Section 4: Deductive validity
An argument is deductively valid if and only if it is impossible for the premises to be true and the conclusion false.
The crucial thing about a valid argument is that it is impossible for the premises to be true at the same time that the conclusion is false. Consider this example:
Oranges are either fruits or musical instruments.
Oranges are not fruits.
.˙. Oranges are musical instruments.
The conclusion of this argument is ridiculous. Nevertheless, it follows validly from the premises. This is a valid argument. If both premises were true, then the conclusion would necessarily be true.
This shows that a deductively valid argument does not need to have true premises or a true conclusion. Conversely, having true premises and a true conclusion is not enough to make an argument valid. Consider this example:
London is in England.
Beijing is in China.
.˙. Paris is in France.
The premises and conclusion of this argument are, as a matter of fact, all true. This is a terrible argument, however, because the premises have nothing to do with the conclusion. Imagine what would happen if Paris declared independence from the rest of France. Then the conclusion would be false, even though the premises would both still be true. Thus, it is logically possible for the premises of this argument to be true and the conclusion false. The argument is invalid.
The important thing to remember is that validity is not about the actual truth or falsity of the sentences in the argument. Instead, it is about the form of the argument: The truth of the premises is incompatible with the falsity of the conclusion.
Inductive arguments
There can be good arguments which nevertheless fail to be deductively valid. Consider this one:
In January 1997, it rained in San Diego.
In January 1998, it rained in San Diego.
In January 1999, it rained in San Diego.
.˙. It rains every January in San Diego.
This is an inductive argument, because it generalizes from many cases to a conclusion about all cases.
Certainly, the argument could be made stronger by adding additional premises: In January 2000, it rained in San Diego. In January 2001. . . and so on. Regardless of how many premises we add, however, the argument will still not be deductively valid. It is possible, although unlikely, that it will fail to rain next January in San Diego. Moreover, we know that the weather can be fickle. No amount of evidence should convince us that it rains there every January. Who is to say that some year will not be a freakish year in which there is no rain in January in San Diego; even a single counter-example is enough to make the conclusion of the argument false.
Inductive arguments, even good inductive arguments, are not deductively valid. We will not be interested in inductive arguments in this book.