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Through The Looking Glass: Logic in the Palm of Your Hand
Ed shows how the logic functions in Lotus 1-2-3 can save your sanity in a logic course as well as in business.
"If we extend credit on the Buskens account, they will feel obliged to accept our bid on the next project. Either we could use a more generous profit margin in making our next bid, or Buskens will not feel obliged to accept our bid, or both. However, it will never happen that we use a more generous profit margin and that our financial situation will get worse. Hence, if we extend credit on the Buskens account, our financial picture will get better."
How would you respond to this argument? Decision makers should never have to face such tortuous reasoning, but all too often they do. Students of logic, however, who daily confront this type of reasoning soon learn that it's not enough to respond with "That sounds right." They learn to prove definitively that such reasoning is either correctly or incorrectly constructed. Truth-table analysis provides a precise procedure for such proofs.
You may remember seeing a truth- table in a college course in logic, but did you ever think that those columns of zeros and ones would find their way into business and research? The fields of operations research, digital circuit design, computer science, and legal contract writing currently use truth-tables. While you won't design a circuit by following this article, you will provide yourself with an intellectually challenging experience with one of the more tangential uses for the Lotus 1-2-3 program built into your Palmtop.
Truth-tables provide a precise path through a logical argument by enabling a logician to follow a line of reasoning step by step. However, such paths are often replete with potential potholes. To ruin a lengthy analysis, you need only one mistake. However, using 1-2-3 to perform truth-table analysis enables you to detect the flaws in arguments.
To Be or Not To Be
The truth-table shown in Screen 1 is familiar to logicians and mathematicians. The table shows the logical meaning of and, or, if ... then and not. Most people are familiar with the common usage of these words. However logicians don't tolerate loose usage because, for them, words must mean precisely what they say.
Screen 1: This truth-table demonstrates the logical meaning of and, or, if...then, and not. The zeros stand for false; the ones stand for true. The first two columns show all the possible states of truth or falsity of the two simple statements: X, Y. The zeros and ones in the rest of the table tell whether the various compound statements composed of these simple statements are true or false.
To read the truth table, first look only at the top row. Reading from left to right, the table says that given any two simple statements assigned the variables X and Y, you can create the following compound statements: X and Y, X or Y, and If X then Y. You can also deny a simple statement: Not X.
In the table, the zeros stand for false, the ones stand for true. The first two columns show all the possible states of truth or falsity of these two simple statements. That is, both can be true, both can be false, or one can be true and one can be false. The zeros and ones in the rest of the table tell whether the various compound statements composed of these simple statements will be true or false.
Take a simple example. Let X stand for the statement "Wolfgang Amadeus Mozart composed music." Let Y stand for the statement "William Shakespeare discovered America." I hope that you all readily agree that statement X is true and statement Y is false. Now look down the first two columns of the truth table. You want to find the row that displays 1 in the X column and 0 in the Y column, in this case, row 3 (See Screen 1).
Consider first the compound statement X and Y. Wolfgang Amadeus Mozart composed music and William Shakespeare discovered America. Look in row 3 of the X and Y column and you'll see that this statement evaluates to 0, or false. The whole statement is false because one of the statements is false.
Now consider the compound statement X or Y. Wolfgang Amadeus Mozart composed music or William Shakespeare discovered America. Looking under the X or Y column, you see the value 1 in row 3. This statement is true because one of the statements is true.
What about the statement Not X: Wolfgang Amadeus Mozart did not compose music. This statement is false. if you look in the sixth and seventh columns of the table, you'll see that if the original statement is true (1), the opposite of that statement is false (0), and vice versa.
But what about the case of If X then Y. If Wolfgang Amadeus Mozart composed music, then William Shakespeare discovered America. Why would this be false? According to the rules of symbolic logic, the entire statement is false because the first part of the compound statement is true, while the second part is false.
Notice that the truth table for two statements is only four rows deep. If you let stand for the number of simple statements in a compound statement, then the number of rows in the truth table is 2^n. (i.e., 2 raised to the power).
Valid or Invalid?
The opening paragraph of this article is a good example of a compound statement made up of several simple statements. Logicians call the whole paragraph a logical argument. The task of the logician is to prove that the argument is correctly constructed. Such a proof establishes the validity or invalidity of the argument.
To prove the validity of this argument, you need look only at its structure; that is, you can throw away the words and examine just the framework of the argument. To begin such an analysis, you pick out all the simple statements in the argument and assign a letter to each. in this case, let's use the following:
C = We extend Credit on the Buskens account.
A = They will feel obliged to accept our bid.
P = We could use a more generous profit margin.
F = Our Financial picture will get better.
Not A = They will not feel obliged to accept our bid.
Not F = Our financial situation will get worse.
Now you can rewrite the structure of the opening argument by using single letters in place of the statements:
if C then A
P or not A
Not (P and not F)
Then if C then F
For this structure to be valid, the first three compound statements must lead, unerringly, to the last statement. Furthermore, they must do so whether each of the simple statements is true or false. To prepare this argument for truth-table analysis you convert it into the following long compound statement:
If [(if C then A) and (P or not A) and (not (P and not F))], then (if C then F).
What you have done is link the first three statements with the word and, and made that lengthy compound statement the first part of an if... then statement. The conclusion of the argument becomes the second part of the if... then statement. The conclusion is an if. .. then statement itself. This conversion to letters will let 1-2-3 do the work of analyzing the argument.
Table the Matter
Now let's put the 1-2-3 logical operators #AND#, #OR#, and #NOT#, as well as the @IF function, to work in determining the validity of the opening argument. As you may know, #AND# and #OR# look at two cells in a worksheet and show either 0 (false) or 1 (true), depending on the values in the two cells. The operator #NOT# looks at only one cell and returns 0 or 1. The @IF function gives you a way to determine whether an if... then compound statement is true or false.
By examining the truth table for "If X then Y" in Screen 1, we can see how to emulate this relationship using the Lotus @IF( ) function. The truth table says in effect "When the value of X is less than or equal to the value of Y, then the value in the fifth column is 1, otherwise the value is 0.' This can be emulated by using the Lotus function @IF(X<=Y,1,0).
To set up the truth-table worksheet as shown in Screen 2, start 1-2-3 on your Palmtop and set a global column width of 5: ([Menu] Worksheet Global Column-Width and enter 5.) Next, enter the letters shown in row 1. These letters stand for the simple statements in the argument. Next, in columns A through D, enter the zeros and ones, which represent all the possible combinations of truth values for these four statements.
Screen 2: Columns A through D constitute a table of all the possible truths values for four statements. By entering formulas (in columns E through K) that represent the premises in the argument presented in this article, you can determine the argument's validity. In this case, column J contains a formula representing the conclusion of the argument. Because this formula returns all ones, which indicate truth, you know that the whole argument is valid.
You already know how to calculate how many possible combinations of truth and falsity will exist for the variables - in this case, namely 2^4 or 16. However, you might wonder how to determine the proper order of the zeros and ones for the truth table. Once you ascertain the total number of combinations, you have all the information you need. For column 1 of the truth table, you enter zeros in the first half of the column and ones in the second half.
Then in the next column, you enter zeros next to half of the zeros that appeared in the previous column, and then ones in the rest. Then you enter zeros opposite half of the ones in the previous column, and ones in the rest. You continue this process until you've filled the first four columns (Screen 2).
Now assign range names by positioning the pointer in cell Al, pressing [MENU] and selecting Range Name Labels Down. indicate range Al..D1 and press Enter. Then enter the following formulas in the indicated cells:
El @IF(C< =A,1,0)
Select: [MENU]Copy and copy range El..Kl to range E2..El7. Now examine the values in column J. They should all be ones. This tells you that whatever the truth values of the individual statements, the end result is that the structure of the argument is always true or valid. If any one of the values in this critical column is a zero, the whole argument will be invalid. That's all it takes, just one zero and it's close but not good enough.
Now that you've learned the fundamentals of truth-table logic, it's time to put your skills to the test. Here are two arguments for you to evaluate. Use the techniques presented in this article to determine whether they are valid or invalid.
1. If you become a programmer, you will be too busy to take vacations. On the other hand, if you do not become a programmer, you will not have enough money to take a vacation. Therefore, whether you become a programmer or you don't, it follows that you will have either no time or no money for vacations.
Use the following abbreviations for the statements in this argument:
E = become a programmer
V = too busy to take vacations
M = have no money
2. If you don't bring me flowers this week, then you want me to think that you don't have a guilty conscience. However, if you want me to think that you don't have a guilty conscience, then you must, in fact, have a guilty conscience. Therefore, you must have a guilty conscience.
Answers: Both arguments are valid.
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