The meaning of Implication in Logic How do I remember Implication Logic $(P \to Q)$ in 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.
Even though the truth table is very simple, I don't want to remember it without knowing its actual meaning.
$$\begin{array}{ |c | c || c | }
  \hline
  P & Q & P\Rightarrow Q \\ \hline
  \text T & \text T & \text T \\
  \text T & \text F & \text F \\
  \text F & \text T & \text T \\
  \text F & \text F & \text T \\ \hline
\end{array}$$

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.     
