
Gentzen's Rules for Natural Deduction - weinzierl
https://blog.plover.com/math/logic/lk.html
======
user2994cb
The paper referenced also defines Gentzen's sequent calculus, an even nicer
formulation of logic:

[http://www.digizeitschriften.de/download/PPN266833020_0039/P...](http://www.digizeitschriften.de/download/PPN266833020_0039/PPN266833020_0039___log32.pdf#page=18)

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laxd
When I click that link in firefox it starts opening like 20-tabs a second.
Chrome works.

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mjd
I also linked the PDF directly from my blog article, at
[https://pic.blog.plover.com/math/logic/lk/Gentzen1934.pdf](https://pic.blog.plover.com/math/logic/lk/Gentzen1934.pdf)
, partly because I found the Digizeitschriften page annoying.

~~~
user2994cb
Indeed, though that's to the entire volume - I was trying to link directly to
the individual Gentzen papers (though by the fairly slow download maybe they
are being created dynamically).

Thanks for the interesting blog piece though - maybe you could write one about
Gentzen's consistency proof?

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RossBencina
For comparison, here are the natural deduction rules expressed in modern
notation:

[https://leanprover.github.io/logic_and_proof/nd_quickref.htm...](https://leanprover.github.io/logic_and_proof/nd_quickref.html)

~~~
chalst
The most significant difference between Gentzen's original calculus and the
way it is most often given today is that discharge of assumptions are
associated with labels in modern calculus, so each discharged assumption is
tied to a rule, whereas in Gentzen's calculus, all instances of that
proposition are discharged.

The innovation is essential to the Curry-Howard correspondence, because this
labelling of assumptions matches variable bindings in typed lambda calculus.

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dwheeler
Very fun.

It turns out that it is easy to implement a natural deduction system using a
system based on Hilbert's approach. A discussion on this for metamath's set.mm
is here:

[http://us.metamath.org/mpeuni/mmnatded.html#natural-
deductio...](http://us.metamath.org/mpeuni/mmnatded.html#natural-deduction-
metamath)

Many steps are an implication phi → ..., and the antecedent mimics the context
(Γ) of most ND systems.

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hamad
another three good writing from same blog
[https://blog.plover.com/math/logic/coherence-
spaces.html](https://blog.plover.com/math/logic/coherence-spaces.html)
[https://blog.plover.com/math/logic/cut-
rule.html](https://blog.plover.com/math/logic/cut-rule.html)
[https://blog.plover.com/math/IL-
contradiction.html](https://blog.plover.com/math/IL-contradiction.html)

~~~
mjd
Thanks for your kind words.

------
chalst
A quibble:

> Gentzen died at age 35, a casualty of the World War.

makes it sound as if GG was a soldier or something. In fact, he was an
academic and Nazi party member who worked in Czechoslovakia during the
occupation. Wikipedia describes his death so:

> Gentzen was arrested during the citizens uprising against the occupying
> German forces on May 5, 1945. He, along with the rest of the staff of the
> German University in Prague was subsequently handed over to Soviet forces.
> Because of his past association with the SA, NSDAP and NSD Dozentenbund,
> Gentzen was detained in a prison camp, where he died of starvation on August
> 4, 1945.

[https://en.wikipedia.org/wiki/Gerhard_Gentzen](https://en.wikipedia.org/wiki/Gerhard_Gentzen)

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trhway
>>a casualty of the World War.

>makes it sound as if GG was a soldier or something

as we're in thread on logic, it worth noting that your statement is logically
incorrect. Given that the most casualties of the WWII were civilians, saying
that somebody was a casualty of that war means that most probably the person
wasn't a soldier or something.

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a1369209993
Pedanticly, "casualty" does actually mean specifically soldiers (or other
military personel). Civilian deaths are called "murders" (or
"executions"/"resisting capture"/not-at-all if your side is doing it). The
term "civilian casualties" is a euphemism that was made up (IIRC _during_
WWII) to obscure/excuse bombing of civilian targets (especially factories). In
practice, the euphemism has largely won, though.

~~~
mjd
No. See my reply to Charles upthread.

