How to prove A → N from A and (A ∨ B) → N in FOL

Question

I have the following premises: A and (A ∨ B) → N

How can I prove A → N from this in a formal proof using FOL?

Answer 1

$\def\fitch#1#2{\begin{array}{|l}#1 \\ \hline #2\end{array}}$

$\fitch{ 1. A \quad Premise\\ 2. (A \lor B) \rightarrow N \quad Premise}{ \fitch{ 3. A \quad \ Assume} {4. A \lor B \quad \lor \ Intro \ 3\\ 5. N \quad \rightarrow \ Elim \ 2,4}\\ 6. A \rightarrow N \quad \rightarrow \ Intro \ 3-5}$

Note that this proof never used premise 1, as indeed $(A \lor B) \rightarrow N$ follows from $A \rightarrow n$ alone; it is an example of the valid inference pattern called Strengthening the Antecedent. Are you sure $A$ was an actual premise?

Answer 2

Proof. Assume $A$ and $(A\lor B)\implies N$ by addition it follows that $A\lor B$ is true and by modus ponens we can reason that $N$ must be true consequently $A\implies N$.

$\blacksquare$