A month ago I posted here a quick comment on some remarks of Warren Goldfarb’s about ‘the contemporary conception of logic’, in particular about the special role (supposedly) given to schemata in the definition of some key logical concepts. I’ve since found that David Bostock in his recent book on Russell’s logical atomism takes pretty much the same line about ‘logic as it is now conceived’. They are, in effect, projecting Quine’s somewhat idiosyncratic views onto the wider logical community.
Their remarks have provoked me into dipping into standard mathematical logic texts, from Mendelson onwards. The headline news is that only one such book I looked it — in fact, Mendelson’s — takes the line that Goldfarb and Bostock think characteristic of modern logic.
Which isn’t to deny that there are important differences between Frege’s approach to logic (Goldfarb’s concern) and Russell’s approach (Bostock’s interest), on the one hand, and most contemporary logicians on the other. But it does suggest that whatever is to be put on our side of the contrast, it isn’t a matter of us moderns typically giving schemata a special role in our very definitions of key logical properties like validity.
I have written up some working notes on this, giving some details of what happens in various canonical modern math. logic texts — with more detail on the first few books chronologically (as they happen to cover most of the available options), and then speeding up as my excitement wanes. You can download the notes here.