Validity
At the start of this text, we said that logic is the theory of inference.
That means that logic should tell us when our inferences are good and when they
aren't. The notion of validity is central to the goal of separating good
from bad inferences. A valid argument is argument where the truth of the
premises guarantees the truth of the conclusion. If the premises are true,
then the conclusion must be true as well. An invalid argument is one where there
is no such guarantee. In an invalid argument it is possible for the premises of
the argument to be true and the conclusion false.
When we speak of the possibility of premises and conclusions being true or false, we're talking of course about truth tables, and specifically about comparing the truth values of the premises and the conclusion on a single row of the truth table. So let's look at the truth-table for an argument to see what's going on.
The argument we'll evaluate first is:
| P ⊃ Q |
| Q |
| P |
You may remember this argument from the opening sections of this text. It's called "affirming the consequent," because the second premise is the consequent of the first premise. Is it valid? Let's do a truth table and see:
| P | Q | (P ⊃ Q) | Q | P |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | F |
Is it possible for the premises to be true and the conclusion false? There are two rows where the premises are true, namely the first row and the third. On the first row the conclusion is also true. So the first row does not show that the argument is invalid. But on the third row the conclusion is false. Hence the argument is invalid. It takes just one row where the premises are true and the conclusion false for the entire argument to be invalid.
Let's look at a valid argument:
| A v B |
| ~A |
| B |
And here's the truth table:
| A | B | (A v B) | ~A | B |
| T | T | T | F | T |
| T | F | T | F | F |
| F | T | T | T | T |
| F | F | F | T | F |
For this argument there's only one row where the premises all come out true, and that's the third row. But the conclusion is also true on that row. Thus if the premises of this argument are true, the conclusion will also be true.
It's worth emphasizing that the crucial rows to hone in on when you're testing an argument for validity are those rows where the premises are all true. The simple question we ask is this: On such rows, is the conclusion true or false? If it's false on any such row, the argument is invalid. If the conclusion is true on all rows where the premises are true, then the argument is valid.
In this exercise, you'll put things together. You will translate an argument and test it for validity using the complete truth-table method.
There's an important relationship between the validity of an argument and the rule of truth for conditionals. The following exercise explores that relationship.
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