WOLFRAM|DEMONSTRATIONS PROJECT

Definitions of Two-Placed Connectives in Lesniewski's Protothetic

​
show the axiom of protothetic
definitions of boolean functions
algebra of boolean functions
choose boolean function
9
boolean operation/relation
intersection
choose first boolean function
15
choose second boolean function
9
show answer
{True,True}
True
{True,False}
False
{False,True}
False
{False,False}
True
equivalence
[pq]:p
q.≡.p≡q
To describe his extended propositional calculus (protothetic), which is a logical system based on a single axiom, Leśniewski invented a special notation for a propositional function of two arguments
f(p,q)
. He used a circle and up to four lines around the circle, which point to those values of arguments of
f
for which the function is true, according to the diagram:
False,False
True,False
False,True
True,True
The combinations of the four lines generate a Boolean algebra of 16 elements. As a consequence, let
g
be the negation of
f
; then the sign for
g
consists of the circle and the four lines that do not belong to the sign of
f
.