I'm having trouble solving this problem using the Fitch system. As I understand Fitch, if the goal has the form $(φ \implies ψ)$, it is often good to assume $φ$ and prove $ψ$ and then use Implication Introduction to derive the goal. Since there are no premises, to prove $(p \implies (q \implies r)) \implies ((p \implies q) \implies (p \implies r))$ with the Fitch system, I'll need to assume the antecedent $(p \implies (q \implies r))$ and use Implication introduction to derive the consequent $((p \implies q) \implies (p \implies r))$. I'm stuck though as using Stanford's Fitch system, assuming $(p \implies (q \implies r))$ results in $p(q)$ and assuming $p \implies q \implies r$ results in $p$. How should I think about this problem so I can resolve the proof?
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Of note: part of my problem was using the symbols (p ⇒ (q ⇒ r)) in the Stanford Fitch system engine and not => – MmmHmm Apr 24 '17 at 19:13
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When I click on your link I get a "Show Answer" button. Don't you? – Pierre-Yves Gaillard Apr 28 '17 at 11:02
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1) $p \to (q \to r)$ --- assumed [a]
2) $p \to q$ --- assumed [b]
3) $p$ --- assumed [c]
4) $q$ --- from 3) and 2) by $\to$-elim
5) $q \to r$ --- from 3) and 1) by $\to$-elim
6) $r$ --- from 4) and 5) by $\to$-elmi
7) $p \to r$ --- from 3) and 6) by $\to$-intro, discharging [b]
8) $(p \to q) \to (p \to r)$ --- from 2) and 7) by $\to$-intro, discharging [b]
9) $(p \to (q \to r)) \to [(p \to q) \to (p \to r)]$ --- from 1) and 8) by $\to$-intro, discharging [a].
As you can see, we have to assume all the antecedents of the conditionals involved.
Mauro ALLEGRANZA
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This helped a lot: "we have to assume all the antecedents of the conditionals involved" especially as it was not covered here: http://intrologic.stanford.edu/notes/chapter_04.html – MmmHmm Apr 24 '17 at 18:53
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1@Mr.Kennedy - be aware, please ... that is not a "formal rule". It works here because we have only conditionals. But in general, it is a good approach whenever we want to prove something of the form: $\varphi \to \psi$. – Mauro ALLEGRANZA Apr 24 '17 at 19:04
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Also very helpful. Can you recommend any resources for better understanding Fitch system? I am struggling with the Stanford online course as the reading material is... less than thorough. – MmmHmm Apr 24 '17 at 19:09
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1@Mr.Kennedy - in general, Chiswell & Hodges is a good textbook based on Natural Deduction. – Mauro ALLEGRANZA Apr 24 '17 at 19:11
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1An you can try with Frederic Brenton Fitch, Symbolic Logic: An Introduction (1952) – Mauro ALLEGRANZA Apr 24 '17 at 19:13
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