Help with understanding logical consequence. We've been tasked with figuring out whether Q is a logical consequence  of these three statements 
P ⇒ Q and ¬Q ⇒ R and P∨ ¬R
Can anyone shed some light on what exactly is meant by logical consequence?
 A: See Logical consequence for the definition.
Regarding propositional logic, the definition amounts to :

"every valuation (or tuth assignment) that causes all the premises to be true also assign true to the conclusion". 

Equivalently : 

"there is no tuth assignment that causes all the premises to be true and the conclusion to be false".

We can use the second definition reasoning by contradiction, i.e. trying to find a truth assignment $v$ for the propositional variables such that the purported conclusion ($Q$) is FALSE and the premise are all TRUE.
Thus, we start with $v(Q)=\text F$.
In order to satisfy the first premise : $P \to Q$, having $v(Q)=\text F$, we are forced to set $v(P)=\text F$.
In order to satisfy the third premise : $P \lor ¬R$, having $v(P)=\text F$, we are forced to set $v(R)=\text F$.
Now, we have $v(Q)=\text F$ and $v(R)=\text F$, that implies : $v(¬Q \to R)= v(\text T \to \text F)=\text F$.
Conclusion : we have shown that it is impossible to find a tuth assignment that causes all the premises to be true and the conclusion to be false.
