Polish Notation

Polish notation, also known as Polish prefix notation or simply prefix notation, is a form of notation for logic, arithmetic, and algebra. Its distinguishing feature is that it places operators to the left of their operands. If the arity of the operators is fixed, the result is a syntax lacking parentheses or other brackets that can still be parsed without ambiguity. The Polish logician Jan Łukasiewicz invented this notation around 1920 in order to simplify sentential logic.

The term Polish notation is sometimes taken (as the opposite of infix notation) to also include Polish postfix notation, or Reverse Polish notation, in which the operator is placed after the operands.

When Polish notation is used as a syntax for mathematical expressions by interpreters of programming languages, it is readily parsed into abstract syntax trees and can, in fact, define a one-to-one representation for the same. Because of this, Lisp (see below) and related programming languages define their entire syntax in terms of prefix notation (and others use postfix notation).

Here is a quotation from a paper by Jan Łukasiewicz, Remarks on Nicod's Axiom and on "Generalizing Deduction", page 180.

"I came upon the idea of a parenthesis-free notation in 1924. I used that notation for the first time in my article Łukasiewicz(1), p. 610, footnote."

The reference cited by Jan Łukasiewicz above is apparently a lithographed report in Polish. The referring paper by Łukasiewicz Remarks on Nicod's Axiom and on "Generalizing Deduction" was reviewed by H. A. Pogorzelski in the Journal of Symbolic Logic in 1965.

Alonzo Church mentions this notation in his classic book on mathematical logic as worthy of remark in notational systems even contrasted to Whitehead and Russell's logical notational exposition and work in Principia Mathematica.

In Łukasiewicz 1951 book, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets and that he had employed his notation in his logical papers since 1929. He then goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus.

While no longer used much in logic, Polish notation has since found a place in computer science.