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I'm taking a course from Stanford in Logic. I'm stuck with an exercise where I'm doing some proof. The Fitch system I'm given only allows

  • $ \land $ introduction and elimination
  • $ \lor $ introduction and elimination
  • $ \to $ introduction and elimination
  • $ \equiv $ introduction and elimination
  • $ \lnot$ introduction and
  • $ \lnot \lnot $ elimination

but not

  • $ \bot $ introduction and elimination

I've been struggling to prove the law of excluded middle (``p ∨ ¬p`) within this system. All of the proofs I've seen online make use of $\bot$ elimination to prove it by contradiction:

| ~(p | ~p)  (assumption)
| | p        (assumption)
| | p | ~p   (or introduction)
| | $\bot$
|
| | ~p       (assumption)
| | p | ~p   (or introduction)
| | $\bot$
|
| (p | ~p)   (proof by contradiction)

Since I can't use $\bot$ elimination, I was trying to do this with $ \lnot$ introduction:

| ~(p | ~p)  (assumption)
| | ???
| ~(p | ~p) => x
|
| | ???
| ~(p | ~p) => ~x
| p | ~p     (not introduction)

but I just can't get it. I can't figure out how to get x and ~x.

Is this provable without $\bot$ elimination? If so, what are the steps?

Background:

What I'm ultimately trying to prove is this: Given p ⇒ q, use the Fitch System to prove ¬p ∨ q. I'm able to prove that p ⇒ ¬p ∨ q and that ¬p ⇒ ¬p ∨ q, but I need to be able to plug p ∨ ¬p into the proof to be able to do 'or elimination'.

Tudor Timi
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    @GitGud I fixed it to not go through Coursera, but directly to the Stanford website. – Tudor Timi Dec 11 '16 at 19:21
  • You have to use double negation elim. – Mauro ALLEGRANZA Dec 11 '16 at 19:25
  • I'm confused. What's wrong with the proof that appears after clicking show answer? – Git Gud Dec 11 '16 at 19:25
  • You do not need both the rules for $\bot$ and $\lnot$... If you have $\bot$, then $\lnot A$ is defined as $A \to \bot$ and thus $\lnot$-elim is simply $\to$-elim. – Mauro ALLEGRANZA Dec 11 '16 at 19:34
  • If you do not use $\bot$ then the rules must be $\lnot$-elim : $\phi, \lnot \phi \vdash \psi$ and $\lnot$-intro : if $\Gamma, \phi \vdash \psi \land \lnot \psi$, then $\Gamma \vdash \lnot \phi$. – Mauro ALLEGRANZA Dec 11 '16 at 19:36
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    The basic idea is to prove $\lnot \lnot (P \lor \lnot P)$, then use double elim. In constructive logic $\lnot \lnot X$ basically means $X$ is consistent, instead of $X$ is true, so those type of theorems are much easier to establish. – DanielV Dec 11 '16 at 19:54
  • @GitGud Silly me, I didn't even notice the Show answer button. I'm not using the exact site I linked, but a different one where I have to compute the answer myself. – Tudor Timi Dec 11 '16 at 21:30
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    @TudorTimi I conjectured so. Well, that settles it, then. – Git Gud Dec 11 '16 at 21:57

2 Answers2

9

1) $\lnot (A \lor \lnot A)$ --- assumed [a]

2) $A$ --- assumed [b]

3) $A \lor \lnot A$ --- from 1) by $\lor$-intro

4) $\lnot A$ --- from the contradiction : 2) and 3) by $\lnot$-intro, discharging [b]

5) $A \lor \lnot A$ --- $\lor$-intro

6) $\lnot \lnot (A \lor \lnot A)$ --- from the contradiction : 1) and 5) by $\lnot$-intro, discharging [a]

7) $A \lor \lnot A$ --- from 6) by $\lnot \lnot$-elim.

For the different "flavours" of the negation rules in Natural Deduction you can see :

Your idea regarding the proof : $P \to Q \vdash \lnot P \lor Q$ is correct: you can use the "derived" rule :

$$\dfrac { } {\lnot P \lor P}$$

Alternatively :

0) $A \to B$ --- premise

1) $\lnot A$ --- assumed [a]

2) $\lnot A \lor B$ --- by $\lor$-intro

3) $\lnot (\lnot A \lor B)$ --- assumed [b]

4) $\lnot \lnot A$ --- from 1) with contradiction : 2) and 3) by $\lnot$-intro, discharging [a]

5) $A$ --- from 4) by $\lnot \lnot$-elim

6) $B$ --- from 0) and 5) by $\to$-elim

7) $\lnot A \lor B$ --- $\lor$-intro

8) $\lnot \lnot (\lnot A \lor B)$ --- from 3) with contradiction : 7) and 3) by $\lnot$-intro, discharging [b]

9) $\lnot A \lor B$ --- from 8) by $\lnot \lnot$-elim

Thus, from 0) and 9) :

$A \to B \vdash \lnot A \lor B$.

Mauro ALLEGRANZA
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  • The Fitch system in my course does not introduction differently than how you did it. A→B, A→¬B ⊢ ¬A. – Tudor Timi Dec 11 '16 at 22:12
  • @TudorTimi - perfect. – Mauro ALLEGRANZA Dec 12 '16 at 06:46
  • In your initial method, what made you say that line 6? I think there is an inherent use of LEM in the proof. You used the fact that A and ~A are the only 2 assumptions and that precisely is LEM. – Black Jack 21 Sep 05 '19 at 18:17
  • In the deduction of LEM, on line 3, shouldn't it read "from 2) by ∨-intro". Also on line 4, shouldn't it read "1) and 3) by ¬-intro"? Thanks – bbarker May 12 '22 at 05:32
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After @GitGud's comment about the answer to my initial question being available here after clicking Show answer, I had a look at the proof and I got inspired on how to do this for p | ~p:

 0.                               (no premises)
 1. *    ~(p | ~p)                (assumption)
 2. **     p                      (assumption)
 3. **     p | ~p                 (or introduction 2.)
 4. *    p => p | ~p              (implication introduction 2., 3.)
 5. **     p                      (assumption)
 6. **     ~(p | ~p)              (reiteration 1.)
 7. *    p => ~(p | ~p)           (implication introduction 5., 6.)
 8. *    ~p                       (negation introduction 4., 7.)
 9.    ~(p | ~p) => ~p            (implication introduction 1., 8.)
10. *    ~(p | ~p)                (assumption)
11. **     ~p                     (assumption)
12. **     p | ~p                 (or introduction 11.)
13. *    ~p => p | ~p             (implication introduction 11., 12.)
14. **     ~p                     (assumption)
15. **     ~(p | ~p)              (reiteration 10.)
16. *    ~p => ~(p | ~p)          (implication introduction 14., 15.)
17. *    ~~p                      (negation introduction 13., 16.)
18.    ~(p | ~p) => ~~p           (implication introduction 10., 17.)
19.    ~~(p | ~p)                 (negation introduction 9., 18.)
20.    p | ~p                     (negation elimination 19.)
Tudor Timi
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