Is entailment biconditional or conditional? When we say a KB entails Q it means that it is never the case that KB is true and Q is false. Does this mean entailment is similar to the conditional statement KB -> Q? I'm confused because our textbook keeps using "if and only if".
 A: Entailment is closest to the material conditional: To say "$KB$ implies $Q,$" or "If $KB$, then $Q$," is to assert $$KB \rightarrow Q$$
To say "$KB \;\text{ if and only if }  \;Q$" is to say: $$KB \rightarrow Q\quad\text{ AND }\quad Q \rightarrow KB$$
In this case, in order for the biconditional to be true, $KB$ and $Q$ must both be true, or must both be false. If their truth-values differ, then exactly one or the other of the two conditionals must be false, and hence, the biconditional is false.
A: Entailment doesn't actually refer to the conditional nor does it refer to the biconditional.  Entailment often refers to semantic consequence.  In symbols, we can write KB entailing Q as
A1: KB |= Q.
The conditional (KB→Q) qualifying as true doesn't refer to semantic consequence in the same way.  If we put (KB→Q) qualifying as true in the same symbolism, we can write
A2: |= (KB→Q).
A2 has a truth-functional connective.  A1 doesn't have such a connective.  In many logical systems A1 and A2 are closely related, because of the rule of detachment (or "modus ponens"), and a meta-theorem which says that if A1 holds, then A2 holds also (a semantic version of a deduction theorem).  So, entailment does often come as related, in some sense, to the conditional.  It is not similarly related to the biconditional.
