........1
0 0.........0
The program PLN3-colors will let you experiment with these strings.

In any logic that is not just two-valued the idea of negation as something reciprocal has to be changed. We can think of PN as representing rotation and PZ as representing inversion. The new truth tables here are:
P || PN | PZ
1.......2.....0
2.......0.....2
0.......1.....1

While the absence of negation in its familiar sense also cancels out the concept of validity, we can still discuss three-value strings as equivalent, and again we can use a truth-table test:
P Q || PQFN | PNQNF
1 1......... 1 2 .... 2.. 2 2
1 2......... 0 1 .....2.. 0 1
1 0..........2 0 .....2.. 1 0
2 1..........0 1 .....0.. 2 1
2 2..........2 0 .....0.. 0 0
2 0..........1 2 .....0.. 1 2
0 1..........2 0 .....1.. 2 0
0 2..........1 2 .....1.. 0 2
0 0..........0 1 .....1.. 1 1

Go on to three-value signals.

Go to algebraic notation in a three-value system.

Go back to the starting page.