# Book Note: Tony Roy, Symbolic Logic, #1

Tony Roy (Philosophy, California State University, San Bernardino) has generously made available his Symbolic Logic: An Accessible Introduction to Serious Mathematical Logic. I’m commenting here on the version of October 6, 2015. The full main text is no less than 746 pages long, and is followed another 180 pages of worked answers to exercises. So this really is a major endeavour. But despite the great length, it doesn’t range widely: the discussion is “ruthlessly directed at core results” in the hope of thereby indeed making them as accessible as possible. That’s a good goal to have.

The text is divided into four parts. First, The Elements (introducing axiomatic and natural deduction presentations of FOL); second, a brief Transition part (including material on mathematical induction); third, Classical Metalogic, getting as far as touching on compactness and the L-S theorems; fourth, Logic and Arithmetic. I will comment about these four parts in four posts.

Roy isn’t very explicit about his intended audience: but reasonably high-flying mathematicians will, surely, find the 327(!) page Part I much too slow and laboured. Rather, this text at least initially goes at the sort of pace, and has the sort and level of coverage, that we expect in first ‘baby logic’ books aimed at non-mathematical philosophers who may have done a critical reasoning course, but otherwise are new to the subject. So what will such a student reader find: how will he or she cope?

The first chapter gives the usual sort of account of the informal notions of validity and soundness. This does the job, though we can quibble about details. For example, I don’t like to define an argument in such a way that almost no mathematical proof is an argument (Roy’s arguments have just premisses and a conclusion, with no room for intervening steps!). I’m not sure what is gained by defining validity in terms of there being no consistent story in which all the premisses are true and the conclusion false. What makes a story consistent other than its ingredient propositions being possibly true together? But if we have that notion of being possibly-true-together, then we can directly define validity in terms of that. Again, the brief section on validity and form needs to say more: students need to get the idea that a form of argument that is not generally valid can have valid instances. But as I say, these are quibbles.

Chapter 2 is on Formal Languages (or rather the syntax thereof). Roy discussed both sentential and quantificational languages here (but you could easily extract from this and following chapters the bits about sentential logic and read all those through before tackling  any quantificational logic). The official choice of languages is austere, with only $\sim, \to, \forall$ as basic, with other connectives and the other quantifier introduced as abbreviations. Not my preference for an introductory book: for one thing, it is better to start with $\sim, \land, \lor$ and show how very nicely things go with these, before trying to sell the material conditional to sceptical philosophy students! For another thing, if philosophers are later to meet non-classical logics where the connectives/quantifiers aren’t interdefinable, it’s better to keep things separate at the outset. But those considerations apart, this chapter is routine and clear enough. It finishes with a first glance at a formal syntax for arithmetic, and this language appears in examples over the coming chapters.

Chapter 3 continues the syntactic theme and tackles Axiomatic Deduction. Roy himself notes that the reader might well want to skip this and return to it later. We get a standard Hilbert System (though unadorned with the Deduction Theorem at this stage, so unfriendly). As e.g. in Mendelson, we end up doing logic at one remove, with derivations all metalinguistic: I don’t myself thing that this is a good line to take, but let’s accept that it is one way to do things which can pass muster  if you make it transparently clear what is happening. But I don’t think Roy pulls this off. For example, he defines consequence (syntactic consequence in a deductive system) as holding between formal wffs. But then he slips unannounced (I think) into talking about consequence as a relation between sentence forms or schemata, and his examples of derivations become lists of schemata. This sort of wobbling isn’t a good example for philosophers to follow!

A more minor thing, but again a sign perhaps that Roy is writing hastily (not surprising perhaps given the length of the text): he sloppily states the modus ponens rule  as $\mathcal{P}$, $\mathcal{P} \to \mathcal{Q}$ $\vdash$ $\mathcal{Q}$ — which on the standard understanding of $\vdash$ is a proposition (or a schema for one) not the articulation of a rule.  More seriously, while in carping mood, the  presentation of the quantification rules and identity rules is, I think, likely to rather too quick for the intended neophyte audience. So overall, not a particularly successful chapter, I think.

Chapter 4 is on Semantics. Wffs of the formal languages are given valuations in terms of assignment of semantic values to elements of the language in the standard way. Though he calls valuations interpretations, Roy doesn’t actually seem at this stage to give the wffs of a formal sentential language, for example, any interpretative content — so the student might indeed wonder whether we (yet) have here any object languages in which we actually can give contentful arguments? Prescinding from general worries of that kind about exactly what is going on, however, I found the details of the treatment of quantificational semantics unhelpfully messy: there are a number of standard textbooks which do things in a more student-friendly way. I’m inclined to say much the same about the 70 page Chapter 5 which, perhaps rather late in the day,  is on Translation. Here we do get introduced to a notion of “interpretation function” (from “ways the world might be” to assignments of semantic values): I think it is questionable how approachable the student reader will find this. Indeed, to my mind, this chapter often makes quite unnecessarily heavy weather of simple things. Students just don’t need to bring to bear the full official apparatus of quantificational semantics in order to learn to work comfortably with translations to and from the formal languages.

Chapter 6 is 120 pp. on Natural Deduction, and presents a Fitch-style system, now with the usual four  connectives and both quantifiers. And Roy also introduces $\bot$ into the system as a new symbol treated as an abbreviation for some contradiction, though this addition doesn’t seem to be handled tidily. The chapter, however, provides a great deal of help on proof-strategy for students using a Fitch-style system and could prove useful. A complaint might be, however, that there isn’t a clean enough separation between (i) getting across a basic understanding of  the rules and of how the system works and (ii) the provision of heuristics for proof-discovery. And I suppose that a real worry is that the mathematically ept won’t need anywhere near so much by way of heuristics, while the philosophers who primarily need to get to understand how proofs work (but needn’t fret so much about learning to roll their own proofs) could get lost in all the details.

Which (I  realize) all sounds a bit ungrateful! But, leaving price considerations aside, there is a lot of competition at this level from some very fine introductory logic books for the non-mathematical which are more polished and to my mind better organized — and brisker. There is a balance to be struck between gently-paced accessibility and downright long-windedness. For my money, Roy too often gets the balance wrong.

To be continued.

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