HELP

Before we discuss the specifics of filling in truth tables, click here if the characters in your truth tables do not look properly aligned in columns.

### Completing a Truth Table

Upon selecting a truth table exercise, one is presented with an incomplete truth table like this one:

 The Power of Logic Exercise 7.3A Problem 25

Complete the truth table!
 A B | ~A -> (~B -> A), B :. ~~A ----|---------------------------- T T | T F | F T | F F |

The top line of the incomplete truth table contains the statement letters contained in the argument, a vertical line, and the sentence or argument for which the truth table is to be constructed. The next line contains a separator -- basically a row of hyphens and a single vertical line. Beginning on the third line, beneath the statement letters appear the 2n possible distinct truth value assignments that can be given for the statement letters. (Because there is such a simple algorithm for constructing these truth value assigments, we decided to relieve you of the tedium of constructing them yourself -- but be sure you know how to construct them on your own! You will need to know how to to do this on your exams.) To complete each row correctly, follow these two guidelines:

1. Enter a truth value (i.e., T or  F) under each logical operator.
2. Enter a truth value beneath an occurrence of a statement letter if, and only if, that occurrence of the statement letter is either one of the premises of the argument or its conclusion .
Thus, following these guidelines a correct first row will look like this:

 The Power of Logic Exercise 7.3A Problem 25

Complete the truth table!
 A B | ~A -> (~B -> A), B :. ~~A ----|---------------------------- T T | F T T T T TF T F | F T | F F |

Filling out each row in similar fashion, a completed truth table will look something like this:

 The Power of Logic Exercise 7.3A Problem 25

Complete the truth table!
 A B | ~A -> (~B -> A), B :. ~~A ----|---------------------------- T T | F T F T T TF T F | F T T T F TF F T | T T F T T FT F F | T F T F F FT

Clicking on the `Check Truth Table Now!' button will do just that, i.e., it will check to see that you have filled in the truth table correctly. If you have not, you should get a helpful response telling you what went wrong, and a copy of your truth table in a new box in which you can make the necessary corrections.

### Determining Validity

Once your truth table is correct, you will be presented with a page that asks you to assess its validity. The relevant portion of the page looks like this:

 A B | ~A -> (~B -> A), B :. ~~A ----|---------------------------- `T T | F T F T T TF ` `T F | F T T T F TF ` `F T | T T F T T FT ` `F F | T F T F F FT `

As noted in Section 7.3 of the text, to test for validity, you look to see whether or not there is a row of the table on which the premises of the argument are true and the conclusion false. If you do not find such a row, the argument is valid, so click on the `Valid' button to complete the exercise. If you have missed such a row you will be so informed and presented with your errant truth table for you to fix. If you do find such a row, check the checkbox to its left, as indicated in our example:

 A B | ~A -> (~B -> A), B :. ~~A ----|---------------------------- `T T | F T F T T TF ` `T F | F T T T F TF ` `F T | T T F T T FT ` `F F | T F T F F FT `

You may then either click on the `Invalid' button (one row with true premises and false conclusion is all you need to establish invalidity) or you may continue to search for further invalidating rows. If the row you check is invalidating, but you miss further invalidating rows, the program will tell you your answer is correct, but will give you the opportunity to find the invalidating rows that remain. In either case, if you have checked a row that does not invalidate the argument, you will again be so informed and presented with your errant truth table. If you are correct, enjoy the praise that will be heaped upon you!

 The Power of Logic Exercise 7.3A Problem 25

# Correct!

 You are correct! Row 3 shows that the argument is invalid! On this row of the truth table, the premises of the argument are true and the conclusion false.

 ```A B | ~A -> (~B -> A), B :. ~~A ----|---------------------------- T T | F T F T T TF T F | F T T T F TF F T | T T F T T FT F F | T F T F F FT ```