I confess, I am in a state of total confusion right now. And I am still struggling to grasp the underlying distinction between the normal material implication, $\Rightarrow$, and the notion of semantic consequence, $\models$.
I've seen stuff like $(P \wedge \neg P) \models Q$ appear in a book I'm currently reading (A First Course in Logic, by Shawn Hedman), but at the end of the day, it had me wondering what the difference is between material implication and semantic consequence. I mean, really.
We all know that the sentence $(P \wedge \neg P) \Rightarrow Q$ is a tautology. And in a material sense (only thinking in terms of truth values and not in terms of how the two formulas P and Q are connected semantically), this is totally acceptable to me. But to write stuff like $(P \wedge \neg P) \models Q$ blows my entire understanding of the difference between $\Rightarrow$ and $\models$ right out the window. I thought the double turnstile symbol is only to be used in a sense that has more to do with the underlying meaning/interpretation behind the sentences? Like, if we say $P \models Q$ (where P and Q are sentences), then wouldn't it mean that we can see a clear logical connection which allows us to accept that Q follows from P?
If translated into words, the sentence "formula Q is a semantic consequence of $P \wedge \neg P$" is incomprehensible to me. It doesn't have that logical connection that I thought must accompany every use of the double turnstile symbol (for instance, I can easily accept $(P \wedge Q) \models P$, because the logical connection is there, after all if its stated that P and Q are both true, then logically, P must be true). If no logical connection is required for the use of $\models$, then how is it different from the normal material implication?
It feels like I'm missing something crucial, like a bigger picture, or a wider generalization of what the double turnstile symbol represents.