Natural Deduction Fitch-style Proof

Question

Find the Fitch Style Proof for the following argument:

$((A \rightarrow B) \rightarrow A) \rightarrow A)$

I am quite unsure on how to apply the famous rules of natural deduction. Would welcome any comprehensive proofs.

Answer 1

Begin at from the ground level

We should prove $((A\to B)\to A)\to A$ by deriving $A$ under assumption of $(A\to B)\to A$ to introduce the conditional . It: a Conditional Proof .

$$\def\fitch#1#2{~~~~\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{}{\fitch{(A\to B)\to A}{~~\vdots\\A}\\((A\to B)\to A)\to A}$$

Now, we might be able to derive $A$ under that assumption if we could derive $A\to B$, but there is no way to do that.   Therefore we can only prove $A$ under that assumption by deriving a contradiction under the further assumption of $\neg A$. Ie: a Proof by Reduction to Absurdity.

$$\fitch{}{\fitch{(A\to B)\to A}{\fitch{\neg A}{~~\vdots\\~\bot}\\\neg\neg A\\A}\\((A\to B)\to A)\to A}$$

Okay, well, that contradiction can only be of one of the assumptions; and as before, we might derive $A$ if we could derive $A\to B$, and now we can do so.

$$\fitch{}{\fitch{(A\to B)\to A}{\fitch{\neg A}{~~\vdots\\A\to B\\A\\~\bot}\\\neg\neg A\\A}\\((A\to B)\to A)\to A}$$

So how might you derive $A\to B$ in the context of those assumptions?