Logical equivalence: De Morgan’s law

Let’s look at an example of logical equivalence. The example we’ll look at here has a name. It’s called De Morgan’s Law, after a famous logician, Augustus de Morgan. De Morgan’s Law says that ‘(P and Q)’ is logically equivalent to ‘not (not P or not Q)’. If it’s logically equivalent, then it should be that ‘(P and Q)’ entails ‘not (not P or not Q)’ and that ‘not (not P or not Q) entails ‘(P and Q)’. Let’s look at this using a truth table. We start by setting out the four ways things could be with the truth values of ‘P’ and ‘Q’.

Then we put: ‘(P and Q)’. The truth

values for that are: true, false, false, false. And now we going to have to do this, [ie. ‘not(not-P or not-Q)] which is a little more complicated. Let’s start with ‘not P’. ‘Not P’ is false, false, true, true. ‘Not Q’ is false, true, false, true. OK. Now,

not-P or not-Q: False and false [gives] false. False and true [gives] true. True and false [gives] true. True and true [gives] true. And now the last operator, the tilde there. If we have false for the plugged-in clause, we’ll get true under the tilde. If we have true for the clause, we’d get false under the tilde. So, that’s the truth table for this compound sentence. And look, ‘(P and Q)’ on the first line of the truth table is true, and so is our complex sentence. On the second row, they are both false. Third row, both false. On the final row, both false.

So you can see the ‘(P and Q)’ is true in exactly the same circumstances as ‘not (not P or not Q)’ and that means they are logically equivalent. Now there are several reforms of the De Morgan’s Law, and you can test some for yourself. One form says ‘(P or Q)’ is equivalent to ‘not (not P and not Q)’. You can see the pattern here. De Morgan’s Law says take a conjunction or a disjunction, change it to the other.

(If it’s a conjunction change it to a disjunction if it’s a disjunction, change it to a conjunction.) Put a tilde in front of the sentences either side, and then put a tilde in front of the whole clause, and you get something logically equivalent.