Polish notation

 ( Mathematical Notation ) 

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 Łukasiewicz, i.e., Łukasiewicz (1), is apparently a lithographed report in Polish language. The referring paper by Łukasiewicz was reviewed by Henry A. Pogorzelski in the Journal of Symbolic Logic in 1965. Heinrich Behmann, editor in 1924 of the article of Moses Schönfinkel, already had the idea of eliminating parentheses in logic formulas. In one of his papers Łukasiewicz stated that his notation is the most compact and the first linearly written parentheses-free notation, but not the first one as Gottlob Frege proposed his parentheses-free Begriffsschrift notation in 1879 already.

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

In Łukasiewicz's 1951 book, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, he mentions that the principle of his notation was to write the Function symbol 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.

can be written in Polish notation as

Assuming a given arity of all involved operators (here the "−" denotes the binary operation of subtraction, not the unary function of sign-change), any well-formed prefix representation is unambiguous, and brackets within the prefix expression are unnecessary. As such, the above expression can be further simplified to

The processing of the product is deferred until its two operands are available (i.e., 5 minus 6, and 7). As with any notation, the innermost expressions are evaluated first, but in Polish notation this "innermost-ness" can be conveyed by the sequence of operators and operands rather than by bracketing.

In the conventional infix notation, parentheses are required to override the standard precedence rules, since, referring to the above example, moving them

or removing them

changes the meaning and the result of the expression. This version is written in Polish notation as

When dealing with non-commutative operations, like division or subtraction, it is necessary to coordinate the sequential arrangement of the operands with the definition of how the operator takes its arguments, i.e., from left to right. For example, , with 10 to the left of 5, has the meaning of 10 ÷ 5 (read as "divide 10 by 5"), or , with 7 left to 6, has the meaning of 7 − 6 (read as "subtract from 7 the operand 6").

The above sketched stack manipulation works—with mirrored input—also for expressions in reverse Polish notation.

The quantifiers ranged over propositional values in Łukasiewicz's work on many-valued logics.

Bocheński introduced a system of Polish notation that names all 16 binary connectives of classical propositional logic. For classical propositional logic, it is a compatible extension of the notation of Łukasiewicz. But the notations are incompatible in the sense that Bocheński uses $L$ and $M$ (for nonimplication and converse nonimplication) in propositional logic and Łukasiewicz uses $L$ and $M$ in modal logic.