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Q1. Can Barber Paradox be proven false in constructive logic?

I am following the lean tutorial by professor Jeremy Avigad et al. One of the exercises in section 4 asks to prove Barber Paradox false. I can prove it using classical logic, but I can't do it using constructive logic.

Specifically, my proof needs classical logic to prove the lemma

lemma contradictionEquivalence (p : Prop) : ¬(p <-> ¬p) := sorry

The proof of the lemma is straightforward. It uses equivalence elimination and excluded middle followed by two short lambda applications.

variable p : Prop

open classical

lemma contradictionEquivalence : ¬(p <-> ¬p) := 
    assume LHS : (p <-> ¬p),
    have Hpnp : (p -> ¬p), from (iff.elim_left LHS),
    have Hnpp : (¬p -> p), from (iff.elim_right LHS),
    have ponp : p ∨ ¬p, from (em p),
    or.elim ponp
        (take Hp : p, Hpnp Hp Hp)
        (take Hnp : ¬p, Hpnp (Hnpp Hnp) (Hnpp Hnp))

With the help of this lemma Barber Paradox can be proven false in a single lambda application.

variables (men : Type) (barber : men) (shaves : men → men → Prop)

variable (H : ∀ x : men, shaves barber x ↔ ¬shaves x x)

theorem barberParadox : false := 
    contradictionEquivalence
        (shaves barber barber)
        (H barber)

If I could prove the lemma contradictionEquivalence in constructive logic, I could answer Q1 in the affirmative.

I suspect that the lemma can be proven in constructive logic, because giving its constructive proof is an exercise in section 3 of the tutorial, but nothing has worked for me so far, so any help would be greatly appreciated.

2 Answers 2

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Assume $p\to\neg p$ and $\neg p \to p$.

  • Asume $p$. Then, since $p\to\neg p$, also $\neg p$. But $p$ and $\neg p$ are a contradiction.
  • Thus, we have proved $\neg p$.
  • Since $\neg p\to p$, also $p$.
  • But $\neg p$ and $p$ are a contradiction.

The assumption $(p\to \neg p)\land(\neg p\to p)$ led to a contradiction, so $\neg((p\to \neg p)\land(\neg p\to p))$.

  • 0
    Ah, I see, it's a double contradiction! That's very helpful, thank you! I'll see how to write it in lean now. – 2017-01-16
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The answer of Henning Makholm is almost perfect, with one exception: according to the tutorial, proof by contradiction by_contradiction can only be used in the classical setting. I have worked around it by using the following lambda application Hpnp Hp Hp. This is a full constructive proof of the lemma:

variable p : Prop

lemma contradictionEquivalence : ¬(p <-> ¬p) := 
    assume LHS : (p <-> ¬p),
    have Hpnp : (p -> ¬p), from (iff.elim_left LHS),
    have Hnpp : (¬p -> p), from (iff.elim_right LHS),
    have Hnp : ¬p, from (assume Hp : p, Hpnp Hp Hp),
    have Hp : p, from Hnpp Hnp,
    show false, from Hnp Hp
  • 1
    [A proof of a negation is not a proof by contradiction](http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/). Henning's proof does not use proof by contradiction. His proof corresponds to the Agda term: example : {P : Set} → ¬ ((P → ¬ P) ∧ (¬ P → P)); example (p-to-not-p , not-p-to-p) = p-to-not-p p p where p = not-p-to-p (λ h → p-to-not-p h h) – 2017-01-16
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    Note that when you read $\lnot p$ as $p \to \bot$ (as you can in intutionistic logic), then Henning's proof works (with slight modifications to the wording), even if you replace $\bot$ by some unspecified $q$. It shows that $((p \to p \to q) \land ((p \to q) \to p)) \to q$ is intuitionistically valid for any $q$ not just $\bot$. – 2017-01-16
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    Unless I'm completely misunderstanding the syntax of your proof assitant, the proof you quote is _exactly_ the one I'm suggesting in my answer. – 2017-01-16
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    Yes it is. As others are pointing out, I just misunderstood your first bullet point. But @RobArthan 's suggestion is very good -- and exactly what I thought was missing from your proof. What's happening there is that we create a lambda expression which takes a term of type p (proof of p), and produces false, i.e. p -> false, i.e. ¬p. But I get it now, so big thanks to both :) – 2017-01-16
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    @RobArthan are you referring to principle of explosion https://en.wikipedia.org/wiki/Principle_of_explosion – 2017-01-16
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    @DerekElkins Thanks, this is a very good read. Frankly, I didn't even know that proving p -> false is called a "proof by negation", but it's good I have a name for it now, it obviously will help me to order things in my head. Unfortunately I don't speak Agda, but I think your (λ h → p-to-not-p h h) is equivalent to my (assume Hp : p, Hpnp Hp Hp), which is the trick I was missing. – 2017-01-16
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    The principle of explosion (or ex falso quodlibet) is a special property of $\bot$ (falsehood). The point of my comment is that this is **not** needed here: $((p \to p \to q) \land ((p \to q) \to p)) \to q$ is intuitionistically valid for any $q$ not just $\bot$: the proof only uses rules about implication and conjunction. – 2017-01-16
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    That's a very neat observation actually. And this snippet does typeset correctly `lemma robsLemma : ((p → (p → q)) ∧ ((p → q) → p)) → q := assume LHS : (p → p → q) ∧ ((p → q) → p), have L : (p → p → q), from (and.elim_left LHS), have R : (p → q) → p, from (and.elim_right LHS), have piq : (p → q), from (assume Hp : p, L Hp Hp), have Hp : p, from (R piq), show q, from L Hp Hp` – 2017-01-16