What is the modern conception of logic? #1

Three and a half years ago, there were a few blogposts here (and also a follow-up document) about whether there is a canonical modern story about how we should conceive of our Ps and Qs, whether we should define validity primarily for schemas or interpreted sentences, and that kind of thing. Just for the fun of discovery (and because I suspect we rush too fast to suppose that there is a uniform ‘contemporary conception’ of such matters) I’m going to return to the issue over some coming blogposts — developing, correcting, adding to, and sometimes retracting what I said before. This kind of nit-picking won’t be to everyone’s taste; but hopefully some might be as intrigued by the variety of views that have been on offer out there in the modern canon of logic texts.

I can’t expect people to remember the previous discussion! —  so I’ll start again from scratch. Here then is Episode #1, even if much of it I have said before.

1. A contemporary conception?

Warren Goldfarb, in his paper ‘Frege’s conception of logic’ in The Cambridge Companion to Frege (2010), 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 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, a 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. (pp. 64–65)

Note an initial oddity here (taking up a theme that Timothy Smiley has remarked on in another context). It is said that a ‘logical form’ just is a schema. What is it then for a sentence to have a logical form? 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 $\lor$ $\neg$\,grass is green’ — is an instance of both the schema $P \lor \neg P$ and the schema $Q \lor \neg Q$. These are two different schemata (if we indeed think of schemata, as Goldfarb describes them, as expressions ‘constructed from logical signs … using schematic letters’): but surely we don’t want to say that the given sentence, for this reason at any rate, has two different logical forms. So something is amiss.

But let’s not worry about this detail for the moment. Let’s ask: is Goldfarb right that contemporary logic always (or at least usually) proceeds by defining notions like validity as applying in the first instance to schemata?

Some other writers on the history of logic take the same line about modern logic. Here, for example, is David Bostock, in his Russell’s Logical Atomism (2012), seeking to describe what he supposes is the ‘nowadays usual’ understanding of elementary logic, again in order to contrast it with the view of one of the founding fathers:

In logic as it is now conceived we are concerned with what follows from what formally, where this is understood in terms of the formal language just introduced, i.e. one which uses ‘P’, ‘Q’, … as schematic letters for any proposition, ‘a’, ‘b’, … as schematic letters for any reference to a singular subject, and ‘F’, ‘G’, … as schematic letters for any predicate. So we first explain validity for such schemata. An interpretation for the language assigns some particular propositions, subjects or predicates to the schematic letters involved. It also assigns some domain for the quantifiers to range over …. Then a single schematic formula counts as valid if it always comes out true, however its schematic letters are interpreted, and whatever the domain of quantification is taken to be. A series of such formulae representing an argument … counts as a valid sequent if in all interpretations it is truth-preserving, i.e. if all the interpretations which make all the premises true also make the conclusions true. …

We now add that an actual proposition counts as ‘formally valid’ if and only if it has a valid form, i.e. is an instance of some schematic formula that is valid. Similarly, an an actual argument is ‘formally valid’ if and it only if it has a valid form, i.e. is an instance of some schematic sequent that is valid. Rather than ‘formally valid’ it would be more accurate to say ‘valid just in virtue of the truth functors and first-level quantifiers it contains’. This begs no question about what is to count as the ‘logical form’ of a proposition or an argument, but it does indicate just which ‘forms’ are considered in elementary logic.

Finally, the task of logic as nowadays conceived is the task of finding explicit rules of inference which allow one to discover which formulae (or sequents) are the valid ones. … What is required is just a set of rules which is both ‘sound’ and ‘complete’, in the sense (i) that the rules prove only formulae (or sequents) that are valid, and (ii) that they can prove all such formulae (or sequents). (pp. 8–10)

Bostock here evidently takes very much the same line as Goldfarb, except that he avoids the unhappy outright identification of logical forms with schemata. And he goes on to say that not only do we define semantic notions like validity in the first place for schemata but proof-systems too deal in schemata — i.e. are in the business of deriving schematic formulae (or sequents) from other schematic formulae (or sequents).

It isn’t difficult to guess a major influence on Goldfarb. His one-time colleague W.V.O. Quine’s Methods of Logic was first published in 1950, and in that book — much used at least by philosophers — logical notions like consistency, validity and implication are indeed defined in the first instance for schemata. Goldfarb himself takes the same line in his own later book Deductive Logic (2003). Bostock’s own book Intermediate Logic is perhaps a little more nuanced, but again takes basically the same line.

But the obvious question is: are Goldfarb and Bostock right that the conception of logic they describe, and which they themselves subscribe to in their respective logic books, is so widely prevalent? I have certainly heard it said that a view of their kind is ‘canonical’: but what does the canon actually say?

Philosophers being a professionally contentious lot, we wouldn’t usually predict happy consensus about anything much! If we are going to find something like a shared a canonical modern conception, it is more likely to be an unreflective party line of mathematical logicians, who might be disposed to speed past preliminary niceties en route to the more interesting stuff. At any rate, what I propose to do here is to concentrate on the mathematical logicians rather than the philosophers. So let’s take some well-regarded mathematical logic textbooks from the modern canon.

How far, going back, should we cast the net? I start with Mendelson’s classic Introduction to Mathematical Logic (first published in 1964), and some books from the same era. Now, you might reasonably say that — although these books are ‘contemporary’ in the loose sense that they are still used, still recommended — they aren’t sufficiently up-to-date to chime with Goldfarb’s and Bostock’s intentions when they talk about logic as it is ‘nowadays conceived’. Fair enough. It could turn out that, beginning with an initially messy variation in approaches in the ‘early modern’ period (if we can call it that! — I mean the 1960s and 1970s, some seventy and more years after the first volume of Grundgesetze), there does indeed later emerge some convergence on a single party line in the ‘modern modern’ period. Well, that will be interesting if so. And it will be interesting too to try to discern whether any such convergence (if such there has been) is based on principled reasons for settling on one dominant story.

So what we’ll be doing to considering e.g. how various authors have regarded formal languages, what they take logical relations to hold between, how they regard the letters which appear in logical formulas, what accounts they give of logical laws and logical consequence, and how they regard formal proofs. To be sure, we will expect to find recurrent themes running through the different treatments (after all, there is only a limited number of options). But will we eventually find enough commonality to make it appropriate to talk of ‘the’ canonical contemporary conception of logic among working logicians? And if so, will it be as Goldfarb and Bostock describe it?

Let’s look and see …

[To be continued]