33.4: Validity
- Page ID
- 95349
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We are going to use tables to test for validity. We discussed validity back in Chapter 2, but as a reminder, an argument is valid just in case if all the premises are true then the conclusion must be true. So far, we have been looking at testing sentences and relationships between sentences. Validity, however, applies only to arguments. To test for validity, we need to construct a truth table with all of the premises and the conclusion on it. So, for the argument:
P v Q
~P
~Q
The table will look like:
.png?revision=1&size=bestfit&width=659&height=169)
Reading a table for validity is slightly more involved than with past concepts. The first step is to restrict our focus to only the rows where all the premises are true. From there, we need to look at the truth value of the conclusion. If there is a row where all the premises are true and the conclusion is false, then the argument is invalid. Under any other circumstances the argument is valid. So, if the conclusion is always true when the premises are true, then the argument is valid, and if there are no row where all the premises are true, then we don’t even need to look at the conclusion; we know the argument is valid. So, back to the table above, the second row has all true premises and a false conclusion. When explaining the results of a table it is also helpful to refer to the row that shows the invalidity. So, we would say the argument is invalid on row 2.
Let’s work an example where the argument is valid.
P v Q
~P
Q
.png?revision=1&size=bestfit&width=657&height=166)
In this case, we have one row in which all the premises are true – row 2 – and on that row the conclusion is also true. So, this argument is valid.
Exercises
Construct truth tables for the following arguments to test for validity.
1. P v ~P
P
~P
2. P → Q
Q
P
3. P v (Q v R)
~Q
P v R
4. P v Q
Q v P
5. P → Q
Q → R
P → R
6. ~P → Q
~Q
P ← → Q
7. ~(P v Q)
~P & ~Q
8. ~(P → Q)
Q
9. Q v ~Q
P v ~P
(Q v ~P) & (~Q v P)
10. P → Q
~Q v P
P
- Selected Answers
-
4. P v Q
Q v P
Valid. There is no row where the premise is true and the conclusion false.
.png?revision=1&size=bestfit&width=526&height=170)
6. ~P → Q
~Q
P ← → Q
Invalid. On row 3 the premises are true and the conclusion is false.
.png?revision=1&size=bestfit&width=694&height=176)