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Why can all truth-functional connectives be defined using the Sheffer stroke?
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Plato (center left) and Aristotle, detail from School of Athens, fresco by Raphael, 1508ā11; in the Stanza della Segnatura, the Vatican. Plato is shown pointing to the heavens and the realm of forms, Aristotle to the earth and the realm of things.
In 1913, the American logician Henry M. Sheffer demonstrated that all truth-functional connectives can be defined using a single operator, now known as the Sheffer stroke.1
This is possible because truth-functional connectives are defined by how the truth-value of a complex proposition depends on the truth-values of its components. Since these dependencies can be systematically represented through truth tables, any connectiveāsuch as conjunction, disjunction, or the conditionalācan be expressed as a combination of other connectives. The Sheffer stroke, which functions as a negated conjunction (often referred to as NAND), provides a sufficient logical basis to reconstruct the truth-functional behavior of all other standard connectives.12