## HOW TO BUILD A TRUTH TABLE

There are four steps to building a truth table.
1.   Determine the number of lines or rows in the table. To do this count the number of different (atomic) propositions in the formula(s) for which the table is being built. This number is also the number of different capital letters in the formula(s).
The number of lines is 2 raised to the power of this number.

 LETTERS 1 2 3 4 and so on LINES 2 4 8 16

2.   Second, the main operator has to be identified. It is never between a left and a right pair of ( ) , [ ] , or { } . One way to find this operator is to pair the parentheses. This can be done by starting at the left end of the formula and moving to the first ) and then backing up to the previous ( . Repeat for each pair of ( ).

3.   Next the basic input values are assigned to each letter. Start with the one on the left. The top half of the lines are assigned T for true and the bottom half F for false. Assign the same values in each column which contains the same letter.

 A T T F F ∨ ( B ⊃ ~ A T T F F )

If there are four lines then the first column consists of two consecutive T's followed by two consecutive F's. If there are eight, then the consecutive number is four...

For each new letter, divide the consecutive number in half. The column for the final letter should contain alternating TF's.

 A T T F F ∨ ( B T F T F ⊃ ~ A T T F F )

4.   The final step is to calculate the values of each logical operator. Keep in mind that each column is used only once to calculate an operator value. Once it is used, do not use it again. On the computer, you can mark a column as having been used by changing its case. Note: If this procedure is followed, then the last column calculated will be the one for the main operator. Below, as on the computer, it is marked within | |

 A T T F F ∨ |T| |T| |T| |T| ( B t f t f ⊃ F T T T ~ f f t t A t t f f )

How to judge the proposition or argument is explained on the terminology screen.