# Rewriting a statement in logical form

I am trying to rewrite the following statement using logical symbols: Regular work is not necessary to pass the course. I know how to do this if the "not" wasn't there: P: Regular work was done. Q: The was course was passed. P is necessary for Q, so Q -> P.

However, the inclusion of the "not" is confusing. How can I, in a manner similar to above, write the statement with the "not" included?

## 2 Answers

It's effectively saying that 'it is not true that regular work is necessary to pass the course', and so you just put a negation in front of the whole conditional, i.e. $\neg (Q \rightarrow P)$ which is equivalent to $Q \land \neg P$, which can be read back as saying that you can pass the course without doing regular work.

• Would also be correct to say that the negation is "If the course was passed, then regular work may or may not have been done?" Or does it have to be "the course was passed yet regular work was not done?" – Wesley Jan 26 '18 at 6:14
• @MilesDavis Actually, here you run into a bit of the paradox of material implication: why should the negation of a conditional mean that the antecedent is in fact true and the consequent is false? And yet, that is exactly how the material conditional, as a mathematically defined truth-functional operator, works. But, if you don't like it, then just stick to the negation of the conditioanl and leave it like that. Also, the sentence "if the course was passed then regular work may or may not have been done" would translate into $Q \rightarrow (P \lor \neg P)$ .... which is a tatutology. – Bram28 Jan 26 '18 at 13:41

$P$ : regular work is done.

$Q$ : the course is passed.

The material implication, $( Q\to P)$, that is "regular work is done if the course is passed," is quite reasonably interpreted as stating "regular work is necessary to pass the course".   (Also written as $P\gets Q$ though not often.)

However, saying "not necessary" is problematic in classic predicate logic.   The simple negation, $\neg(Q\to P)$, is (classically) equivalent to $\neg P\wedge Q$, which is "regular work is not done yet the course is passed."   Yet we merely want to assert that this may happen, rather than that it will happen; that it is "possible".

Trying to express a negation duality between "necessarily" and "possibly" is exactly the inspiration for developing "modal logics".

"It is not necessarily so that regular work is done whenever the course is passed", "It is possibly so that regular work is not done yet the course is passed."

$$\neg\Box(Q\to P)\iff \Diamond(\neg P\wedge Q)$$