Logic 5: Complex Truth Tables

Now that we have the basic truth tables for the five connectives, we can examine complex truth tables, truth tables that involve combinations of connectives.  The truth table method allows us to evaluate expressions made of several statements one step at a time until we reach the set of truth values for the expression in all possible cases.

We have seen that the expression A ^ B is true only when both A and B are true.  What about the negation of this expression, ~ (A ^ B), or “It is not the case that A and B”.  It is easy to see, particularly when we create a truth table, to see that we get the inverse or opposite of the truth values for A ^ B.

Notice that the final truth values for the whole expression are now on the left, underneath the NOT symbol.  First, we figure out the truth values for the part of the expression inside the parentheses (A ^ B), and then, for our second step, we take those truth values to figure out the truth values for the whole expression.

Now, what about the expression ~ A ^ ~ B, “It is not the case that A and it is not the case that B”.  Does it have the same truth values as ~ (A ^ B)?  We can create a truth table to show that it does not.  First, we negate the truth values from our original four cases underneath ~ A and ~ B, such that underneath ~ A we have F, F, T and T, and underneath ~ B we have F, T, F and T.  Then, taking the new truth values and remembering that AND is only true when both sides are true, we evaluate the expression to find the truth values for the entire expression.  Notice that the final truth values for ~ A ^ ~ B are the same in the first case (F) and fourth case (T), but not the same in the second and third case.  This shows us that we cannot distribute the NOT within the parentheses without changing the expression.

We have now used NOT inside parentheses and outside parentheses.  What about expressions in which NOT is both inside and outside of parentheses?  Let us construct a truth table for the expression ~ (~ A ^ ~ B).  Within the parentheses, it is the same truth table as our last, for ~ A ^ ~ B.  Then, as we did for ~ (A ^ B), we will negate the expression within the parentheses to arrive at the final truth values for the whole expression.

Notice the surprising result, that ~ (~ A ^ ~ B) has the same final truth values as A OR B.  As we will see in the next chapter, this means that ~ A ^ ~ B is equivalent to ~ (A v B), such that saying, “It is not the case that A and it is not the case that B” is the same thing as saying, “It is not the case that A or B”.  We can show this, one half of DeMorgan’s Theorem, using truth tables.

Now, let us construct a truth table in which two expressions that are joined together with connectives are themselves joined together with connectives.  Let us say we want to know in what cases the expression, “Either I have an apple and a banana or I have an apple or a banana”, which we can symbolize as (A ^ B) v (A v B).  First we must find the truth values for both of the expressions within the parentheses.

Then, as our second step, we use the truth values for each of the expressions within the parentheses to arrive at the final truth values for the expression.  With negation, we worked from the inside to the outside, but here, we work from the outside inward.  Interestingly, the entire expression has the same truth values as and is the equivalent of A v B.  Now let us add in negations together with connectives that connect other connectives.

Let us construct a truth table for the expression, “Either I have an apple and a banana or I have an apple and I do not have a banana”, (A ^ B) v (A ^ ~ B).  First, we must find the truth values for both expressions within the parentheses, as we did with the last truth table, but this time, we must remember to put negated truth values underneath the ~ B.

Now, taking the truth values for both of the expressions within the parentheses, and remembering that OR is true when there is a T on either side, we can find the truth values for the entire expression.  Notice that just as we take the truth values on either side of a connective within the parentheses to figure out the truth value for that connective, we find the truth values for the entire set within the parentheses, the truth value underneath the connective within the parentheses, to figure out the truth value for the connective between the two sets within parentheses.

Let us construct one final complex truth table.  Let us say we want to evaluate the expression, “If I do not have both an apple and a banana, then I do not have a banana”, which we can symbolize as ~ (A ^ B) > ~B.  First, we must find the truth values for the expression within the parentheses, as well as negate the original truth values for B and place them underneath the ~B.

For our second step, we negate the truth values for the expression inside the parentheses, and then use these negated values along with the truth values underneath ~B to find the final truth values for the entire expression, remembering that IF-THEN is true unless the truth value to the left is true and the truth value to the right is false.

Exercise 4: For each of the following sentences, symbolize the sentence in SL and evaluate the expression by constructing a truth table.

1)  I do not have an apple or a banana.

2)  If I do not have an apple, I have a banana.

3)  If I have an apple or I have a banana, then I have both an apple and a banana.

4)  If I have a banana, then I do not have both an apple and a banana.

5)  If I do not have both an apple a banana, then I have an apple or I have a banana.

6)  If I have an apple then I do not have a banana, and I have an apple.

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