The truth of a claim can fall from its station,
When this occurs, the true thing’s the negation.
Negation is the simplest operator, because it functions on a single proposition. Today I’m going to cover what it means, some of the symbols for it, as well as the truth table and why truth tables are useful for thinking about operators.
Negation is often symbolized as ~ or ¬, and when accompanying a proposition looks like ~P or ¬P. Its function, as you may have guessed by now, is to negate the proposition. So if P is “The cat is on the mat,” then ~P is “The cat is not on the mat”. The reason why we use an operator, rather than counting the negation of the statement as a separate proposition, is because when we say that the cat is on the mat, we’re saying “It’s true that the cat is on the mat”. When we think that the cat isn’t on the mat, we’re negating it by essentially saying “It is true that it is false that the cat is on the mat.” Both P and ~P have separate conditions under which they’re true, and we can see those in a truth table.
This table outlines all the possible conditions under which ~P can be true, by exploring all possible true/false combinations. These are going to become really important when dealing with operators, so you can look forward to more of them. In the first column, it’s concerned with the truth of P, and shows that P will either be true or it’ll be false. The second column, which is concerned with ~P, shows that if P is true, then ~P has to be false. If the cat is on the mat, then it’s not true that the cat is not on the mat. Similarly, if P is false, then ~P is going to be true. This is also an example of what logicians call the Law of Non-Contradiction, which states that it can’t be true that both P and ~P at the same time. The cat can’t be both entirely on the mat and not on the mat.
So that’s negation, the first and easiest operator, mostly because it only works on one proposition. Next week I’ll talk about conjunction, which is a little trickier, but the truth table will make it clear. If you’ve got any question, leave a comment here, or ask me on twitter at @ConceptCrucible.