Two more episodes to follow very soon.

Earlier instalments can be found here.

The previous episodes are available:

1. Episode 1, Incompleteness — the very idea (version of Oct. 16)
2. Episode 2. Incompleteness and undecidability (version of Oct. 26)
3. Episode 3. Two weak arithmetics (version of Nov. 1)
4. Episode 4. First-order Peano Arithmetic (version of Nov. 1)

Here are the previous episodes:

1. Episode 1, Incompleteness — the very idea (version of Oct. 16)
2. Episode 2. Incompleteness and undecidability (version of Oct. 26)
3. Episode 3. Two weak arithmetics (version of Nov. 1)

As explained, I’m writing these notes as just-after-the-event handouts for weekly lectures. And each week I’ll be checking through the previous handout (and no doubt finding small corrections to make) before I give the next lecture. So here’s the latest version of Episode 1, dated 16 October.

This really is a poorly written book, and it is pretty difficult to imagine for whom it is written. Although it is subtitled “An introduction to its syntax and semantics”, no one who hasn’t already done some modal logic is going to get anything much out of the opening chapters. For this is written in that style of hyper-formalization and over-abstraction that philosophers writing logic books still too often affect. Why? Who is it supposed to impress? (It is as if the authors are trying to prove that they aren’t really weedy soft-minded philosophers, but can play tough with the big boys. The irony is that the big boys, the good mathematicians, don’t play the game this way.)

Here’s a trivial example. If you or I were introducing a suitable language for doing propositional modal logic, we might say: OK, we need an unlimited supply of propositional atoms, and here they are, P, P’, P”, P”’, etc.; we want a couple of propositional connectives, say → and ¬; and the Box as a necessity operator. Then we’d remark, parenthetically, that of course the precise choice of symbolism is neither here nor there. Job done. For of course, sufficient unto the day is the rigour thereof.

But Cocchiarella and Freund are having none of this. In fact they don’t tell us what any actual modal language looks like. Rather they introduce some metalinguistic names for the atoms, whatever they are; and then there are three other symbols named c, n and l, whatever they might be, to serve as a conditional, negation and necessity operator. And the rest of the discussion proceeds at one remove, without us ever actually meeting an object language modal sentence. (Well, actually there’s another problem: for on their account it would be jolly hard to meet one, as for them a modal sentence is a set of sets of sets of numbers and symbols. Despite their extreme pernicketiness about formal matters, they are cheerfully casual about identifying set-theoretic proxies with the real thing — but let that pass.)

OK, what does their formalistic fussing get us? Nothing that I can see. The surface appearance of extra generality is spurious. And in fact, Cocchiarella and Freund soon stop any pretence at generality. For example, when the wraps are off, they require any logistic system based on the conditional and negation to have a bracket-free Polish grammar, where logical operators are prefix. And they require any derivation in such a system to be in linear Hilbert style, without rules of proof or suppositional inferences. Those requirements combined make most modal logical systems you’ve ever seen not count as such according to them.

Consider your old friend, von Wright’s M. As we all learnt it in the cradle from Hughes and Cresswell, and ignoring the fact that they go for particular modal axioms and a rule of substitution rather than using axiom schemata, their system has two rules of inference, modus ponens and a rule of necessitation that allows us to infer BoxA if we’ve proved A from no assumptions. But such a rule of course isn’t allowed if derivations all have to be Hilbert style, with conclusions always being derived by the application of rules to previous wffs, not to previous (sub)proofs. This means that Hughes and Cresswell’s M is not a modal system according to Cocchiarella and Freund. And when they talk about M, since they only have modus ponens as an inference rule, they have to complicate the axioms, by allowing us to take any of Hughes and Cresswell’s axioms and precede it by as many necessity operators as you want. They then prove what they call the rule of necessitation, which tells us that if there is a proof of A from no assumptions in their system M, then there is also a proof of BoxA in their system. But note, the C&F “rule of necessitation” is quite different from H&C’s rule. In fact the C&F rule stands to H&C’s rule pretty much as the Deduction Theorem stands to Conditional Proof.

Now, I don’t particularly object to Cocchiarella and Freund doing things this way. But I do object to their doing it this way without bothering to tell us what they are doing, how it relates to the more familiar way, and why they’ve chosen to do things their way. Why is the reader left trying to figure out which deviations from the familiar might be significant, and which not?

Anyway, we certainly weren’t impressed. The grad students — a very bright and interested bunch — uniformly found the style rebarbative and entirely off-putting. There was no general will to continue. And democracy rules in the reading group!

Later: I’ve already replaced the first version with a slightly better one …

This is fascinating stuff for philosophers of maths, in both form and content — though I don’t begin to pretend to be following all the ins and outs. In form, because it’s always intriguing to see mathematical work-in-progress, exploring ideas, guesses, dead-ends (live mathematics as an activity, if you like, as opposed to the polished product presented according to the norms for “proper” publication). And in content, because you begin to get a sense of why something that initially seems as though it ought to be easy to settle (P = NP?) is really hard.