# theoretical question regarding deduction and relation between $\vdash$ and $\vDash$

i have a very basic and theoritical question to help me understand the basics of logics and deduction systems. trying to understand the basics between the difference and deduction of $\vdash$ $\vDash$ and the relation and deduction between them.

D is an deduction system of proposition calculus. it's axioms are all the propositions that are not tautologies and it's only deduction rule is: $\frac{a \lor b}{a \land b}$.

1)does it apply for every set of proposition x, so that $x \vdash a \Rightarrow x \vDash a$?

2)does $x \not \vDash a \Rightarrow x \vdash a$ ?

thank you very much for aiding me learn.

You have to prove a sort of soundness theorem for your calculus.

if $\Gamma \vdash a$, the $\Gamma \vDash a$.
Assume that $\Gamma \vdash a$ and consider the ususal cases :
(i) $a \in \Gamma$; then obviously $\Gamma \vDash a$, because every assignment $v$ that satisfies every formula in $\Gamma$ will satisfies also $a$.
(ii) $a$ is an axiom; this means that $a$ is not a tautology. Consider the case $a := p$.
According to the definition of derivation, we have $\{ q \lor r \} \vdash p$, but obviously : $\{ q \lor r \} \nvDash p$.
(iii) $a$ is derived with the inference rule from a previous formula in the derivation.