How can I prove that $⊢ (F→ ¬F)→ ¬F$
If it is known that
$(A1):F→(G→F);$
$():(F→(G→H))→((F→G)→(F→H));$
$():(¬G→¬F)→((¬G→F)→G).$
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I assume your only rule of inference is Modus Ponens? Can you use the Deduction Theorem? What did you try? – Bram28 Nov 12 '17 at 18:46
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it is possible to use a deduction theorem – Student Nov 12 '17 at 18:49
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I have tried to prove it same way as F -> F – Student Nov 12 '17 at 18:50
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Well, it's going to be a good bit more complicated than $F \rightarrow F$ ... but the Deduction Theorem will be a life saver! .... OK, start with trying to prove $\neg \neg F \vdash \neg F \rightarrow G$ – Bram28 Nov 12 '17 at 18:54
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what will it give if I don't have G? – Student Nov 12 '17 at 19:00
1 Answers
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Hint
We need some prliminary results:
A) $\vdash \lnot \lnot F \to F$
B) $\vdash F \to F$
C) $A \to B, B \to C \vdash A \to C$.
Now for the proof:
1) $F \to \lnot F \vdash \lnot \lnot F \to \lnot F$ --- using A) and C) above
2) $\vdash (¬¬F → ¬F) → ((¬¬ F → F) → ¬F)$ --- axiom (A3)
3) $F \to \lnot F \vdash (¬¬ F → F) → ¬F$ --- from 1) and 2) by MP
4) $F \to \lnot F \vdash ¬F$ --- from A) and 3) by MP
5) $\vdash (F \to \lnot F) \to ¬F$ --- from 4) by DT.
Note: for the proof of some of the results above, you can see here, as well as in many posts in this site.
Mauro ALLEGRANZA
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