## Logical Methods — on propositional logic

I have now had a chance to read the first part of Greg Restall and Shawn Sandefer’s *Logical Methods*, some 113 pages on propositional logic.

I enjoyed this well enough but I am, to be frank, a bit puzzled about the intended readership. The book’s Preface starts “Welcome to *Logical Methods*, an introduction to logic for philosophy students …”. And the text does indeed seem to start right from scratch. But Restall’s web-page for the book says “The text was developed through years of teaching intermediate (second-year) logic at the University of Melbourne.” While their Amazon blurb says “suitable for undergraduate courses and above.” Which suggests a rather unstable focus. And indeed, a significant amount of the material here, as we’ll see in a moment, is at what strikes me as a decidedly non-introductory level.

Certainly, things that can (and often should!) give pause to a philosophy student encountering formal logic for the first time are often skated over at speed. For example, when we do propositional logic, just what is the relation between the formal systems and our everyday inferences using the ordinary-language connectives? So, exactly what are these dratted “p”s and “q”s doing? On p.8 we are told that “declarative sentences express propositions”, and that we are going to be looking at propositional languages “where there are declarative sentences”. But then are also immediately told that our formal language is just designed “to express the *forms* of propositions combined with [the connectives]” (my emphasis). So do the “p”s and “q”s get interpretations as expressing propositions or not?

On p. 9 we are baldly told that “disjunctions will always be inclusive in this text” without a moment’s discussion of how things might or might not stand in ordinary language. And later, the much more vexed question of how the logician’s conditional might be related to the ordinary language conditional is relegated to a “challenge question” on p. 32. I wonder: if we don’t say rather more about the ordinary-language logical apparatus, how do we rack up a persuasive score sheet of the costs and benefits of various alternative formal choices? (Teachers using this book with real beginners might well be adding quite a bit of appropriate classroom chat on such matters as they go along — but I’m thinking here of a student reader taking the book “neat”.)

Again, the beginning reader is given just *one* worked example of a truth-table test for validity in action. And nothing is said e.g. about standard heuristics to speed things up (as in “you don’t need to work further on a line where the conclusion is true because that can’t give us a counterexample”) Yes, yes, of course truth-table testing complicated examples is as boring as heck. But surely(?) we do want our beginning students to be just a bit more au fait with how things can work out in practice.

So already, I’m not sure how well this is going to work with real beginners. But there are more serious worries. Restall and Sandefer advertise their book as presenting “proof construction on equal footing with model building” — but in fact that briskness over truth-tables is just one sign that their presentation is really skewed to emphasize proof-theoretic ideas. And so, long before we ever hear about the classical truth-functional interpretation of the connectives, we are tangling with why we might want detour-free proofs in a Gentzen-style natural deduction system. (By the way, much as though I like the elegance of Gentzen trees, I’m yet to be really persuaded that they trump Fitch-style proofs for introducing ND to students.)

And now, not only is the — I agree! — reasonably intuitive idea of a detour-free proof canvassed, but we actually get a full-on, ten-page, proof of normalizability for intuitionistic propositional logic (starting as early as p. 53 in the book). I honestly can’t imagine too many thinking that *this* is where they want their beginning philosophy students to be concentrating, so early in their logical encounters!

Now, I don’t want to carp, so let’s now recalibrate our expectations, and think of this as in fact a second-level text with some brisk reminders of the more elementary stuff. Then, on positive side, it can be said that the normalization proof and other parts of the discussion of Gentzen style ND are very accessibly done. So I can e.g. well foresee the relevant sections getting into the next edition of the Study Guide as warmly recommended reading on entry-level proof theory. But yes, for me at least, that is where this material really belongs, a step or two up from a first introductory text for philosophers. Call me old-fashioned!

I note that the text was typeset by the authors (and some of their aesthetic choices are a bit wonky!). But that does raise a question. I do wonder why, in 2023, since they have a nice PDF to hand, they have gone done the route of conventional publication when they could have got the book into so many more students’ hands by going down the free-PDF-plus-cheapo-print-on-demand route? Just saying.