Self-membership,
and self-application are terms
associated with the reasons for the creation of the branch of logic designated
by the expression Combinatory Logic (Bimbo
2016, para. 6).
Self-membership is the
central topic of the Russell’s Paradox, and this paradox is considered
unsolvable by many experts (Sion 2017, p. 130).
Combinatory Logic has
been invented by Shönfinkel, and it was
developed by Curry in the 1920s (Baker 2019, para. 2). Both of them are
classified as mathematician-logician [(Fracademic 2019), (Seldin 2019)].
Moses Schönfinkel Haskell Brooks Curry
(1889-1942) (1900-1982)
Born in Dnipro,
Ukraine Born in
Massachusetts, US
[(Pngkey.com 2018), (Fracademic 2019)] (Seldin 2019)
In
Combinatory Logic, any expression can be combined with any other expression
(Baker 2019, para. 2).
Its special symbols are ‘B’, ‘I’, ‘K’, ‘S’, and ‘W’ [(Seldin 2019, para. 34),
(Baker 2019, para. 2)], and these letters seem to originate in the German language:
Identität/Identity (I), Wiederholen/Repeat (W), Konstanten/Constant (K),
Substitutionsprozesse/Substitution processes (S), and Beherbergen/Take in (B)
[(Curry, H 1930), (Google.com 2019)].
It also uses
parenthesis, and variables, and its variables are single characters, such as ‘x’, and ‘y’ (Baker 2019, para. 2).
I is the identity operator,
and it is defined to be \x x [(Seldin
2019, para. 4), (Pryor 2015, para. 3)].
K is the constancy operator,
and it is defined to be \x y. x: K eliminates its second a
rgument [(Seldin
2019), (Pryor
2015, para. 4)].
S is the distributor, and it
is defined to be \f g x. f x (g x): S copies its third argument, and
distributes it over the arguments that precede it [(Seldin 2019), (Pryor
2015, para. 5)].
B
is the composition, and it is defined to be \f g x. f (g x): B
changes f into a function, and then g, and x into its arguments (Pryor 2015,
para. 8).
W
is defined to be \f x. f x x: W doubles x (Pryor 2015, para. 11).
References
Bimbo,
K 2016, Combinatory Logic, Stanford,
viewed 1 August 2019, <https://plato.stanford.edu/entries/logic-combinatory/#ReduEquaTheiForm>
Barker, C 2019, Combinatory Logic Tutorial, viewed 1
August 2019, <http://www.nyu.edu/projects/barker/Lambda/ski.html>
Pryor, J 2015, Week3 Combinatory Logic, Combinators and
Combinatory Logic, viewed 1 August 2019, <http://lambda.jimpryor.net/topics/week3_combinatory_logic/>
Pngkey.com 2018, File – Schonfinkel – Moses Schonfinkel,
viewed 1 August 2019, < https://www.pngkey.com/detail/u2e6a9t4w7y3q8u2_file-schonfinkel-moses-schnfinkel/>
Fracademic 2019, Moses Schonfinkel, viewed 1 August 2019,
<https://fracademic.com/dic.nsf/frwiki/1190889>
Sion, A 2017, Paradoxes
and Their Resolutions: A Thematic Compilation, self-published, Geneva,
viewed 1 August 2019, <https://books.google.com.au/books?id=nqg-DwAAQBAJ&pg=PA130&lpg=PA130&dq=paradox+self-membership&source=bl&ots=6ea0W6hs7v&sig=ACfU3U1IqRx8Hf5AkwDLXbVexvTOvM9gNQ&hl=en&sa=X&ved=2ahUKEwibz9bK6eDjAhWUWX0KHbZJBa8Q6AEwC3oECAgQAQ#v=onepage&q=paradox%20self-membership&f=false>
