LESSON #21

Truth Functions

Reading Assignment: 6.2 (pp. 313-323)

Click here to skip the following discussion and go straight to the assignments.

Each of the five connectives can be defined in terms of a corresponding truth function. These truth functions are represented by the little charts, or truth tables that capture all the possible combinations of the truth or falsity of the component simple statements.

You will need to know this section very well. Do not go on to Lesson #22 until you are quite familiar and comfortable with the truth functions corresponding to all five connectives. These should be memorized!

You can see the truth table for each connective on pages 304 through 308, and, in a slightly more condensed form, inside the front cover of the text. Here is a shorthand way of summarizing the table. This will be much like a chant to say to yourself. (It should probably be put it to music!--extra credit offered to anyone who sends me a tape...(!))

You will also need to be able to determine the truth values of complex statements. All you do for this is assign the truth values to the letters, then progressing from the most deeply embedded components (what is "most inside" the parentheses) "unpack" the parentheses until you finally reach the operator that is completely outside of any parentheses, determining the truth value of each part step by step. Hurley gives a good explanation of this process on pp. 309-311. More and more practice will be of much help here. Go slowly.

The main operator of a complex statement is what is entirely outside the parentheses (or brackets or braces). Note that if a "~" and another connective are both entirely outside parentheses, the "~" will "lose out" to the other connective (i.e. the other connective will be the main operator). A "~" is understood as attaching itself to whatever it is negating. If the "~" is the only connective entirely outside any parentheses, then it is the main operator.

Logic Coach Assignment: I 6-10, III 10-20 (do some of the earlier ones first if you are confused)

Assignment 1: (Five points each)

A. Identify the main operators in the following propositions.

1. ~(A · B)

2. (C v D) É E

3. [~(F v G) · H] º I

4. J v [(K É L) · (M º N)]

5. ~(O v P) · (Q É S)

 

B. Determine the truth values of the following symbolized statements. A, B & C are true and X, Y, and Z are false. Show your work.

1. (B ·  Z) v (~X · A)

2. (Z º ~X) v (A º B)

3. ~[(A · ~B) É (~X v C)]

4. ~A · (Y É C)

5. C É (C É ~B)

6. ~B v (Y É A)

7. (A v Y) · (Z v ~C)

8. (B É X) · ~(B É ~X)

9. (A v ~B) É (C É X)

10. ~[(X · Y) É B]

ASSIGNMENT 2:

A. Write the following compound statements in symbolic form, then use your knowledge of the historical events referred to by the simple statements to determine the truth of the compound statements. (10 points each)

1. Nixon resigned the presidency and Lincoln wrote the Gettysburg address.

2. It is not the case that Custer was killed by the Indians, unless both Nixon resigned the presidency and Edison invented the telephone.

3. Either Lindburgh crossed the Atlantic and Edison invented the telephone or both Nixon resigned the presidency and it is false that Edison invented the telephone.

B. Find a compound statement on your own and determine its truth value as you did above. I won't make you submit a photocopy, but do try to use one you have found, not just one you have copied from elsewhere in the text. --No cheating!! (20 points)

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