I'm currently solving some of the assignments from my Mathematical Logic course and I'm solving some exercises of the form $T,A \vDash B$ and I'm wondering how having an Unsatisfiable statement A would affect my reasoning.
For example: Prove if the following is true or false.
$A,B \vDash C \iff A \vDash (B \to C) $
$(\Rightarrow)$ This direction is trivial.
$(\Leftarrow)$ Let $\alpha$ be the truth function. if $\alpha(A)=1$ then $\alpha(B\to C)=1$ Then it follows that:
- $\alpha(B)=0$
- $\alpha(B)=\alpha(C)=1$
So B could be Unsatisfiable and $A \vDash (B \to C)$ would be true even in the event that $\alpha(C)=0$. However, how does the case of B being unsatisfiable affect my reasoning for the rest of the proof? Since it is in the premises of $A,B \vDash C$ do I treat B like it could be satisfied? Would this fall in case 2 and make me assume that $\alpha(C)=1$?
Simply put, should I care about a premise being unsatisfiable? I know that when it comes to deductions there's a distinction between valid and sound arguments, do I make a similar distinction here?
No you don’t need to care about when $B$ is not satisfied $A,B\models C$ means that for every interpretation $\phi$, if $\phi(A)=1$ and $\phi(B)=1$ then $\phi(C)=1$. For any interpretation if $B$ is false then the statement is vacously true.