This reminds me of W. V. Quine's commentary on Moses Schönfinkel's landmark 1924 paper "Building Blocks of Logic". Schönfinkel pioneered the reduction to K and S, with parentheses, and further suggested that he could reduce the combinators to just one called J, where S = (J J) and K = (J S).
Then Quine observed that one could use a preponent binary operator "o" to dispense with parentheses, at which point ~"" All Schönfinkel's sentences build of "J" and parentheses go over unambiguously into strings of "J" and "o".""
Therefore all valid forms can be represented as binary numerals as well, though the converse is not true: not all binary numerals represent valid forms -- unlike Iota's bijective mapping.
Comments
This reminds me of W. V. Quine's commentary on Moses Schönfinkel's landmark 1924 paper "Building Blocks of Logic". Schönfinkel pioneered the reduction to K and S, with parentheses, and further suggested that he could reduce the combinators to just one called J, where S = (J J) and K = (J S).
Then Quine observed that one could use a preponent binary operator "o" to dispense with parentheses, at which point ~"" All Schönfinkel's sentences build of "J" and parentheses go over unambiguously into strings of "J" and "o".""
Therefore all valid forms can be represented as binary numerals as well, though the converse is not true: not all binary numerals represent valid forms -- unlike Iota's bijective mapping.
See also: One combinator basis for stack based language http://comments.gmane.org/gmane.comp.lang.concatenative/1915
Two combinator basis for Joy: http://tunes.org/~iepos/joy.html