Does empty set entails everything?

Question

I am new to propositional logic, and I am trying to figure out whether the below statement is correct:

[a |= c] or [ b |= c] → [a^b |= c]

First of all, it is obvious that

a^b ⊆ a and a^b ⊆ b.

Moreover, because either a |= c or b|= c are correct(or both), it seems that a^b must also entail c. In other words:

M_{a ^ b} ⊆ M_a . Also, when we say that a |= c,

we have M _a ⊆ M_c.

But what if the intersection of a and b is empty? Does this argument still hold?

Answer 1

$a\land b\vDash c$ means that there are no interpretations(aka models) which value $a\land b$ as true but $c$ as false.

If there are no interpretations which value $a\land b$ as true, then the above holds. (See also vacuous truths)

The intersection of the models $\mathbf M_a$ and $\mathbf M_b$ being empty means there are no models which simultaneously value $a$ and $b$ as true.   Thus $\mathbf M_{a\land b}$ will be empty.   Therefore...