 But although Frege distinguished between logical levels, he did not isolate the portion of his quantificational system that ranges only over variables of the first order as a distinct system of logic: nor would it have been natural for him to have done so. In this respect, there is a significant contrast with Peirce. Frege’s project was to show that arithmetic could be grounded in the laws of logic: for him, there was only one logic, and logic necessarily included the logic of higher-order concepts. Peirce, in contrast, rejected the notion of a single, over-arching logic, instead thinking in terms of logics that vary according to the “universe of discourse”. Largely for this reason he came closer in his 1885 paper to isolating the propositional calculus, the “logic of first intention”, and the “logic of second intention” as distinct systems, each worthy of study in its own right: in that regard, he was closer to modern conceptions than was Frege. There is a further and subtler difference. Peirce’s \(\Sigma\) and \(\Pi\) notation for the quantifiers was explicitly conceived of in terms of (possibly infinite) conjunctions and disjunctions of propositions about individuals. This is a highly suggestive conception that is hard to represent in Frege’s system of notation. Löwenheim was to exploit it in his early work in model theory, leading to technical discoveries that were ultimately to call attention to first-order logic. But all this work lay decades in the future, and neither Frege nor Peirce can be credited with a modern understanding of the difference between first-order and higher-order logics.