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

.