Logic: Same premises and different conclusion

Question

Suppose that there are $n$ premises, say $p_1,\cdots, p_n$.
Then, is it possible to obtain two different conclusion?
That is, Can we obtain two conclusions $q_1$ and $q_2$ using the rule of inferences if we have the same premises $p_1,\cdots, p_n$?

For example, if $p_1 \rightarrow p_2$ and $p_2 \rightarrow p_3$, then the rule of inferences says that $p_1 \rightarrow p_3$ is valid. In this situation, can we get another valid argument?

Answer 1

For example, if $p_1 \rightarrow p_2$ and $p_2 \rightarrow p_3$, then the rule of inferences says that $p_1 \rightarrow p_3$ is valid. In this situation, can we get another valid argument?

Sure. Since $$(A→B) ∧ (B→C) \implies (A∧B → C) ∧ (A→C),$$ the arguments $$A→C\tag{A1}$$ and $$A∧B → C\tag{A2}$$ are both derivable from the premises $$A→B$$ and $$B→C.$$ Furthermore, since $$(A∧B→C) \kern.8em\not\kern-.8em\iff (A→C),$$ arguments $(A1)$ and $(A2)$ are not logically equivalent.