LESSON 2 - Understanding Truth Tables

Assignment 2


The most important idea I'd like you to have at the beginning is that whenever we have a well formed string in PLN we can think of it as having one or another value.  Usually we work with a two-value logic.  This is what we have with an electric switch when the current will either be on or off, and it is what we have when we are thinking of a world of facts in which the things we say will be either true or false.

Because I would like you to still to be thinking of logic as a game, I am going to use colors--red and blue.  Imagine that every variable and every signal could be one or the other, and the color that defines the string itself is the one that goes with the final letter.  We will also use the number 1 to represent the red and the number 0 to represent the blue.

For instance, P and Q are variables.  If we use the signal N we are reversing the color we had before.  We can set up what we call a truth table to show what happens.
 

P
N
  
Q
N
1
0
  
1
0
0
1
  
0
1

We can also see what happens when we use the signal A to show that the string will be true only when both its elements are true.
 
 

P
Q
A
1
1
1
1
0
0
0
1
0
0
0
0

Here is a special note for those students interested in computer science.  If you are really looking carefully, you'll notice that the numbers in the third column above are the product of the numbers in the first two columns.  We can think of conjunction as logical multiplication, and another way of expressing the relationship would be to use an algebraic notation so that PQA is rewritten as PQ.  Negation would be the same as adding 1 in a modular 2 way of counting (like a clock with just two points), and PN would then be P+1.  This actually is the key to having a computer calculate the truth values of a string, as in a program that you will be able to use later on.

You've already seen that in PLN we work with five signals for the logical relationships we can have with P and Q.  The table below shows what happens with each as the values of P and Q change.
 

P
Q
PQA
PQO
PQC
PQB
PQM
1
1
     1
      1 
     1
    1
    0
1
0
    0
     1
    0
    0
    1
0
1
     0
    1
    1
    0
    1
0
0
     0
    0
    1
    1
    0

To evaluate a string we work left to right using the key patterns above.  For instance, PQNARSOC expresses a conditional relationship between the string ending in A and the string ending in O.  Let's say we are insterested in what happens when we know P and Q are both red but R and S are both blue.  We are going to use a count above the string and indicate the truth values below.  What we will find is that the entire string becomes red (always looking for the final letter).
 

1
2
2
1
2
3
2
1
P
Q
N
A
R
S
O
C
1
1
0
0
0
0
0
1

The elements for A are P and N, the elements for O are R and S, and the elements for C are A and O.
The rule for working here tells us that we look back to the last letter that has the same count as the signal and use that with the last letter before the signal itself.

If we wanted to see all our possibilities we would need a truth table with sixteen lines to allow for every possible combination of red and blue with the variables.

For those with PC's and a Microsoft operating system, I recommend downloading and working with the following files (they can be used to develop the answers for the next assignment):
Introducing PLN
Evaluating strings
Equivalent strings
Please note that similar files, including a version of the second program that works with colors, are available at http://groups.yahoo.com/group/symboliclogic/files/
 
 

Go on to Assignment 2.