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

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?
• See this related post Commented Sep 25, 2019 at 6:42

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.