Math logic - determine whether an inference exist This is the first time I see this kind of question.
Ok, I have:
$\{\neg A  \vee B, B \to C, A \vee C \} \models B \vee C$
I have to determine whether an inference exist or not.
How do I do so? please help.
 A: HINTs:
Note that on the left of the double turnstile $\models$, you have $\lnot A\lor B$, and you have $A\lor C$.
$$\lnot A \lor B \equiv A\rightarrow B\tag{1}$$
$$A \lor C \equiv \lnot A\rightarrow C\tag{2}$$
Now, we know that for any proposition, $A$, we have that $A \lor \lnot A$ is a tautology (always true).
So, given that we have $A\lor \lnot A$, $(1)$, and $(2)$, what can we say about $B\lor C$?
A: You used $\models$, which often, but not always, means semantic entailment.
If that is what is meant, then all we need to do is to show that if the sentences on the left are true (under some assignment of truth values to atomic sentences if you are doing propositional logic), then the sentence on the right is true under the same assignment of truth values to atomic sentences.
Suppose that $\lnot A \lor B$ is true   If $\lnot A$ is true, then $A$ is false. But then since $A\lor C$ is true, we get that $C$ is true. It follows that the sentence $B\lor C$ on the right is true.
If $\lnot A$ is false, then since $\lnot A\lor B$ is true, it follows that $B$ is true. But then $B\lor C$ on the right is true. 
Note that we did not need or us $B\rightarrow C$.
Remark: If the double turnstyle denotes derivability, then the above, in the absence of general theorems, is inadequate. Derivability is always with respect to a specific set of axioms and/or rules of inference.  The axioms and rules of inference used vary quite widely. All the common ones one sees are equivalent. However, if one has to cite specific axioms and rules of inference, I cannot do it without knowing the fine details of the presentation of logic in your course.  
