2013

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.

Warren Goldfarb, in his paper ‘Frege’s conception of logic’ in The Cambridge Companion to Frege, announces that his ‘first task is that of delineating the differences between Frege’s conception of logic and the contemporary one’. And it is not a new idea that there are important contrasts to be drawn between Frege’s approach and some modern views of logic. But one thing that immediately catches the eye in Goldfarb’s prospectus is his reference to the contemporary conception of logic. And that should surely give us some pause, even before reading on.

So how does Goldfarb characterize this uniform contemporary conception? It holds, supposedly, that

the subject matter of logic consists of logical properties of sentences and logical relations among sentences. Sentences have such properties and bear such relations to each other by dint of their having the logical forms they do. Hence, logical properties and relations are defined by way of the logical forms; logic deals with what is common to and can be abstracted from different sentences. Logical forms are not mysterious quasi-entities, à la Russell. Rather, they are simply schemata: representations of the composition of the sentences, constructed from the logical signs (quantifiers and truth-functional connectives, in the standard case) using schematic letters of various sorts (predicate, sentence, and function letters). Schemata do not state anything and so are neither true nor false, but they can be interpreted: a universe of discourse is assigned to the quantifiers, predicate letters are replaced by predicates or assigned extensions (of the appropriate arities) over the universe, sentence letters can be replaced by sentences or assigned truth-values. Under interpretation, a schema will receive a truth-value. We may then define: a schema is valid if and only if it is true under every interpretation; one schema implies another, that is, the second schema is a  logical consequence of the first, if and only if every interpretation that makes the first true also makes the second true. A more general notion of logical consequence, between sets of schemata and a schema, may be defined similarly. Finally, we may arrive at the logical properties or relations between sentences thus: a sentence is logically true if and only if it can be schematized by a schema that is valid; one sentence implies another if they can be schematized by schemata the first of which implies the second.

Note an oddity here (something Timothy Smiley has complained about in another context). It is said that a ‘logical form’ just is a schema. So what is it then for a sentence to have a logical form (as you can’t have a schema): presumably it is for the sentence to be an instance of the schema. But the sentence ‘Either grass is green or grass is not green’ — at least once we pre-process it as ‘Grass is green $latex \lor\ \neg$grass is green’ — is an instance of both the schema $latex P \lor \neg P$ and the schema $latex Q \lor \neg Q$. These are two different schemata: but surely no contemporary logician when thinking straight would say that the given sentence, for this reason at any rate, has two different logical forms. So something is amiss.

But let’s hang fire on this point. The more immediate question is: just how widely endorsed is something like Goldfarb’s described conception of logic? For evidence, we can take a look at some well-regarded math. logic textbooks from the modern era, i.e. from the last fifty years — which, I agree, is construing ‘contemporary’ rather generously (but not to Goldfarb’s disadvantage). We’d need to consider e.g. how the various authors regard formal languages, what they take logical relations to hold between, how they regard the letters which appear in logical formulae, what accounts they give of logical laws and logical consequence, and how they regard formal proofs. To be sure, we might expect to find many recurrent themes running through different modern treatments (after all, there is only a limited number of options). But will we find enough commonality to make it appropriate to talk of `the’ contemporary conception of logic?

Of course, I hope it will be agreed that this question is interesting in its own right: I’m really just using Goldfarb as a provocation go on the required trawl through the literature. I’ve picked off my shelves a dozen or so textbooks from Mendelson (1962) to (say) Chiswell and Hodges (2007), and it will be interesting to see how many share the view of logic which Goldfarb describes.

Preliminary report: to my surprise (as it isn’t how I remembered it) Mendelson’s conception of logic does fit Goldfarb’s account very well. At the propositional level, tautologies for Mendelson are a kind of schema (so aren’t true!); logical consequence is defined as holding between schemata; Mendelson’s formal theory is a theory for deriving schemata. Likewise, charitably read, for his treatment of quantificational logic. Moreover Mendelson avoids the unnecessary trouble that Goldfarb gets himself into when he talks of logical form: Mendelson too talks of logical structure, but he supposes that this is ‘made apparent’ by using statement forms, not that it is to be identified with statement forms. So 1/1 for Goldfarb.

So far so good. But chronologically the next book I’ve looked at is ‘little Kleene’ , i.e. Kleene’s Mathematical Logic (1967). And Goldfarb’s account doesn’t apply to this. For a start, Kleene’s $latex P \lor \neg P$ is not schematic but picks out a truth in some object language fixed in the context, as it might be Jack loves Jill or Jack doesn’t love Jill or $latex 3 < 5 \lor 3 \not< 5$  Which (to cut the story short) makes the score 1/2.

I’ll let you know what the score is when I’ve looked at the other texts on my list …