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.
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 thereof 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
changes the meaning and the result of the expression. This version is written in Polish notation as
The above sketched stack manipulation works—with mirrored input—also for expressions in reverse Polish notation.
Note that the quantifiers ranged over propositional values in Łukasiewicz's work on many-valued logics.
The number of return values of an expression equals the difference between the number of operands in an expression and the total arity of the operators minus the total number of return values of the operators.