Is there a relation between logical consequence, validity (tautology), and formula modeling?

Question

I am new to propositional logic and I have noticed that these 3 things use the same symbol (double turnstile) and I am wondering if they are related to each other somehow.

to say that a formula B is a logical consequence of another formula A we write: A ⊨ B.

also if we want to say that a valuation V models the formula A we write: V ⊨ A.

and if we want to say that the formula A is valid we write: ⊨ A.

so, here are my questions:

1. is there a reason they all use the same symbol?
2. can we define validity and formula modeling in terms of the logical consequence?
3. can a formula be a logical consequence of a valuation?

Answer 1

We start from the satisfaction relation: $V \models A$ means that $A$ is true under valuation $V$.

Logical consequence, $A \models B$, means that, for every valuation $V$, if $V \models A$, then $V \models B$. Since logical consequence is defined in terms of satisfaction, it makes sense to extend the meaning of the double turnstile.

As for validity, $\models A$ means that for every valuation $V$, $V \models A$. Hence, once again, it makes sense for the notation for validity to be based on the one for satisfaction.

For the second question, you could try to take logical consequence as the primitive notion, and define the other two in terms of it, but it goes against the principle that the semantics of sentences are defined in terms of the satisfaction relation.

For the third question, I'd say is a matter of terminology. We say that a formula is satisfied by a valuation, or that a valuation is a model for a formula. Saying that a formula is a logical consequence of a valuation would be another, somewhat awkward way to say the same thing. So, we usually don't.