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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!

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!

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!

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.

Your 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
				  

Appearance Problems in Netscape

If you are using Netscape, and statement letters, vertical bars, and truth values (T's and F's) do not seem to be lining up properly in columns, then Netscape is probably not paying attention when it is being told to use a "nonproportional" font in a text area. The problem is that Netscape by default likes to use a "proportional" font in text areas, that is, a font whose characters come in different widths. Thus, in a proportional font like Times or Helvetica, an "m", for example, will usually be a lot wider than an "i" or a "-". To fix this problem, click on the "Edit" menu option and then click on "Preferences...". In the dialog box that comes up, click on the little icon next to "Appearance" so you can see its suboptions, and then click on "Fonts". Depending on whether you are using Netscape 4 or Netscape 6, do one of the following:
  • In Netscape 4, on the right side of the dialog box you should find that the button next to the option to "Use my default fonts, overriding document-specified fonts" has been clicked. Click on either of the other options which begin "Use document-specified fonts".
  • In Netscape 6, on the right side of the dialog box you will simply find a checkbox option to "Allow documents to use other fonts". Check that option.
This procedure should allow Netscape to obey when it is told to use a nonproportional font, and hence should now render beautiful truth tables!

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