Truth Tables for Propositions
Reading Assignment: 6.3 (pp. 325-331)
Click here to skip the following discussion and go straight to the assignments.
We can construct truth tables for statements that will show every possible combination of truth values for the component parts (letters). The number of lines (rows)* in a truth table depends upon the number of different components in the statement (i.e., how many different letters there are).
You will not be required to do more than a 16 line truth table, so just remember that:
|2||different letters requires||
|3||different letters requires||
|4||different letters requires||
Refer to the text for detailed instructions on setting up a truth table.
1. Write out your symbolized complex proposition in one line, with the correct number of lines below it.
2. Make a column for each letter and operator in the proposition. Ignore the parentheses for now, but don't obscure them, because you will need to read them later.
3. Beginning with the left-most letter, fill in the column below with truth values.
4. Continuing to the right, making a new column for each new letter and ignoring the operators. NOTE: Repeats of the same letter get the same configuration of T's and F's
In setting up the table, you make the first column* (no matter whether a 4, 8 or 16 row table) always half true (1st half) and half false. The next column you put half as many consecutive T's than in the first column before you switch to F's.
Each column will be half as many consecutive T's and F's than the previous column. Your last column will always alternate 1T and 1F, 1T and 1F.
So, for example, in an eight row table you will first begin with four consecutive T's (because four is ½ of eight), then in the next column you will begin with two consecutive T's (because two is ½ of four) and your last column will have only one T (because one is ½ of two). Your last column should always be one T and one F alternated.
5. After the initial set-up of a truth table it is simply a matter of following the rules for connectives (truth functions). Memorizing the "chants" from Lesson #20 will be very helpful here. Here again, you will "unpack" from the inside of the parentheses out.
· Starting with the innermost parentheses (or one of them, if there are more than one), work down the column, looking at the columns immediately adjacent to it for the truth values of the simple propositions.
· You may want to lightly stroke out "used" columns, when you are done with them.
· When one column is completed, you will use it and the next unused column to fill in the empty column of next more basic operator, then stroke it out.
· When you have filled in the last column highlight it somehow. It is the main operator--it is what you'll be "reading".
Reading Truth Tables
Once a truth table is completed, we can interpret the table and learn something about our statement.
First you will learn to classify single statements. You do this by looking at the column under the main connective.
|Statement is . . .||when main connective column shows:
|Contingent||At least 1 T and 1 F
Secondly, you will learn to compare two statements. Set up the truth table with the statements side by side, using a double line to separate them. Fill in the table and compare the main connective columns of each statement. NOTE: The same letter gets the same column configuration in BOTH statements.
|Statement is . . .||when main connective column shows:|
|Equivalent||the same on every row|
|Contradictory||the opposite on every row|
|Neither||the same on some and opposite on some (at least one of each)|
|Consistent||At least one line that is T on both truth tables|
|Inconsistent||No line that is T on both tables.|
Logic Coach Assignment: I 1-10, II 1-10
Assignment: (10 points each)
NOTE: Always check to make sure you have copied the problem down correctly!
A. Use truth tables to determine whether the following symbolized statements are tautologus, self-contradictory or contingent.
1. (A · B) v C
2. F º (G É ~H)
3. P É (R É P)
4. (R É S) · (R · ~S)
5. (N É P) v (P É N)
B. Use truth tables to determine whether the following pairs of statements are logically equivalent, contradictory or neither. Also state whether they are consistent or inconsistent.
1. ~(P · Q) ~P v ~Q
2. P É Q Q É P
3. P · (Q v R) (P · Q) v (P · R)
4. A É ~B A · B
5. ~(S v T) ~(S · T)
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