Natural Deduction proof - Is it correct?

Question

Let me know if this proof is correct. This is in french.

Translation French --> English

prémisse = premise

supposition = assumption

I now know that this is 100% incorrect. Does anyone know how to resolve it mathematically?

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Edit 1 : Thanks to @Mauro Curto here is what I came up with

Answer 1

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

Not quite.

Conditional Introduction deduces a conditional statement whose antecedent is the assumption of a subproof and whose consequent is the conclusion. Thus when you assume $p$ with the aim of deriving $r$, you are intending to introduce $p\to r$ outside the context of that subproof.

Conversely, Conditional Elimination requires a conditional statement and its antecedent to both be accessible in the context where you intend to derive the consequent. Thus all such eliminations using the assumed conditionals of $p\to \textsf{whatever}$ must take place inside a context where $p$ is available.

That is in the innermost nest of the suproofs, which derives $q$ and $q\to r$, and from these another Conditional Elimination derives $r$. And so...

$$\fitch{~~1.~p\to(q\to r)}{\fitch{~~2.~p\to q}{\fitch{~~3.~p}{~~4.~q\\~~5.~q\to r\\~~6.~r}\\~~7.~p\to r}\\~~8.~(p\to q)\to(p\to r)}$$

Answer 2

No, not correct.

Line 5 cannot refer to line 3, since that is part of a subproof that was closed right after line 3

Line 6, 7, and 8 are all not correct: you introduce a conditional by closing a subproof, inferring a conditional with as its antecednt the assumption of that subproof, and as its conseuqent the last line of that subproof (or at the very least, a line contained within that subproof). On lines 6 and 7 you are not closing any subproofs, and line 8 is not referring to the assumption of the subproof you just closed

To be harsh this but frank: it's a mess! Please study the rule for conditional introduction more carefully ... not just syntactically, but also conceptually and indeed purely logically. Nothing you do here makes any logical sense if you step back from the strings of symbols and think about what you do here.

Answer 3

Your derivation is incorrect. Your supposition in step 2 is not the adequate supposition. After you colse a supposition you can't use the supposition anymore so your step 7 is wrong, nad therefore the step 8 is also wrong.

Since the main connective of the consequent is a conditional you have to use the "practical rule" of the introduction of the conditional wich states that you have to assume the antecedent and then try to derivate the consequent. You have to use this practical rule twice in this derivation.

A derivations looks like:

$1). p → (q → r) - premise$

$2). p → q - supposition$

$3). p - supposition$

$4)....$

So now you close supposition 3 after you derivate $r$, and then you close supposition 2 after you derivate $p → r$