The meaning of implication in logic

How to remember implication logic by remembering a simple english.

I read some sentence like

if P,then Q
P only if Q
Q if P


But i am unable to correlate these sentences with the following logic. Although truth table is very simple but i don't want to just remember it without it's actual meaning.

P Q P=>Q
0 0 1
0 1 1
1 0 0
1 1 1

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It may help to read the implication "$P \implies Q$" as "buy $P$, get $Q$ free (whether you want it or not!)". Buying $P$ means that you also get $Q$, so you have both; not-buying $P$ doesn't rule out the possibility of getting (or not-getting) $Q$ by other means; however, not-getting $Q$ does rule out having bought $P$. –  Blue Jan 20 '12 at 6:04
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6 Answers

If you start out with a false premise, then, as far as implication is concerned, you are free to conclude anything. (This corresponds to the fact that, when $P$ is false, the implication $P \rightarrow Q$ is true no matter what $Q$ is.)

If you start out with a true premise, then the implication should be true only when the conclusion is also true. (This corresponds to the fact that, when $P$ is true, the truth of the implication is the same as the truth of $Q$.)

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Here's your truth table:

\begin{align}\mathbf P & \mathbf Q & \mathbf{P \implies Q} \\ 1 & 1&1 \\ 1&0&0 \\ 0&1&1 \\ 0&0&1\end{align}

$1$ means true and $0$ means false.

What does logical implication mean? "If $\Phi$ then $\Psi$" can be written as $\mathbf{\Phi \Rightarrow \Psi}$. Albeit, it's much more defined than real life. Remember that Mathematical thinking is different than general thinking.

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Remember that "implies" is equivalent to "subset of". It works in exactly the same way: "if an element is in the subset (e.g A), it MUST also be in the superset (e.g. B)". By definition, it is impossible that an element is in the subset, but not in the superset. That's the P=1, Q=0; P=>Q = 0 case. In fact, "A ⊆ B" means that a ∈ A implies that a ∈ B. If a is not in subset A then you can't draw any conclusions on whether a is in the superset B. That's how I keep remembering it.

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Not really as an answer but as an anecdote I'll sketch the following real life situation (in abstract terms to avoid controversy).

A politician $s$ declares, in a menacing voice: "if we would do $P$ then $Q$ will ensue!" where $P$ is something like electing his adversary, or not adopting the Draconic measures he proposes, and $Q$ are catastrophic events like people losing their jobs and the country plunging into a deep crisis. Now suppose $s$ is lucky and manages to avoid $P$, but that then $Q$ happens anyway. Now does this show that $s$ lied? Since $P$ is false and $Q$ is true, we are in the second line of your table, you can read off that $P\Rightarrow Q$ is deemed true in this case. In fact since $s$ prophesized about a circumstance $P$ that did not happen, later events could not have shown him a liar either way. An this in spite of the fact that by common sense the statement he made was either false (if doing $P$ would actually have prevented $Q$) or irrelevant (if $Q$ would have happened independently of $P$, or depending on other conditions than those of $P$).

You see how smart politicians are? (Of course $s$ can be shown to be a liar if he does not manage to avoid $P$, but then being out of office anyway, $s$ probably won't care much about being proven a liar as well.)

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This anecdote leads to the answer; or at least the explanation of why the truth table for "If P then Q" doesn't quite match what some people expect. In everyday situations, when we say "If ... then ...", this actually implies some degree of causation; that is, that P somehow causes Q, or makes Q more likely. Of course, propositional logic doesn't express causation or likelihood; so the propositional logic meaning of "If P then Q" differs ever-so-slightly from the everyday meaning. –  user22805 Jan 19 '12 at 9:21
Thanks Marc for nice explanation. –  P K Jan 19 '12 at 17:54
thanks Marc! This really helped me understand implication :) –  ambertch Aug 5 '12 at 23:25
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"if P then Q" is equivalent to P=>Q
"P only if Q" is also equivalent to P=>Q (example here)
"Q if P" is same as "if P then Q" and equivalent to P=>Q

However, note (following statement which is not given in the original question): P if Q is equivalent to "if q then p" or Q=>P

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linked example makes sense to me. thanks. –  P K Jan 18 '12 at 23:25
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In your truth table, look only at the lines where $P\implies Q$ holds (is $1$), i.e., drop the third line. In the remaining lines, for each line where $P$ holds (i.e., the last one) $Q$ holds as well. Moreover, the column $P\implies Q$ has $1$s in all lines possible where this property holds (i.e., if we made its unique entry $0$ in the third line a $1$ as well, the property would fail).

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