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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.

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

Hint about soundness :

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.

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