Tableaux or not tableaux, #2

Exactly what you choose to put into a first formal logic course for philosophers, and the level of coverage you try to achieve, will be constrained by so many considerations which vary from university to university that there can’t possibly be a one-size-fits-all way of doing things. Have the kids already had an “informal logic/critical reasoning” module? How many lecturing hours have you got? Is there a system of frequent small-group tutorial classes backing up the lectures to go through exercises (and how much extra teaching can be done by TAs in that way)? Is this the only formal logic course offered in the first couple of years or is there a follow-up course in the second year (so it isn’t so much a question of what to teach as when to teach it)? What proportion of the class can be expected to have the equivalent of a top grade in A-level (high school) maths? What proportion are likely to be rather symbol-phobic humanities students? And so it goes …

I suppose, though, that we can agree — can we? — that part of what we’ll want to cover at some level in first course consists in topics such as these (put very roughly, so as not to prejudge too many choices on details of approach):

• Basic ideas about deductive validity (and related notions like logical consistency and logical necessity) and the contrast between deductive and good-but-not-deductively valid inference. Ideas about the way deductively valid arguments can fall into patterns sharing the same form, etc.
• The initial case of arguments relying on the connectives and/or/not; the ideas of a tautology and of tautological entailment; truth-table testing. You’ll want to connect this to important informal ideas like proof-by-cases, reductio-ad-absurdum, etc.
• Relatedly, we need to explain a tidy formal language for regimenting this propositional logic, getting over the idea of strict syntactic formation rules, and the idea of interpretations/valuations for the language.
• Along the way, we’ll have to say something about use and mention, quotation conventions and so on. And about the use of object-language variables vs schematic variables added to our English meta-language.
• Now add informal reflections on varieties of conditionals, and considerations about which inferences are intuitively valid or otherwise for different kinds of conditional (e.g. when can we contrapose?). Also informal thoughts about ‘only if’ and ‘if and only if’. Then consider prospects for adding a workable conditional to our truth-functional language, expanding the truth-table test etc.
• Now broadening our scope, we want to consider quantificational arguments. So we need to explain and motivate the usual sort of quantificational language QL (initially without identity and functions), again explaining the syntax very carefully and at least giving an informal understanding of the idea of an interpretation of the language (and hence of quantification validity as truth-preservation on any interpretations).
• We want students to be confident in handling the language QL when they meet it in use elsewhere, so we are going to need to spend quite some time getting them happy with translation/transcription/regimentation (whatever you want to call in) from English into the formal language and back again.
• Some informal discussion of examples of valid and invalid inferences of QL, and relation to vernacular arguments.
• Now add identity. Discussion of what we can now translate using identity. Russell’s theory of descriptions. Ideally something about functions too.

I should add (prompted by David Makinson’s congenial comment below) that we’ll also want students early on to get a smidgin of knowledge about set notation and basic ideas, and perhaps a few ideas about probability too. If you have to wrap these topics too into the first “logic” course (in my case, I didn’t have to do that, as there were separate mini-courses on them), then things are getting even more crowded.

Now note that, as yet, we’ve said nothing about a formal deductive system, whether natural deduction (Fitch-style, Lemmon-flavoured, Gentzen style), tableaux, or — perish the thought! — axiomatic. Yet we have already got a pretty substantial menu to get through. And if you are going to cover it properly and in an interesting way, keeping the majority of your class on board, then the material I’ve mentioned so far is necessarily going to take up a goodly amount of time. In fact, I guess this part of the menu used to occupy two-thirds of my lecturing time in a 24 lecture course. So the question “what formal system to use” became for me “what can we usefully do in maybe seven or eight lectures”

Now, I was ridiculously fortunate in my last years of logic teaching. I was in Cambridge, where we can do our very best to pick the brightest/most promising students, and where we can demand (and get) very intense levels of term-time work, and set vacation work too. The students got a lot of first year logic lectures, with back-up examples classes taught by enthusiastic grad students. Even so, and despite the fact that I intentionally aimed the lectures at the non-mathematical majority, and we didn’t go out of our way to set nasty tripos questions at the end of the year, it was clear that quite a few even of these students find even the stuff I’ve mentioned (before we tackle a formal system) surprisingly tough. Lots of course sail through, but I learnt that, even in Cambridge, it is only too easy to under-estimate how foreign all this kind of thing is to many students, however smart they are. So the question in fact became “what can I usefully do in maybe seven or eight lectures, with students many of whom find thinking formally quite tough, without entirely losing too many people?”

To be continued.