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The statement $P\rightarrow Q$ means: if $P$ then $Q$.

p | q | p->q
_____________
T | F |  F
F | F |  T
T | T |  T
F | T |  T

Lets say: if I'm hungry $h$ - I'm eating $e$.

p      | q      | p->q
_______________________
h      | not(e) |  F
not(h) | not(e) |  T
h      | e      |  T
not(h) | e      |  T    // ?????

If I'm not hungry, I'm eating? (This does not make any sense...)

Can you please explain that for me?

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5  
There is an informal and not completely precise meaning to the English phrase "if $\dots$ then $\dots$." That meaning cannot be fully captured by a connective which is truth-functional. To put it another way, the English phrase "A implies B" carries the connotation of a causal connection between A and B. That cannot be fully captured by a truth-functional connective: the informal meaning of A implies B involves more than whether A and/or B are true or not. But the logical connective implies does a not terrible job of translation. –  André Nicolas Apr 21 '12 at 14:54
1  
It means that you can eat when you are not hungry (e.g. some people eat when stressed), but you cannot afford not-eating if hungry (because <put your reason here>, e.g. "for you will die of exhaustion"). –  dtldarek Apr 21 '12 at 15:30
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4 Answers

up vote 5 down vote accepted

Rather than your example about food — which is not very good, there is a lot of people starving and not eating —, let consider a more mathematical one : if $n$ equals 2 then $n$ is even.

I — How to interpret the truth table ?

Fix $n$ an integer. Let $p$ denote the assertion “$n = 2$”, and $q$ the assertion “$n$ is even”. These two assertions can be true or false, depending on $n$. What does mean that the assertion “$p \to q$” is true ? This means precisely :

  • $p$ true and $q$ false is not possible ;
  • $p$ false and $q$ false is a priori possible (e.g. with $n = 3$) ;
  • $p$ true and $q$ true is a priori possible (e.g. with $n = 2$) ;
  • $p$ false and $q$ true is a priori possible (e.g. with $n = 4$).

II — Some common errors

The assertion “($n$ is divided by 2) $\to$ ($n$ is divided by 3)” is definitely not true for all $n$, but it can be true, for example for $n=3$, or $n=6$.

The fact that “false implies true” should not be read as “if not $p$ then $q$”. Indeed “false implies false” also holds, so if $p$ is false then either $q$ is false, either it is true, which is not a big deal.

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The fact that some people starve doesn't mean that the property of not starving is a bad example. (In fact, not being necessarily true arguably makes it a better example of things one can express using logic -- it is vitally important for understanding logic to understand that simply being able to write something down does not make it true, and writing down untruths can sometimes be part of a fruitful argument). –  Henning Makholm Apr 21 '12 at 16:28
    
I agree with you, I just meant that it is a fuzzy example, not suitable for explanations. –  Lierre Apr 21 '12 at 16:47
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The best way I have found of thinking about those lines on the truth table is that if $p$ is false, you can't say anything at all about the truth of $q$. The point of $p\Rightarrow q$ is to get you from the truth of $p$ to the truth of $q$. If $p$ is false, then you're dead in the water; $p\Rightarrow q$ could be true or it could also be false; you can't say anything about it.

Here's my favorite example. Let $P$ be "It's raining" and $Q$ be "It's cloudy." The proposition $P\Rightarrow Q$ is, "If it's raining, then it's cloudy." If it's not raining, then you can't say anything at all about whether or not it's cloudy; it could be not raining and still cloudy, or not raining without a single cloud in sight. Either way, "if it's raining, then it's cloudy" is still true.

In your example, if you're not hungry, you still might be eating. Or maybe you're not eating. If all you know is that you're not hungry, then you don't have any information about whether or not you're eating. You only get information if you know you're hungry (in which case you know you're eating) or if you know you're not eating (in which case you are definitely not hungry).

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The truth table for logical implication can be confusing until you realize that it is really designed to be used together with quantification. The general scheme is something like this:

Imagine that we're attempting to consider a number of different situations at once (since in mathematics we always try to save work by doing the work form many cases only once). The different "situations" may be different points in time, different places, different parameters to a function, different functions or (more vividly but less concretely) different "possible worlds".

What we then often want to do is to express a statement of the form "Every situation that has the property $P$ also has the property $Q$". For example: "Every day when I am hungry is a day when I eat". That statement only claims something about days when I'm in fact hungry -- it might still be the case that there are non-hungry days where I nevertheless eat; that does not contradict my statement about the hungry days.

Now, such a claim "in every situation where $P$ holds, $Q$ holds too" is a bit cumbersome to reason formally about, because then we need to do reasoning that applies to only some situations, and we'd need a mechanism for keeping track of which situations we're considering at a particular point in the argument. The way we handle that is by a trick by which we transform our statement to one that is about every situation. Start with

On every day when I'm hungry I eat.

Trick, part one: Instead of claiming that this is true, claim that it is not false:

It is not the case that I'm sometimes hungry without eating.

Or, explicitly mentioning days again:

There is no day such that on that day I'm hungry and on the same day I don't eat.

Part two: Note that "there is no day such that $X$" is the same as "every day is a not-$X$ day".

On every day it is not true that (I'm hungry and I'm not eating).

Now we've expressed our original claim about some days as a statement about every day, where the thing that must be true on every day is built from "I'm hungry" and "I'm not eating" using AND and NOT connectives, which I assume you're already familiar with. We can do this in general: Instead of "In all situations such that $P$ it also holds that $Q$", we write: "In all situations it holds that $\neg(P\land \neg Q)$". Then we don't need any specific logical machinery to reason about "all situations such that", but can make do with knowledge about AND and NOT.

Now it turns out that this construction is so useful and common that it is very convenient to have an abbreviation for $\neg(P\land \neg Q)$ so we can recognize the standard construction easily without needing to check that all the $\neg$s and $\land$s are in the right place. You know the abbreviation already: it is "$P\to Q$", and you can check that the truth table of "$P\to Q$" is indeed the same as that of $\neg(P\land \neg Q)$.

It only remains to justify the convention that we pronounce $P\to Q$ as "if $P$ then $Q$". (Note that this is really only a pronunciation convention: the meaning of $\to$ is whatever its truth table says it is, regardless of pronunciation). Ultimately, this is just a convention that one has to learn.

This convention does match how "if..then" is sometimes used in everyday English -- for example you can say "if the report is not done by Friday, then I will fire you", and it may happen that the report is done on Friday and you fire the employee anyway (say, because he assaulted a coworker), which doesn't make a lie of your earlier threat to fire if the report didn't get done.

But there are other ways to use "if...then" in Engliosh that do not match the logical semantics of $\to$. That is neither a problem for English nor for logic, unless you fool yourself into thinking that the two ought to be the same.

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If I'm hungry(h), then I'm eating(e)... [your original premise]

If I'm not hungry, then I'm eating ? (does not make any sense...)

You are right. It doesn't make sense, because it is wrong. The truth table tells us that if you are not hungry, then your original premise is true whether or you are eating not.

In mathematics, where there is no notion of causality, I find it is helpful to simply define $P\rightarrow Q$ as $\neg (P \wedge \neg Q)$. Then, if $\neg P$ is true, it is easy to see that $\neg (P \wedge \neg Q)$ is also true, whatever $Q$ may be.

Using this definition, your example would translate: I cannot be both hungry and not eating (however unrealistic that may be). If you are not hungry, then it would still be the case that you are not both hungry and not eating.

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