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My textbook says that in the deductive system of natural deduction every hypotesis must be discarded by a rule of inference and after being discarded, it cannot be used again in the deduction... But then it presented me to this derivation for $A \rightarrow (B \rightarrow (A \land B))$:

$[A]^1$ $[B]^2$

________________$(\land I)^2$

$A \land B$

________________$(\rightarrow I)\times2$

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

By $"(\rightarrow I)\times2"$, I'm assuming he firstly did $(\rightarrow I)^2$ and then $(\rightarrow I)^1$.

But that confused me. Shouldn't $B$ be discarded after $(\land I)^2$? Does only the implication insertion rule discard hypothesis?

Jonas
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    When you prove a theorem $T$ under the hypotheses $H$, what you have proved is that "T holds, provided that H is true". To discard the hypotheses or assumption is to move from above result to the new theorem "if H then T". – Mauro ALLEGRANZA Feb 13 '23 at 14:25
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    Wrt to the example above, YES, the assumptions (in square brackets) are discharged by the two applications of $\to$-Intro rule (see the $\times 2$ notation). Thus, you have something like: $((\to \text I) \times 2) ^{[2][1]}$ – Mauro ALLEGRANZA Feb 13 '23 at 14:27
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    See also the post What does discharging an assumption mean in Natural Deduction? – Mauro ALLEGRANZA Feb 13 '23 at 14:30
  • @MauroALLEGRANZA thanks, but why wasn't the assumption B discarded before after the application of $(\land I)^2$? – Jonas Feb 13 '23 at 14:43
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    Maybe we have to review the notation used by your textbook... But the issue is that $(\land \text I)$ does not discharge assumptions. Thus, the sequence must be: $(A \land B)$ by $(\land \text I)$ followed by $B \to (A \land B)$ by $(\to \text I)$ discharging [2], followed by $A \to (B \to (A \land B))$ by $(\to \text I)$ discharging [1]. – Mauro ALLEGRANZA Feb 13 '23 at 14:53
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    You are right, I made a mistake. It doesnt use $(\land I)^2$ but $(\land I)$. Does only the implication introduction discharge assumptions? – Jonas Feb 13 '23 at 15:18
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    @Jonas Yes, only some rules discharge assumptions. And typically, it is treated such that you are allowed to discharge the assumptions, but don't have to if you want to leave them open for some reason. – Natalie Clarius Feb 13 '23 at 15:25
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    No; see Natural Deduction rules: also $(\lnot \text I), (\lor \text E)$ and in Classical Logic also $(\text {RAA})$. – Mauro ALLEGRANZA Feb 13 '23 at 15:28

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