Peano's backward C. AI

Grok

1 Peano used a C for implies, or ph a backward C, which Russell adopted. What word did Peano's C originally represent?

Peano's original symbol for implication was a backward C (Ɔ), which represented consequens (Latin for "consequence" or "that which follows"). In his notation, p Ɔ q meant "p is a consequence of q" (or equivalently, "q deduces p" or "p is deduced from q," aligning with the deductive sense where the consequent follows from the antecedent).

He used a normal C for the reverse relation in set inclusion ("is contained in"), drawing from earlier sources like Gergonne's "C" for "contient" (French for "contains"). The backward C (Ɔ) inverted it for the converse, dual-purposed for deduction/consequence in propositions.

Russell adopted and stylized this as the horseshoe ⊃ in Principia Mathematica, where p ⊃ q means "p implies q" (with the antecedent on the left implying the consequent on the right). This flipped the directional reading from Peano's original but preserved the symbol's origin tied to consequence/deduction.

Sources confirming the Latin root include historical analyses of Peano's works (e.g., his 1889–1908 notations and Formulario Mathematico), where the symbol is explicitly linked to "consequens" or "consequential."