Intuition of implication in propositional logic

Solution 1:

The example that was most memorable for me when learning this was the following: let $p$ represent "you eat your vegetables", and $q$ represent "you get your dessert". When we're talking about "if $p$ then $q$", imagine I am your parent and I've stated that. When am I lying (false result)?

If you eat your vegetables (true), I've got to give you your dessert (true). This is fine (true result).

If you eat your vegetables (true), and I don't give you your dessert (false), then I've lied to you (false result. shame on me).

But if you don't eat your vegetables (false), I'm not lying (true result) whether I choose to give you dessert or not (true or false).

Explanation of the last paragraph: If I don't give you your dessert, you can hardly call me unfair. But if I do give you your dessert, perhaps this wasn't because of the vegetable thing. Maybe you did something else, like the washing up, that merited dessert.

Solution 2:

In the first case, the supposedly obvious idea is that cheese isn't red. Hence red objects can't be made of cheese.

In the second case, this isn't handled very well by propositional calculus. $p,q$ need to be true or false, not true some of the time and false some of the time. To handle such situations you need quantifiers, which are probably coming up soon in your text.

$p\rightarrow \neg q$ must be either true or false. If it's true, then whenever your phone rings nobody is calling. If it's false, then if your phone rings people may be calling or not.

Solution 3:

The 'obviousness' that the book is talking about refers to the following interpretation fo the connective $\Rightarrow$:

$$ p\Rightarrow q \textrm{ means `If } p \textrm{ then } q \textrm{'} $$

In most cases, 'If $p$ then $q$' corresponds quite well with the mathematical statement $p\Rightarrow q$. Recall that $p\Rightarrow q$ is false only when $p$ is true and $q$ isn't. If $p$ is true and $q$ isn't, then the statement 'if $p$ then $q$' obviously isn't true. By convention, we say that it is true in all other cases.

\begin{align} p&\textrm{ - The moon is red.}\\ q&\textrm{ - The moon is made of cheese.} \end{align}

Clearly, if the moon is red, then it is not made of cheese. So we say that $p\Rightarrow q$ obviously.

\begin{align} p&\textrm{ - My telephone is ringing.}\\ q&\textrm{ - Someone is calling me.} \end{align}

There are two reasons why the author might have said that $p\Rightarrow\textrm{not }q$ is obviously false. The first, and most likely, is that he (I decided the sex of the author by tossing a coin, by the way) was not being particularly careful about other situations in which your phone might be ringing without somebody calling you. In that case, since the phone ringing means someone is calling you (most of the time) it is false that $p\Rightarrow\textrm{not }q$.

The other reason is that he is not talking about mathematical logic, but about a wider form of logic in which statements do not have a definite truth value. For example, we might know the phone is ringing, but then we do not immediately know whether someone is calling us or not. In that situation, $p\Rightarrow q$ means 'in all possible situations where $p$ is true, $q$ is true. Then $p\Rightarrow\textrm{not }q$ is not true, since it is possible that $p$ is true and $q$ is as well.

Solution 4:

I think most intuitive explanation is as an if-then statement; e.g., If I am hungry then I will eat.