What does mean weaker and stronger terms in logic? I want to understand what does mean the terms stronger and weaker which I'm reading in Programming: The Derivation of Algorithms: 
When $[P\Rightarrow Q]$ holds then $P$ is called stronger than $Q$ and $Q$ is called weaker than $P$. For example, $x\geqslant 2$ is stronger than $x\geqslant 1$ and $x^{2}\geqslant 0$ is weaker than $x\geqslant 0$. The weakest predicate is predicate true, since $[P\Rightarrow true]$ for all $P$, and the strongest predicate is predicate false, since $[false \Rightarrow P]$ for all $P$.
I would like to hear examples from daily common language which apply these two terms.
 A: Basically, "stronger" means "tells you more" or "has more implications." If $P\implies Q$ holds, then everything $Q$ tells you, $P$ also tells you.
For instance, "It will rain tomorrow" is a much stronger statement than "It will rain sometime in the future."
Examples of this being used in natural language might be a bit hard to come by. One "near miss" is the phrase "more specific than": saying "it will rain tomorrow" is more specific than saying "it will rain sometime in the future," and frequently "more specific than" is interchangeable with "stronger than."

In general, though, I wouldn't look too much to natural language for explanation of formal concepts. Here's an example of where the phrase "stronger than" can be used in natural language in a very misleading (for this context) way: I've heard people describe "Candidate A will win in a landslide" as a stronger prediction than "Candidate A will narrowly win," despite the fact that formally of course the two statements are contradictory. Why is "stronger" an appropriate term to use in this context, from a natural language perspective? Well, it could be because the former statement is in some sense more "extreme" than the latter; or it could be that each statement can naturally be thought of as a comment on the strength of Candidate A, and the former asserts more strength. It could even be connected to the assumed state of mind of the speaker: the former statement suggests that the speaker has high (=strong) confidence in Candidate A's chances of winning, while the latter suggests less. Regardless, this usage does not line up with the formal one, but is in fact something I've heard used in real life.
Tl;dr - natural language is weird. Don't rely on it too much for intuition.
