Prefix Polish (Łukasiewicz) Notation

 

Polish mathematician/logician Jan Łukasiewicz (1878-1956), whose major mathematical work was focused on mathematical logic, developed an alternative system of logical notation in 1920.  Known as Łukasiewicz notation or prefix Polish notation, this system eliminates the need for any groupers when dealing with propositional logic.  Moreover, this system served as the foundation of the recursive stack a last-in, first-out computer memory storage devise that made the digital revolution possible.  In Polish notation the logical operators are placed before the statement letters rather than between them.  In stead of logical  symbols, Polish notation uses capital letters to represent logical operations and lower case letters to represent statements.

 

Standard symbol

Polish Notation

Negation

~

N

Conjunction

K

Disjunction

q

A

Conditional

C

Biconditional

E

Statement Letters

A, B, C.

a, b, c . . .

Groupers

(, [, {

Not Used

One clear advantage of Polish notation is that the main connective in a formula is obvious--is is simple the first connective in the string.

 

Standard symbolization

Polish symbolization

Negation

~P

Np

Conjunction

P Q

Kpq

Disjunction

P Q

Apq

Conditional

P Q

Cpq

Biconditional

P Q

Epq

The basic idea behind Polish notation is elegant:  imagine expressing a formula in a complex English statement identifying all the components of the formula and then symbolizing that expression.  For example, the formula "[P (Q   R)] is a CONDITIONAL whose antecedent is P and whose consequent is the ALTERNATION (disjunction) of Q with R.  This becomes CpAqr.

Eliminating groupers makes it easier to interpret complex statements.  Consider the following examples:

Standard symbolization Polish symbolization
 {(P R)   (P R)}    CKprApr
 {(P ~Q)  (Q S)}    ACpNqAqs
 {~(P R) [P (R  S)]}    ENCprKAprs
 {[~(P (Q R))]   ~[P (R S)]}    CNKpAqrNCpArs
 {~[P (Q R)] [R (P ~S)]}    KNCpAqrErNKpNs
 { P ~(R S) [(P Q) ~[(P R)  S]]}    KCpNArsCKpqNAKprs

WFF's in Polish Notation        Return to Tutorials Index

Historical Note:  Polish notation and its derivative recursive stack memory storage, led to the development of the English Electric multi-programmed KDF9 computer system of 1963, which had two such hardware register stacks. A similar concept underlies the Reverse Polish notation (RPN), also known as postfix notation.  RPN is an arithmetic formula notation, derived from the Łukasiewicz's polish notation by Australian philosopher and computer scientist Charles Hamblin in the mid-1950s, to enable zero-address memory stores.  As a user interface for calculation, the notation was first used in Hewlett-Packard's desktop calculators from the late 1960s and then in the HP-35 handheld scientific calculator launched in 1972. In RPN the operands precede the operator, thus dispensing with the need for parentheses. For example, the expression 3 * ( 4 + 7) would be written as 3 4 7 + *, and done on an RPN calculator as "3", "Enter", "4", "Enter", "7", "+", "*".