,

,

each of which can be used in the protothetic as the definition of the conjunction "." by means of the equivalence "≡". Note the use of dots instead of parentheses. Without this usage, the second definition would be:

.

Here the expression

is the universal quantifier. The definitions of True and False can be

and

.

Sobociński discovered that conjunction can be defined by means of equivalence, the universal quantifier, and a variable propositional functor of two arguments.

The first of Sobociński's definitions of conjunction is

.

In more modern notation, the definition could be

.

So

means the universal quantifier for all

and all

. The range of propositional variables is the set

or

. The expression

means for all

, where the range is the set of all 16 Boolean functions of two arguments. Note also that variables in Sobociński's definitions are not separated by commas.

This Demonstration shows all 12 of Sobociński's definitions and an expression that is not a definition of conjunction.

[1] B. Sobociński, "An Investigation of Protothetic,"

*Polish Logic 1920-1939* (S. McCall, ed.), Oxford: Oxford University Press, 1967 pp. 201–206.

[2] J. T. J. Srzdnicki and Z. Stachniak, eds.,

*S. Leśniewski's Lecture Notes in Logic*, Dordrecht: Kluwer Academic Publishers, 1988 pp. 2–28.