To continue, then, with the comments on Tony Roy’s *Symbolic Logic: An Accessible Introduction to Serious Mathematical Logic, *Part II is called “Transition: Reasoning about Logic”, and is almost another 100 pages, though it consists of just two chapters.

Chapter 7 is a curious affair. It’s called “Direct Semantic Reasoning”, and the aim is to get the student to be able to use their understanding of the official semantics to demonstrate that, e.g., (i) \(\exists xFx \vDash \exists y(Fx \lor Gy)\) or more challengingly (ii) \(\exists x \forall y Rxy \vDash \forall y \exists x Rxy\), while also being able to show e.g. (iii) \(\forall x \exists y Rxy \nvDash \exists y\forall x Rxy\). And yes, we of course want students to be able to do these things. Or rather more carefully, we want a student to able to produce informal but rigorous proofs of entailments given the definition of semantic consequence, and to produce counterexamples to witness non-entailments. This amounts to learning how to do some bits of (pretty elementary) mathematical reasoning, and students coming from a non-maths background need to be given some exemplars and models of this sort of informal reasoning done with the right amount of detail to pass muster.

But Roy is keen to replace informal proofs with *formal* derivations set out in natural deduction style, and the results seem to me to be rebarbative and consequently unhelpful. It is difficult to believe that students will find their abilities at informal reasoning actually improved by the sort of messy regimentations we find in this chapter. Have a look at p. 364 and ask if this is likely to help a student who doesn’t immediately see why, morally, (ii) should be true.

In his Preface, Roy tries to make a case for his approach in this chapter, but I’m not convinced. And to the extent that the idea is to give students an example of working formally in reasoning about logic, then we have to hand a framework for formalizing semantic reasoning which is much, much more elegant — i.e. use semantic tableaux, which are designed for the job!

Chapter 8 is on Mathematical Induction. This starts with a general discussion of induction, with some simple arithmetical and geometrical examples. Roy then goes on to give some applications of induction to prove syntactic and semantic facts about his formal languages. The chapter seems pretty clear, though as before things are perhaps rather stretched out. However, philosophy students needing to get their heads round the idea of induction and struggling with brisker presentations could very well find this a useful compendium.

*To be continued.*

Tony RoyEven though (perhaps because) they are not terribly positive, I very much appreciate Peter’s comments on my book. I will definitely pay close attention as the manuscript is revised. In some cases, however, it may be that comments reflect differences about ends.

Peter is right that the text (especially in the beginning) is aimed at students without mathematical background. The aim is to begin where they are, and gradually bring them to increasing levels of sophistication. My students are not admitted to Cambridge! With this in mind,

Peter’s Comment #1

1. I agree that chapter 3 (on axiomatic deduction) is too quick for a neophyte audience. As Peter observes I do suggest that students not attempt the chapter until they have worked all the way through natural derivations. At that stage, they aren’t quite neophytes. The chapter’s location immediately after presentation of syntax (and its prickly nature) is intended to make vivid that derivations are defined purely with respect to form.

2. Again, I agree that chapter 4 (on semantics) remains entirely at the level of valuations (what I call ‘interpretations’) so that formal languages are not tied to anything in ordinary language. Even the standard valuation of the language for arithmetic is presented as one among others. But that is the point: I am concerned to present the formal languages “as themselves” and without connection to ordinary language. This is supposed to mitigate the objections as, “the table for ‘’ is wrong insofar as it does not always correspond to the ordinary ‘if. . . then. . .’”

3. Chapter 5 (on translation) finally makes connections to ordinary language. I do wheel in the semantics – but that is because it’s only relative to the semantics that I have any account about how the formal language works. (The approach may mitigate the objection that tilde, caret and wedge are better primitive symbols than tilde and arrow – at this stage all the usual operators are defined, and we can begin with the cases that make the best fit to ordinary expressions.) Also, I do have an elaborate method for translations especially in the sentential case – but, theory to the side, my experience is that beginning students struggle with translations and take the method as a lifeline (in contrast to the way many of us learned).

4. Peter is right that chapter 6 (on natural deduction) has more by way of heuristics than the is required by the mathematically ept, and more detail than required for a bare account of how proofs work. But, again, the chapter is meant to bring along students who struggle at the simplest of derivations. The “strategies” are offered as a lifeline to students who would otherwise drown.

Peter’s Comment #2

5. In chapter 7 (on elementary mathematical reasoning) I do introduce a formal system in a natural deduction style. But I am not keen to replace informal proofs with formal derivations! Rather the formal derivations are intended as an aid for producing informal reasoning. Again, I have not found that simple provision of examples is generally sufficient. And the reason for natural rather than tableaux formalization is precisely because the formal derivations are supposed to aid (natural) informal reasoning.

6. Peter suggests that chapter 8 (on mathematical induction) is a bit stretched out – though it might be useful to students struggling with brisker presentations. Perhaps some of this comes down to the difference between students at my institution and those at Cambridge (and so a contrast with the way most of us learned)!