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.

Most interesting. Patience was not tested.

This inquiry is opening a subject of which I have not noticed reviews. Perhaps it could be called logical hermaneutics to avoid the exhausted prefix meta.

It would be an unfortunate unintended consequence,however, to taint Gadamer and his successors with a lack of rigor.An explicit hermaneutics of logic might be better.

If your inquiry is extended beyond general logic texts to logical systems, a different answer might be found.

The on-going discussion on FOM about first-order logic should be grist for your mill. Here, for example, is what one poster said:

I’m reading your Notes with great interest and pleasure.

Your connection from Quine’s Methods of Logic (1st ed, 1950) to Goldfarb’s Deductive Logic (2003) is illuminating.

The influence of Quine is deep; I think that we must take also into account Quine’s Philosopy of Logic (1970). How much of the “mainstream” of logic textbooks reflects an implicit use of Quine’s philosophy of logic as a “tacit” background ?

Should be interesting trying to find other “philosophy of logic” books that are not consonant with Quine, in order to trace back their possible influence (if any).

What about Putnam ? And the recent book by John Burgess, Philosophical Logic (2009) ?

May we say that the recent book of Richard Kaye, The Mathematics of Logic (2007) is not in the “mainstream” tradition ?

See page 24 : “Formal systems are kinds of mathematical games with strings of symbols and precise rules. They mimic the idea of a ‘proof’. ” and page 63 (propositional logic) : “the statements in our system will be elements of the boolean algebra – or rather terms representing elements in the boolean algebra.”

If I’m not wrong, there a no schemata; logic is about formal systems, i.e.mathematical structure based on expressions.

mauro