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$(p\rightarrow \bot)\rightarrow \bot \vdash p$

I need to prove this using natural deduction.

I tried assuming that $\neg p$ is true, so I can prove $p$ by contradiction. I do not have $\neg p$ defined as $p\rightarrow \bot$.

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  • 2
    $\begingroup$ What have you tried so far? Where do you get stuck? $\endgroup$ – Taroccoesbrocco Feb 11 at 17:46
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    $\begingroup$ i assumed that ~p is true, so i can prove p by contradiction $\endgroup$ – Nick Kakl Feb 11 at 17:49
  • $\begingroup$ What set of rules are you using ? Is $\lnot p$ defined as $p \to \bot$ ? $\endgroup$ – Mauro ALLEGRANZA Feb 11 at 18:01
  • $\begingroup$ no it is not :( $\endgroup$ – Nick Kakl Feb 11 at 18:03
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Using the set of rules of Michael Huth & Mark Ryan, Logic in Computer Science : Modelling and Reasoning about Systems (Cambridge UP, 2004), page 27.

1) $(p→⊥)→⊥$ --- premise

$\quad$2) $\lnot p$ --- assumed [a]

$\qquad$3) $p$ --- assumed [b]

$\qquad$4) $\bot$ --- from 2) and 3) by $\lnot$-elim

$\quad$5) $p \to \bot$ --- from 3) and 4) by $\to$-intro, discharging [b]

$\quad$6) $\bot$ --- from 1) and 5) by $\to$-elim

7) $p$ --- from 2) and 6) by $\lnot \lnot$-elim, discharging [a].

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  • $\begingroup$ Thank you :) :) $\endgroup$ – Nick Kakl Feb 11 at 18:11

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