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Why is it that an assumption need not be discharged for the first implication introduction in natural deduction?

For example,

$$\dfrac{\dfrac{[p]^1}{q\to p}{{\to}\mathrm I}}{p\to(q\to p)}{{\to}\mathrm I^1}$$

Why is it q need not be discharged like p?

Example 1

Example 2

Graham Kemp
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Mordecai
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  • We are allowed but not forced t discharge assumptions, with rules that allow tjhe discharging – Mauro ALLEGRANZA Dec 10 '20 at 09:23
  • @MauroALLEGRANZA meaning that, we discharge if possible but don't if not? – Mordecai Dec 10 '20 at 09:31
  • There are rules that do not allow discharging, like e.g. the rules for $\land$ and $\forall$ and rules that allow it: $\to$-Intro, $\lor$-elim and $\exists$-elim. See Natural Deduction rules – Mauro ALLEGRANZA Dec 10 '20 at 09:33
  • Welcome to the group, Mordecai! If I understand your notation correctly, $p$ is your initial assumption, and you would obtain $p\to (p\to(q\to p))$ when you finally discharged it. Nothing wrong with that. FWIW as a proof, it would seem unfinished to me if you did NOT do this. – Dan Christensen Dec 11 '20 at 02:34
  • I think this question should be reopened. I don't think the supposed duplicate addresses the issue here, i.e. whether or not you must discharge all assumptions to truly finish a proof. If you only want to establish that $P \to (Q\to P)$, see my finished proof there with all assumptions/premises discharged. – Dan Christensen Dec 11 '20 at 03:02
  • The issue is not why an assumption was not discharged, but why it was not raised.$$\dfrac{\dfrac{[p]^2}{q\to p}{{\to}\mathrm I}}{p\to(q\to p)}{{\to}\mathrm I^2}\quad\text{versus}\quad\dfrac{\dfrac{~[q]^1\quad[p]^2~}{q\to p}{{\to}\mathrm I^1}}{p\to(q\to p)}{{\to}\mathrm I^2}$$The rule allows us to discharge any assumptions of the antecedent occurring above the rule-line -- not necessarily all, but as many occurrences as we wish, including none. Since $q$ has was to be immediately discharged without being referenced by any other derivation, it is economical to not raise it at all. – Graham Kemp Feb 11 '21 at 00:38

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