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So, in all the books on propositional logic, I feel unsatisfied with the "intuition" about the meaning of the implication connective. I completely understand how the mechanics work via truth tables, but whenever an anecdote is given in natural language, I completely do not follow.

For example, in my current book, they give the atomic formulae:

p = the moon is red
q = the moon is made of cheese

And the compound formula:

p => not q

And they state,

obviously p => not q is true.

This is not obvious to me. I understand that the following truth valuations of p and q induce the implication to be true:

p q not q
---------
1 0  1
0 1  0
0 0  1

So, the author thinks that one or more of the above cases is obvious? Can someone walk me through this?

The author then changes p and q for the statements:

p = My telephone is ringing
q = Someone is calling me

The author then claims

p => not q is obviously false

However I can think of many cases of why the telephone is ringing (p=1) and someone is not calling me (q=0). E.g. I'm fixing a phone, or I'm setting a ring-tone, etc.

Can someone help me understand this "obviousness"?

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See this question: math.stackexchange.com/questions/232309/… –  Francis Adams Jun 28 '13 at 14:20
2  
"In all the books on propositional logic". All??? Some are rather more careful about this stuff. See, for a random example, P*t*r Sm*th's Introduction to Formal Logic (CUP). –  Peter Smith Jun 28 '13 at 14:45

5 Answers 5

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?

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

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

But if you don't eat your vegetables, I'm not lying no matter which of $q$ or not $q$ I do. 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.

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1  
This is a great way to explain that a false statement implies everything. I will remember it. Thanks ! –  justt Jun 28 '13 at 19:14
2  
Another analogy is: If you don't eat your vegetables, you don't get your desert. If you don't eat the vegetables, I keep the promise and I won't give you the desert. If you do eat them, I don't have to do anything (because the premise is false). Neither way I'm obliged to give you the desert. –  Petr Pudlák Jun 29 '13 at 14:02

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.

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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.

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

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You can interpret the material conditional in two-valued logic as meaning "it is not the case that both p and not q are true." In such a case, your first example becomes, "it is not the case that the moon is red and the moon is not made of cheese." I think you agree it's obvious that the moon is not red, so (p => not q) is true.

Your second example becomes "it is not the case that my telephone is ringing and someone is not calling me." Alright, so I'll guess that the author got things backwards here. It makes more sense to say "it is not the case that someone is calling me and my telephone is not ringing," because someone calling you (and neither you nor anyone else having picked up the phone yet, and your phone is working as it should) necessitates that your telephone is ringing. Examples often have all sorts of hidden conditions like this. They serve to illustrate how theoretical things work. If they don't help you understand how theoretical things work, look for other examples, or ask for them.

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