Do you see the problem with these two?
Do you see the problem with these two? In the first one (denying the antecedent), even a false antecedent (if Maria will not walk to work) doesn’t mean that she will not be late. Maybe she will sit at home and be late, or be late for some other rea- son. When the antecedent is denied, the premises can be true and the conclusion false—clearly an invalid argument. In the second argument (affirm- ing the consequent), even a true consequent (if Maria will be late) doesn’t mean that she will walk to work. Some other factor besides her walking could cause Maria to be late. Again, the premises can be true while the conclusion is false—definitely invalid.
Consider one last form, the hypothetical syllo- gism (hypothetical means conditional; a syllogism is a three-statement deductive argument):
If Maria walks to work, then she will be late.
If she is late, she will be fired.
Therefore, if Maria walks to work, she will be fired.
If p, then q.
If q, then r.
Therefore, if p, then r.
The hypothetical syllogism is a valid argument form. If the premises are true, the conclusion must be true.
Obviously, if modus ponens, modus tollens, and the hypothetical syllogism are always valid, then any arguments you encounter that have the same form will also be valid. And if denying the antecedent and affirming the consequent are always invalid, any arguments you come across that have the same form will also be invalid. The best way to make use of these facts is to memorize each argument form so you can tell right away when an argument matches one of them—and thereby see immediately that it is valid (or invalid).
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But what if you bump into a deductive argu- ment that does not match one of these common forms? You can try the counterexample method. This approach is based on a fundamental fact that you already know: it is impossible for a valid argument to have true premises and a false conclusion. So to test the validity of an argument, you first invent a twin argument that has exactly the same form as the argument you are examining—but you try to give this new argument true premises and a false con- clusion. If you can construct such an argument, you have proven that your original argument is invalid.
Suppose you want to test this argument for validity:
If capital punishment deters crime, then the number of death row inmates will decrease over time.
But capital punishment does not deter crime.
Therefore, the number of death row inmates will not decrease over time.
You can probably see right away that this argu- ment is an example of denying the antecedent, an invalid form. But for the sake of example, let’s use the counterexample method in this case. Suppose we come up with this twin argument:
If lizards are mammals, then they have legs.
But they are not mammals.
Therefore, they do not have legs.
We have invented a twin argument that has true premises and a false conclusion, so we know that the original argument is invalid.
IMPLIED PREMISES
Most of the arguments that we encounter in every- day life are embedded in larger tracts of nonargu- mentative prose—in essays, reports, letters to the editor, editorials, and the like. The challenge is to pick out the premises and conclusions and evaluate