4.3 Not to Neglect Negation: ~
You've learned two connectives already: & and v. Now we learn a third, ~, called negation, tilde, or not.
Together these three connectives are called Boolean connectives, because they were studied by George Boole 150 years ago. They also have some important logical features which we'll learn about soon.
First we must learn to use negation. Here's how we say Pis a not guilty: ~P
As you can see, English has many ways of expressing negation. We symbolize them all with ~.
In English it's a bit odd to have multiple negations in one sentence, but it is perfectly fine in BOOL. For example, ~~~~P is a perfectly good sentence.
We've discovered another use for negation: sometimes in English one word is the negation of another, like "guilty" and "innocent". "Innocent" just means "not guilty."
If we are already using P to mean Pia is guilty, then we can capture the logical connection between those words by writing ~P for Pia is innocent.
It wouldn't be impossible to use a different letter, like I, to mean Pia is innocent, but then we would lose a structural relation between innocent and guilty.
Our goal in developing the logical system BOOL is to study those structural relations, so that is why we want to capture them in our translations whenever we can.
Next, we'll look at another important feature of negation.
The only difference in those translations is the parentheses. But you can see that the parentheses matter: those sentences in English do not say the same thing.
It is too messy to write parentheses when negation is around an atomic sentence: ~(P). So we will use this rule: negation always applies to the smallest possible sentence. Thus in the sentence ~P&Q, the negation is just around the P.
If you want to negate a complex sentence, you always need parentheses.
Here is a glimpse ahead: the scope of a connective is how much of the sentence it governs.
If a connective governs a small part of the sentence, it has narrow scope. If it governs a lot of the sentence it has wide scope.
So this is another way we can put the rule: By default negation has narrow scope, and you must use parentheses if you want it to have wider scope.