On 7/23/11 9:37 AM, Ting Lei wrote:
I know the Reverse Polish is not a couple of hundred years old. I have an impression of reading something about people writing natural deduction systems using only dots in place of parenthesis. And it is said that it was "natural" in those pre-historic times.
The dot notation was used in Principia Mathematica (1910--1913), where differing numbers of dots represented differing levels of grouping (N.B., not quite the same as depth of grouping, i.e. count of parentheses, as indicated by expressions such as p . = . q V r where, because the right dot is required for grouping the disjunct, we must also introduce the left dot in order to ensure that the p, the equality symbol, and the disjunctive expression are all on the same level). The dots were also used to represent logical conjunction. However parentheses, brackets, and braces do show up occasionally without much explanation as to their meaning. Whitehead and Russell claim to have taken the dot notation from Peano. In fact, the dot notation is the origin of the period in the current notation for binding forms like lambda, Pi, Sigma, forall, and exists; as well as, IIRC, the origin of the colon for typing expressions and set comprehensions. The PM notation for quantifications were (x) . P (exists x) . P and more-modern philosophical texts retain this notation sans the period. It is by far the most baffling notation IMO. The other popular notation in modern philosophical texts is to replace the period by bracketing, leading to forall x[P] exists x[P] Note that an apparent variant latter with the forall elided, is actually a notation for expressing an open term P (N.B., not an abstraction) which dates back to Frege. -- Live well, ~wren