ABSTRACT
In view of the foregoing reflections, it is extremely difficult to draw a nonarbitrary line between logic and mathematics. Some feel that this line should be identified with the line between first-order logic and second-order logic; but, as we have just seen, this has the awkward consequence that the notions of validity and implication. Frege, and also Russell and Whitehead, counted not only second-order logic but even higher-order logic as logic; this decision amounts to saying that there is no line "between" mathematics and logic; mathematics is part of logic. If one wishes an "in-between" view, perhaps we should take the one between second- and third-order logic to be the "line" in question. The philosophical questions the authors discuss which affect the philosophy of mathematics as much as the philosophy of logic.
