  In 1873 Peirce published a long paper (completed in 1870) “Description of a Notation for the Logic of Relatives” in which he introduced (six years before Frege’s  Begriffsschrift was published) a complete syntax for the logic of relations of arbitrary adicity (or: arity). In this paper the notion of the variable (though not under the name “variable”) was invented, and Peirce provided devices for negating, for combining relations (basically by building upon de Morgan’s relative product and relative sum), and for quantifying existentially and universally.  By 1883, along with his student O. H. Mitchell, Peirce had developed a full syntax for quantificational logic that was only a very little different (as was mentioned just above) from the standard Russell-Whitehead syntax, which did not appear until 1910 (with no adequate citations of Peirce).