What is the fastest way for a person to check whether the following propositional logic formula is a logical consequence of the theory?

Question

I have the following propositional logic theory:

T = {$(A\vee B)\wedge\neg C, B\rightarrow (D\vee C), A\leftrightarrow B$}

I have to answer what of the following formulas is a logical consequence of this theory:

1. $\neg B$
2. $\neg C$
3. $C$
4. $A$

I know I can check it by doing semantic trees for every formula. But that takes a long time since I have to do four trees. I cannot allow so much time wasted since it is a question from a test in school and we are limited by time. How would you solve it faster than?

Answer 1

The first gives $\lnot C$ and $A \vee B$. As $A \leftrightarrow B$ you have $A$ and $B$. So the answers are $2,4$ You can also derive $D$, but were not asked for that.

Answer 2

Because $A \iff B$, you can consider instead: $$\{(A\lor A)\land\neg C, A\implies (D\lor C)\}=\{A\land\neg C, A\implies (D\lor C)\}.$$ Now we have both $A$ and $\neg C$ from the first proposition, and $D\lor C$ from the second proposition. Now $\neg C$ implies $D$. Because $A \iff B$, we also have $B$. In summary, $A \land B \land \neg C \land D$, so choices 2 and 4 are correct.

Answer 3

Since you know trees, you can try to do just one tree with only the givens, and see if any of the literals 'fall out'. And in this case, that's exactly what's going to happen: you will end up with one open branch where $A$, $B$, and $D$ are all True, but $C$ is False.

Note that you can always use this method when the possible answers are literals .. if you have more complex statements as possible answers, then you'd be lucky if you do get an open branch with any of those complex statements. But since trees 'break down' statements to their literals, it would actually work really well in this case.

Also note that when you get multiple open branches, then you could still look to see if some literal occurs in all the open branches: if so, then it is forced. Finally, if all branches close, then you know the givens form an inconsistent set, from which any statemwnt logically follows.