LESSON #24
Indirect Truth Tables for Arguments
Reading Assignment: 6.5 (pp. 338-342--you may ignore the section on testing consistency)
Indirect truth tables are a kind of short-cut way of determining validity/invalidity. They are especially useful when there are a large number of components in an argument. Six different letters would require an unwieldy 64-row regular truth table! But as for any shortcut, there are potential detractions. Indirect truth tables can be more confusing if you are not quite careful and precise.
Recall that an argument is invalid if it is possible for it to have all true premises and a false conclusion at the same time. Indirect truth tables work by trying to force the argument into that situation. We will set up the argument assuming that the premises are all true and the conclusion is false. Then we will work to see if that is indeed possible. In other words, we will try to figure out what the truth values of the components would need to be to make the argument have true premises and a false conclusion. If we cannot do this, if we run up against a contradiction that we cannot get away from--then the argument is valid. If we are able to assign truth values to the components such that it makes the argument have all true premises and a false conclusion, then we have proven that the argument is indeed invalid. (PLEASE don't do all the work right and then get confused about what you have just proven. It'll make me cry to take points off in that case, but, I'll have to)
First, write your argument out in one line exactly like you would for a standard truth table.
Next you should "set" the premises true and the conclusion false. This you do by putting a T (or F) under the main operator of each statement.
Model:
| A | É | (B | · | C) | / | (B | v | D) | É | C | // | A | É | (B | v | D) |
| T | T | F |
Now you start working through the table. To see it in real time click here.
Begin with the conclusion. If there is only one way for the conclusion to be false, your work will be simple. Fill in the correct truth values under the letters that make the conclusion false.
· "v" is only false when both sides are false.
· "É " is only false when the antecedent is true and the consequent is false.
Your table will only have one line if it is one of these compound statements, or a similar one.
· "~" simply changes the truth value to its opposite.
Your table will only have one line if it is a negated conjunction.
And so on.
Model:
| A | É | (B | · | C) | / | (B | v | D) | É | C | // | A | É | (B | v | D) | |
|
T | T | T |
F |
|
F | |||||||||||
Continue filling in truth values, in whatever order you like, based on your knowledge of the connectives, until you find a contradiction. To see the steps from here on click here.
Valid
If you are not forced to fill in a T or F in a particular column, leave it alone. If you can fill in the entire rest of the chart with no contradiction the argument is invalid no matter what you put there. If not, you can come back and fill it in later.
If there is more than one way for the conclusion to be false
In this case you will need two or three lines for your indirect truth table. (You'll never need more than that).
Write columns under the letters in the conclusion with all the possibilities. Carry the "T"s under the main operators in the premises down the necessary number of lines.
Then, using the first line only, work the truth table as before. If you reach no contradiction, you're lucky. Stop. It's invalid.
If you reach a contradiction on the first line, go on to the second, and so on, till they're all used up.
It is important to understand that you should not give up trying to make the argument have true premises and a false conclusion until you are sure that all the options have been tried. Once, on the other hand, you have shown that it is possible to have all true premises and a false conclusion, then you can stop. You have then proven the argument invalid.
NOTE: This is a slightly different approach than Hurley's. He tells you to decide how many lines you need by choosing whether looking at the premise or the conclusion will be fewer lines. I think my way--always determine the truth values and number of lines by looking at the conclusion-- is simpler and less confusing. Sometimes you will have to do an extra line this way, though.
Be sure to be consistent. Go slowly and carefully.
I have noticed that people who are new to this process have an amazing capacity to fill in a contradiction without noticing or being bothered by it. This is a shame, because then, though they have done all but the last step carefully, they think a valid argument is invalid. Don't let this happen to you!! Cultivate your capacity to be bothered by contradictions. (Soon you'll be a real philosopher--contradictions are one thing we don't hate , we ENJOY being bothered by them--we find it stimulating--we call them "dilemmas". . .)
Logic Coach Assignment: 6.5 I 1-12
Assignment: (10 points each)
Use indirect truth tables to determine whether the following symbolized arguments are valid or invalid. Be sure to copy the problem down correctly!
1. (A É B)2. ~(C · D)
~D v ~E
~E
3. F É ~G
~G É H
H É
~I
F É ~I
4. J º (K É L)
~K v L
~K
~J
5. M É N
N É (~M · O)
P É Q
Q
6. A É (B v C)
C É (D · E)
~B
A É ~E
7. ( A v B) · (D
v E)
C É ~E
A É (F · ~D)
~C v ~A
A É B
8. E É (C
v D)
C É (B v D)
D É B
~B v (E · C)
C º E
9. (A v B) É ( C
v D)
C É (D · E)
(E v F) É G
A É G
10. A É (B v C)
B É (D v E)
C É (F v D)
(F v E) É D
A É D