Definitions of Two-Placed Connectives in Lesniewski's Protothetic

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 . He used a circle and up to four lines around the circle, which point to those values of arguments of for which the function is true, according to the diagram:

The combinations of the four lines generate a Boolean algebra of 16 elements. As a consequence, let be the negation of ; then the sign for consists of the circle and the four lines that do not belong to the sign of .

The names of the propositional functions are taken from [1, pp. 19–20]. Disjunction and alternation are reversed with respect to modern usage. The protothetic axiom from 1926 was copied from [2, pp. 129].

References

[1] J. J. T. Srzednicki and Z. Stachniak, eds., S. Leśniewski's Lecture Notes in Logic,New York: Springer, 1988.

[2] S. McCall, ed., Polish Logic, 1920–1939, Oxford: Oxford University Press, 1967.