7.  UNDERSTANDING TRUTH FUNCTIONS

Being able to symbolize various things we say would not be particularly useful unless we are able to do something with these expressions.  What we do with them is look to see how some in fact have the same logical meaning as others and how some follow necessarily from one or more others.  In other words, we are interested in what we call equivalences (what happens when we say that if it not's true that logic is both easy and fun then what must be true is that logic is either not easy or not fun) and implications (if we know that logic is fun unless it is not easy but we also know that it really is easy, so we can say for sure that it has to be fun).  These are all examples of what we mean by arguments--using the truth of one or more statements as support for saying that still another statement is true).

All of this hinges on the idea that the statements we work with can be thought of as definitely true or false, even if we might not know which.  Atomic statements, such as P or Sa, can be seen as always one or the other (true or false) while the statements that result from using our connectives (including quantifiers). such as P->Q and ~Sa and (x)Sx, have a truth value that depends on the value of the atomic statements and how connectives are used.  We call these all these expressions truth-functional, and we examine the possibiities in what are called truth tables.

The persistence of the word "truth" shows the historical grounding of symbolic logic in how we work with language, even though in electronics we also talk about truth tables to indicate current being on or off.  In part to show the more general applications of what we are working with, I am going to use the numerals 1 and O in place of the T's and F's typically found in a logic textbook.  (Those interested in the inner dynamics of what is going on might want to note that in essence we are working with a mod 2 number system so that every expression could also be rewritten as a formula in what we might call an algebraic notation).

Let's look at a  basic set of truth tables.  If we have P and Q as example atomic statements and we then use binary connectives (representing our "and" and "or" and "if" and "if and only if"), we will have four lines that we need to account for.  We start with the fact that both P and Q can be either true or false and then attempt to see what would have to be true from that point on.
P
 Q
 ~P
 ~Q
 P & Q
 P v Q
 P -> Q
 P <-> Q
1
 1
 0
 0
 1
 1
 1
 1
1
 0
 0
 1
 0
 1
 0
 0
0
 1
 1
 0
 0
 1
 1
 0
0
 0
 1
 1
 0
 0
 1
 1
Note that a conjunction is true only when both parts are true, a disjunction is false only when both parts are false, red flag a conditional is false only when the lefthand part (the antecedent) is true and the righthand part (the consequent) is false, a biconditional is false only when both parts are different.  You should memorize this basic chart, especially the pattern for using our arrow.

(For those interested in the idea of algebraic notation, note that conjunctions can be seen just as multiplication of the original values (PQ), negations as adding 1 (P+1), disjunctions as multiplication plus addition (PQ+P+Q),  implications as multiplication plus the antecedent plus 1 (PQ+P+1), and equivalences as addition plus 1 (P+Q+1).  This is once used to develop a program to evaluate mechanically the truth value of an expression when working with a postfix literal notation.  For more see  my web page on algebraic notation from an earlier version of an online textbook.)

Obviously, we want to be able to use any letter at all and also use expressions that result from previous combinations of these letters (what are called substitution instances).  Supposing we need to determine the truth value of (P v Q) & ~R under all conditions.  Since we have three letters in use, we need to double the number of lines in play.  What we are going to do is work from inside out until we arrive at our main connective.  What we see when we do is that the expression can be true only under three conditions: (1)  P and Q are both true while R is false, (2) P is true while both Q and R are false. and (3) Q is true while both P and R are false.  Do note that it makes things easier to follow by first setting up the columns for the variables in alphabetical order.
P
 Q
 R
 ( P v Q)
 &
 ~R
1
 1
 1
 1
 0
 0
1
 1
 0
 1
 1
 1
1
 0
 1
 1
 0
 0
1
 0
 0
 1
 1
 1
0
 1
 1
 1
 0
 0
0
 1
 0
 1
 1
 1
0
 0
 1
 0
 0
 0
0
 0
 0
 0
 0
 1

Truth tables for simple expressions in predicate logic can be prepared in just the same way (imagine we had (Fa v Gb) & ~Hc in place of P and Q and R).  When we bring in quantifiers, however, things will become more complicated, and for this reason we are going to employ a rather different technique as we go on through the course.


EXERCISES


Complete the following table, filling in just for the main connectives.   Afterwards check your work against the answer key.
P
 Q
 R
 P->(Q & R)
 P->(Q v R)
 (P  v Q)  -> R
 (P &~Q) -> ~R
 ~P & (Q <-> R)
 (P v~Q) v (R & ~P)
1
 1
 1
          
           
                    
                   
            

1
 1
 0





1
 0
 1





1
 0
0





0
 1
 1





0
 1
 0





0
  0
1





0
 0
 0