I know there are multiple questions about this, but I still dont get it. $⊢$ is the syntactic consequence. Syntactic in this case would mean that we only care about the form of the sentence and not the meaning.
While $⊨$ is called semantic consequence. Now my question is, how is this a semantic consequence if I've read on multiple books, and it's really stressed out, that we don't have to take in consideration the meaning of the sentence to find a valuation for it but only have to care about the logical form of the sentence? That should be what distinguishes logical consequence from material consequence no?.
The explenation I always find for $A, B⊨C$ is roughly if its impossible (in any world) for $A$ and $B$ to be true and $C$ to be false. While for $A, B⊢C$ if you can derive C with a proof system from A and B. But isn't that exactly what I have to do to show that $A, B⊨C$?
What's "semantic" about $⊨$?
I know that in prepositional and predicate logic they are equivalent (soundness and completeness) and that's why I guess It's confusing but I don't really really understand the difference. Could you please explain me without mentioning models and structures since I've never studied model theory? And I only know basic PL and QL?