TRUTH TABLE

A truth table is a breakdown of a logic function by listing all possible values the function can attain. Such a table typically contains several rows and columns, with the top row representing the logical variables and combinations, in increasing complexity leading up to the final function.

In a logic function, there are three basic operations: NOT (also called inversion or negation and symbolized -), OR (also called disjunction or addition and symbolized +), and AND (also called conjunction or multiplication and symbolized *). The values of the functions are normally assigned as logic 0 = false and logic 1 = true. Thus, the following rules apply:

If A = 0, then -A = 1
If A = 1, then -A = 0
A+B = 1 except when A = 0 and B = 0
A+B = 0 if A = 0 and B = 0
A*B = 0 except when A = 1 and B = 1
A*B = 1 if A = 1 and B = 1

HOW TO MAKE TRUTH TABLE?

Truth tables provide a useful method of assessing the validity or invalidity of the form any argument. We can use the table to determine whether the entire form of the argument is true or false, based on one very simple rule: Any argument that allows for a set of all true premises with a false conclusion must be invalid. This elegant process provides us with a means of providing a logical, deductive proof that an argument form is valid. In addition, this process allows us to identify which forms are invalid, allowing for refutations by logical analogy: a process wherein one may use a ridiculous version of an opponents argument, using the precise form of his argument, to show where it the argument form reaches illogical conclusions.

To use a truth table to test an argument:

1. Make a column for each of the components used in the argument. There must be a line for each possible combination of truth values for these components. Each additional component will double the number of lines needed. A single component will need two lines. Thus, if there are n components there must be 2ⁿ lines. By using a set pattern we can be sure to have included all the possible combinations and that no combination occurs more than once. The set pattern is to alternate true with false in the first column, in the second column (if needed) alternate 2 trues with 2 falses, in the third column (if needed) alternate 4 trues with four falses, and so on (doubling the number kept together for alternation with each column one adds) until the final component's column consists of the top half true and the bottom half false.

2. Add a column for each premise and for the conclusion. This may require additional columns to enable the calculation of complex premises or conclusions. Calculate the truth values for each line in the premises' and conclusion's columns.

3. Identify the lines where the conclusion is false and check those lines to see whether there is at least one false premise on that line.

4. If there is at least one false premise on every line where the conclusion is false, the argument is valid. Otherwise, you have demonstrated the possibility of all the premises being true at the same time as the conclusion is false, which is the mark of an invalid argument.