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Leśniewski's protothetic is an extended propositional calculus in which all Boolean functions can be defined. The theory also includes a universal quantifier. In 1922, Alfred Tarski established that by employing functional variables and quantifiers, all the functions of propositional calculus can be defined using equivalence as the sole primitive function.

Contributed by: Izidor Hafner (March 2016)
Open content licensed under CC BY-NC-SA


Tarski proved:



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.


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