Hi Dmitriy, "So, is it possible to derive ET from DN in calculus of

constructions? If it is, i'd appreciate not a direct answer, but a hint on how to do this.”

Yes, it is possible and below is a hint (in Coq). You can play this in online Coq compiler [2]. Definition DN_implies_EM : (forall q : Prop, ~~q -> q) -> (forall p : Prop, p \/ ~p). Proof. refine (fun (Ha : forall q : Prop, ~ ~ q -> q) (p : Prop) => Ha (p \/ ~ p) ((fun Hb : ~ (p \/ ~ p) => Hb _ ) : ~ ~ (p \/ ~ p))). Admitted. The remaining goal is Ha : forall q : Prop, ~ ~ q -> q p : Prop Hb : ~ (p \/ ~ p) ============================ p \/ ~ p So can you see the proof now? Spoiler alert: complete proof [1]. Best, Mukesh [1] https://gist.github.com/mukeshtiwari/f3de83335830ce4e287f231499028d6f [2] https://jscoq.github.io/scratchpad.html

On 4 Nov 2024, at 21:24, Albert Y. C. Lai wrote:

Heh, I find it amusing. But I have had prior experience from figuring out a similar proof. I used Pierce's Law (aka call/cc), but it too unfolds into the same story when evaluated.

On 2024-11-02 17:02, Brent Yorgey wrote:

...

You may find this helpful: https://www.cs.cmu.edu/~cmartens/if/dem.html <https://www.cs.cmu.edu/ ~cmartens/if/dem.html> Actually, you probably won't find it helpful, but at least it is amusing. And after you figure out your proof, you can go back and figure out what this story has to do with it. -Brent On Fri, Nov 1, 2024 at 8:45 AM Dmitriy Matrosov mailto:sgf.dma@gmail.com> wrote: Hi. I've reading "Type theory and formal proof, an introduction" book and in chapter 7.4 authors say, that in constructive logic plus either excluded third law (ET) or double negation law (DN) we can derive the other. And then authors derive DN from ET in calculus of constructions. But they didn't say anything (yet?) about the vice versa: deriving ET from DN in calculus of constructions. I've tried to do this, but the best i can come up with is just case analysis: - assume, that 'a' is true, then 'a or not a' is also true (by 'or-intro' rule). - or if after assuming that 'a' is true we can derive bottom, then 'not a' is true (by 'not-intro' rule). Then using the 'or-intro' rule we again end up with 'a or (not a)' being true. - assume 'not a' and if we can derive bottom again, then 'not (not a)' is true. Then by using DN we again end up with 'a' being true. etc. I.e. I may reduce any (not.. (n times) .. not a) into either 'a' or 'not a' by using DN and function composition. Thus, i probable can derive ET from DN using induction, but i can't code neither induction, nor above case analysis in calculus of constructions. So, is it possible to derive ET from DN in calculus of constructions? If it is, i'd appreciate not a direct answer, but a hint on how to do this. And if it is not, well, probably, authors will explain this later in the book and I should just continue reading. Thanks. _______________________________________________ Haskell-Cafe mailing list To (un)subscribe, modify options or view archives go to: http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe Only members subscribed via the mailman list are allowed to post. _______________________________________________ Haskell-Cafe mailing list To (un)subscribe, modify options or view archives go to: http://mail.haskell.org/cgi-bin/mailman/listinfo/haskell-cafe Only members subscribed via the mailman list are allowed to post.

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