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).<p>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".""<p>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 <a href="http://comments.gmane.org/gmane.comp.lang.concatenative/1915" rel="nofollow">http://comments.gmane.org/gmane.comp.lang.concatenative/1915</a><p>Two combinator basis for Joy: <a href="http://tunes.org/~iepos/joy.html" rel="nofollow">http://tunes.org/~iepos/joy.html</a>