As we have seen, propositions can affirm or deny predicates of subjects. It’s important to keep in mind that affirming a predicate of a subject doesn’t get you a true proposition in all cases, just as denying a predicate of a subject doesn’t get you a false proposition in all cases. Consider the next example:
Aristotle is female.
This proposition would be true, according to our earlier standard, if and only if it actually is the case that Aristotle is female. However, Aristotle is male. Therefore, “Aristotle is female” is false and its negation, “Aristotle is not female” is true. It should be clear by now that negation is a kind of switch between the two truth-values: take a true proposition and negate it – you’ll get a false proposition; take a false proposition and negate it – you’ll get a true proposition. Furthermore, we can also see now that whenever we take a proposition and its negation, if the first is true, the second has to be false; and if the first is false, the second has to be true. We’ll have more to say about this result at the very end of our course.
At this point, you might be asking yourself about what would happen if we took a proposition in which a predicate is denied of a subject and then negated that. Look at the following:
2. Aristotle is not female.
According to our earlier explanation, the negation of example 4 would look something like this: “Aristotle is not not female” or “It’s not the case that Aristotle is not female.” These look quite strange, but luckily, we can simplify them to a more familiar expression. Take the former. We know that “Aristotle is not female” is true and we also know that negation switches truth-values. Therefore, we know that “Aristotle is not not female” – the negation of “Aristotle is female” – is false. But we know that “Aristotle is female” is also false from example 3, that its negation is exactly “Aristotle is not female,”. So, when we say “Aristotle is not not female,” we’re just flipping back to “Aristotle is female.” in a roundabout way.
The above is an instance of the kind of reasoning we sometimes do in logic: we have some information and we leverage it to get other, hopefully clearer information. What we learned from example 4 is simple to state: two negations get you an affirmation. In logicians’ talk, this is called the law of double negation.