Natural Deduction and Lambda Calculus

Asked Sep 03 '24 at 13:23

Active Sep 03 '24 at 13:23

Viewed 127 times

Could somebody elaborate in details the meaning of following statement (source ):

[...] under the propositions as types correspondence, Hilbert systems correspond to combinatory logic in the same way that natural deduction/sequent calculus corresponds to lambda-calculus.

Essentially, the question reduces to in which sense in this context natural deduction corresponds/ in which sense related to lambda-calculus?
A naive guess: Can lambda- calculus be regarded as certain model/"realization" of natural deduction system?

(The question is from content related also to this one

asked Sep 03 '24 at 13:23

user267839

9,217

1

See for instance https://www.cse.chalmers.se/~abela/esslli2016/lecture1-nd.pdf, in particular parts 1.5 and 4 (where the formalism used for discharged hypotheses, with explicit variables, makes the link with λ-calculus quite clear). The introduction and elimination rules of → correspond to λ-abstraction and application. – sparusaurata Sep 03 '24 at 13:35
What do you mean by ''being a model/realization of natural deduction''? – Taroccoesbrocco Sep 04 '24 at 02:23
@Taroccoesbrocco: Think it was a totally wrong guess of mine. I thought there of a "model" in this context in sense of "a model of a theory", but I see now that this is nonsensish as a deduction apparatus is only a part of a theory – user267839 Sep 04 '24 at 05:58
@sparusaurata: in this script is used several time the term "discharged hypothesis". What does this mean? – user267839 Sep 04 '24 at 06:04
1

https://math.stackexchange.com/questions/3527285/what-does-discharging-an-assumption-mean-in-natural-deduction – Naïm Camille Favier Sep 04 '24 at 07:54
@NaïmFavier: Great, thanks! – user267839 Sep 04 '24 at 10:26
1

Maybe useful: Philip Wadler, Propositions as types. – Mauro ALLEGRANZA Sep 06 '24 at 07:10

Natural Deduction and Lambda Calculus

0 Answers0