Avigad MLC — 6: Arithmetics

Chs 1 to 7 of MLC, as we’ve seen, give us a high-level and often rather challenging introduction to core first-order logic with a quite strongly proof-theoretic flavour. Now moving on, the next three chapters are on arithmetics — Ch. 8 is on primitive recursion, Ch. 9 on PRA, and Ch. 10 on richer first-order arithmetics.

I won’t pause long over Ch. 8, as the basic facts about p.r. functions aren’t wonderfully exciting! Rather characteristically, Avigad dives straight into the general definition of p.r. functions, and then it’s shown that the familiar recursive definitions of addition, exponentiation, and lots more, fit the bill. We then see how to handle finite sets and sequences in a p.r. way. §8.4 discusses other recursion principles which keep us within the set of p.r. functions, and §8.5 discusses recursion along well-founded relations — these two sections are a bit tersely abstract, and it would surely have been good to have more by way of motivating examples. Finally §8.6 tells us about diagonalizing out of the p.r. functions to get a computable but not primitive recursive function, and says just a little about fast-growing functions. All this is technically fine, it goes without saying; though once again I suspect that many students will find this chapter rather more manageable if they have had a preliminary warm-up with a gentler first introduction to the topic first.

But while Ch. 8 might be said to be relatively routine (albeit quite forgiveably so!), Ch. 9 is anything but. It is the most detailed and helpful treatment of Primitive Recursive Arithmetic that I know.

Avigad first presents an axiomatization of PRA in the context of a classical quantifier-free first-order logic. Hence

  1. The logic has the propositional connectives, axioms for equality, plus a substitution rule (wffs with variables are treated as if universally quantified, so from A(x) we can infer A(t) for any term).
  2. We then have a symbol for each p.r. function — and we can think of these added iteratively, so as each new p.r. function is defined by composition or primitive recursion from existing functions, a symbol for the new function is introduced along with its appropriate defining quantifier-free equations in terms of already-defined functions.  
  3. We also have a quantifer-free induction rule: from A(0) and A(x) \to A(Sx), infer A(t) for any term.

§§9.1–9.3 explore this version of PRA in some detail, deriving a lot of arithmetic, showing e.g. that PRA proves that if p is prime, and p \mid xy then either p \mid x or p \mid y, and noting along the way that we could have used an intuitionistic version of the logic without changing what’s provable.

Then the next two sections very usefully discuss two variant presentations of PRA. §9.4 enriches the language and the logic by allowing quantifiers, though induction is still just for quantifier-free formulas. It is proved that this is conservative over quantifier-free PRA for quantifier-free sentences. And there’s a stronger result. Suppose full first-order PRA proves the \Pi_2 sentence \forall x \exists y A(x, y), then for some p.r. function symbol f, quantifier-free PRA proves A(x, f(x)) (and we can generalize to more complex \Pi_2 sentences).

By contrast §9.5 weakens the language and the logic by removing the connectives, so all we are left with are equations, and we replace the induction rule by a rule which in effect says that functions satisfying the same p.r. definition are everywhere equal. This takes us back — as Avigad nicely notes — to the version of PRA presented in Goodstein’s (now unread?) 1957 classic  Recursive Number Theory, which old hands might recall is subtitled  ‘A Development of Recursive Arithmetic in a Logic-Free Equation Calculus’.

All this is done in an exemplary way, I think. Perhaps Avigad is conscious that in this chapter he is travelling over ground that it is significantly less well-trodden in other textbooks, and so here he allows himself to be rather more expansive in his motivating explanations.

The following Ch. 10 is the longest in the book, some forty two pages on ‘First-Order Arithmetic’. Or rather, the chapter is on arithmetics, plural — for as well as the expected treatment of first-order Peano Arithmetic, with nods to Heyting Arithmetic,  there is also a perhaps surprising amount here about subsystems of classical PA.

In more detail, §10.1 briefly introduces PA and HA. You might expect next to get a section explaining how PA (with rather its minimal vocabulary) in fact can be seen as extending the PRA which we’ve just met (with all its built-in p.r. functions). But we have to wait until  §10.4 to get the story about how to define p.r. functions using some version of the beta-function trick. In between, there are two longish sections on the arithmetical hierarchy of wffs, and on subsystems of PA with induction restricted to some level of the hierarchy. Then §10.5 shows e.g. how truth for \Sigma_n sentences can be defined in a \Sigma_n way, and shows e.g. that I\Sigma_{n+1} (arithmetic with induction for \Sigma_{n + 1} wffs) can prove the consistency of I\Sigma_{n}, and also — again using truth-predicates — it is shown e.g. that I\Sigma_{n} is finitely axiomatizable.

There’s a minor glitch. In the proof of 10.3.5 there is a reference to the eighth axiom of Q — but Robinson arithmetic isn’t in fact introduced until Chapter 12. Otherwise, so far, so good. The material here is all stuff that is very good to know. Yes, the discussion is a bit dense in places; but it should all be pretty manageable because the ideas are, after all, straightforward enough.

However, the chapter ends with another ten pages whose role in the book I’m not so sure about. §10.6 proves three theorems using cut elimination arguments, namely (1) that I\Sigma_{1} is conservative over PRA for \Pi_2 formulas; (2) Parikh’s Theorem, (3) that so-called B\Sigma_{n+1} is conservative over I\Sigma_{n} for \Pi_{n+2} wffs. What gives these results, in particular the third, enough interest to labour through them? They are, as far as I can see, never referred to again in later chapters of the book. And yet §10.7 then proves the same theorems again using model theoretic arguments. I suppose that these pages give us samples of the kinds of conservation results we can achieve and some methods for proving them. But I’m not myself convinced they really deserve this kind of substantial treatment in a book at this level.

Avigad MLC — 5: More on FOL

Following on from the very interesting Chapter 6 on cut-elimination, there’s one further chapter on FOL, Chapter 7 on “Properties of First-Order Logic”. There are sections on Herbrand’s Theorem, on the Disjunction Property for intuitionistic logic, on the Interpolation Lemma, on Indefinite Descriptions and on Skolemization. This nicely follows on from the previous chapter, as the proofs here mostly rely on the availability of cut-elimination. I’m not going to dwell on this chapter, though, which I think most readers will find quite hard going. Hard going in part because, apart from perhaps the interpolation lemma, it won’t this time be obvious from the off what the point of various theorems are.

Take for example the section on Skolemization. This goes at pace. And the only comment we get about why this might all matter is at the end of the section, where we read:

The reduction of classical first-order logic to quantifier-free logic with Skolem functions is also mathematically and philosophically interesting. Hilbert viewed such functions (more precisely, epsilon terms, which are closely related) as representing the ideal elements that are added to finitistic reasoning to allow reasoning over infinite domains.

So that’s just one sentence expounding on the wider interest  — which is hardly likely to be transparent to most readers! It would have been good to hear more.

Avigad’s sections in this chapter are of course technically just fine and crisply done, and can certainly be used as a good source to consolidate and develop your grip on their cluster of topics if you already have met the key ideas. But to be honest I wouldn’t recommend starting here.

OK: The chapters to this point take us some 190 pages into MLC — they form, if you like, a book within the book, on core FOL topics but with an unusually and distinctively proof-theoretic flavour. So this is well worth having. But to repeat what I’ve said more than once before, I suspect that a reader who is going to happily navigate and appreciate the treatments of topics here will typically need rather more background in logic than Avigad implies.

So when I get round to revising the Beginning Mathematical Logic Study Guide, these opening chapters will find their place in the suggested more advanced supplementary readings.

Avigad MLC — 4: Sequent calculi vs tableaux?

Jeremy Avigad’s book is turning out to be not quite what I was expecting. The pace and (often) compression can make for rather surprisingly tough going.There is a shortage of motivational chat, which could make the book a challenging read for its intended audience.

Still, I’m cheerfully pressing on through the book, with enjoyment. As I put it before, I’m polishing off some rust — and improving my mental map of the logical terrain. So, largely as an aide memoire for when I get round to updating the Beginning Mathematical Logic Guide next year, I’ll keep on jotting down a few notes, and I will keep posting them here in case anyone else is interested.

MLC continues, then, with Chapter 6 on Cut Elimination. And the order of explanation is, I think, interestingly and attractively novel.

Yes, things begin in a familiar way. §6.1 introduces a standard sequent calculus for (minimal and) intuitionistic FOL logic without identity. §6.2 then, again in the usual way, gives us a sequent calculus for classical logic by adopting Gentzen’s device of allowing more than one wff to the right of the sequent sign. But then A notes that we can trade in two-sided sequents, which allow sets of wffs on both sides, for one-sided sequents where everything originally on the left gets pushed to the right of sequent side (being negated as it goes). These one-sided sequents (if that’s really the best label for them) are, as far as I can recall, not treated at all in Negri and von Plato’s lovely book on structural proof theory; and they are mentioned as something of an afterthought at the end of the relevant chapter on Gentzen systems in Troelstra and Schwichtenberg. But here in MLC they are promoted to centre stage.

So in §6.2 we are introduced to a calculus for classical FOL using such one-sided, disjunctively-read, sequents (we can drop the sequent sign as now redundant) — and it is taken that we are dealing with wffs in ‘negation normal form’, i.e. with conditionals eliminated and negation signs pushed as far as possible inside the scope of other logical operators so that they attach only to atomic wffs. This gives us a very lean calculus. There’s the rule that any \Gamma, A, \neg A with A atomic counts as an axiom. There’s just one rule each for \land, \lor, \forall, \exists. There also is a cut rule, which tells us that from \Gamma, A and \Gamma, {\sim}{A} we can infer \Gamma (here {\sim}{A} is notation for the result of putting the negation of A in negation normal form).

And Avidgad  now proves twice over that this cut rule is eliminable. So first in §6.3 we get a semantics-based proof that the calculus without cut is already sound and complete. Then in §6.4 we get a proof-theoretic argument that cuts can be eliminated one at a time, starting with cuts on the most complex formulas, with a perhaps exponential increase in the depth of the proof at each stage — you know the kind of thing! Two comments:

  1. The details of the semantic proof will strike many readers as familiar — closely related to the soundness and completeness proofs for a Smullyan-style tableaux system for FOL. And indeed, it’s an old idea that Gentzen-style proofs and certain kind of tableaux can be thought of as essentially the same, though conventionally written in opposite up-down directions (see Ch XI of Smullyan’s 1968 classic First-Order Logic). In the present case, Avigad’s one-sided sequent calculus without cut is in effect a block tableau system for negation normal formulas where every wff is signed F. Given that those readers whose background comes from logic courses  for philosophers will probably be familiar with tableaux (truth-trees), and indeed given the elegance of Smullyan systems, I think it is perhaps a pity that A misses the opportunity to spend a little time on the connections.
  2. A’s sparse one-sided calculus does make for a nicely minimal context in which to run a bare-bones proof-theoretic argument for the eliminability of the cut rule, where we have to look at a very small number of different cases in developing the proof instead of having to hack through the usual clutter. That’s a very nice device! I have to report though that, to my mind, A’s mode of presentation doesn’t really make the proof any more accessible than usual. In fact, once again  the compression makes for quite hard going (even though I came to it knowing in principle what was supposed to be going on, I often had to re-read). Even just a few more examples along the way of cuts being moved would surely have helped.

To continue (and I’ll be briefer) §6.5 looks at proof-theoretic treatments of cut elimination for intuitionistic logic, and §6.6 adds axioms for identity into the sequent calculi and proves cut elimination again. §6.7 is called ‘Variations on Cut Elimination’ with a first look at what can happen with theories other than the theory of identity when presented in sequent form. Finally §6.8 returns to intuitionistic logic and (compare §6.5) this time gives a nice semantic argument for the eliminability of cut, going via a generalization of Kripke models.

All very good stuff, and I learnt from this. But I hope it doesn’t sound too ungrateful to say that a student new to sequent calculi and cut-elimination proofs would still do best to read a few chapters of Negri and von Plato (for example) first, in order to later be able get a lively appreciation of §6.4 and the following sections of MLC.

Avigad, MLC — 3: are domains sets?

The next two chapters of MLC are on the syntax and proof systems for FOL — in three flavours again, minimal, intuitionstic, and classical — and then on semantics and a smidgin of model theory. Again, things proceed at pace, and ideas come thick and fast.

One tiny stylistic improvement which could have helped the reader (ok, this reader) would have been to chunk up the sections into subsections — for example, by simply marking the start of a new subsection/new subtheme by leaving a blank line and beginning the next paragraph  “(a)”, “(b)”, etc. A number of times, I found myself re-reading the start of a new para to check whether it was supposed to flow on from the previous thought. Yes, yes, this is of course a minor thing! But readers of dense texts like this need all help we can get!

So in a bit more detail, how do Chapters 4 and 5 proceed? Broadly following the pattern of the two chapters on PL, in §4.1 we find a brisk presentation of FOL syntax (in the standard form, with no syntactic distinction made between variables-as-bound-by-quantifiers and variables-standing-freely). Officially, recall, wffs that result from relabelling bound variables are identified. But this seems to make little difference: I’m not sure what the gain is, at least here in these chapters, in a first encounter with FOL.

§4.2 presents axiomatic and ND proof systems for the quantifiers, adding to the systems for PL in the standard ways. §4.3 deals with identity/equality and says something about the “equational fragment” of FOL. §4.4 says more than usual about equational and quantifier-free subsystems of FOL, noting some (un)decidability results. §4.5 briefly touches on prenex normal form. §4.6 picks up the topic (dealt with in much more detail than usual) of translations between minimal, intuitionist, and classical logic. §4.7 is titled “Definite Descriptions” but isn’t as you might expect about how to add a description operator, a Russellian iota, but rather about how — when we can prove \forall x\exists! yA(x, y) — we can add a function symbol f such that f(x) = y holds when A(x, y), and all goes as we’d hope. Finally, §4.8 treats two topics: first, how to mock up sorted quantifiers in single-sorted FOL; and second, how to augment our logic to deal with partially defined terms. That last subsection is very brisk: if you are going to treat any varieties of free logic (and I’m all for that in a book at this level, with this breadth) there’s more worth saying.

Then, turning to semantics, §5.1 is the predictable story about full classical logic with identity,  with soundness and completeness theorems, all crisply done. §5.2 tells us more about equational and quantifier-free logics.  §5.3 extends Kripke semantics to deal with quantified intuitionistic logic. We then get algebraic semantics for classical and intuitionistic logic in §5.4 (so, as before, A is casting his net more widely than usual — though the treatment of the intuitionistic case is indeed pretty compressed). The chapter finishes with a fast-moving 10 pages giving us two sections on model theory. §5.5 deals with some (un)definability results, and talks briefly about non-standard models of true arithmetic. §5.6 gives us the L-S theorems and some results about axiomatizability. So that’s a great deal packed into this chapter. And at a sophisticated level too — it is perhaps rather telling that A’s note at the end of the chapter gives Peter Johnstone’s book on Stone Spaces as a “good reference” for one of the constructions!

What more is there to say? I’m enjoying polishing off some patches of rust! — but as is probably already clear, these initial chapters are pitched at what many student readers will surely find a pretty demanding level, unless they bring quite a bit to the party. That’s not a criticism (except perhaps of Avigad’s initial advertising pitch, saying that this book will be accessible to advanced undergraduates without actually requiring a logical background): I‘m just locating the book on the spectrum of texts. I’d have slowed down the exposition a bit at a number of points, but that (as I said in the last post) is a judgement call, and it would be unexciting to go into details here.

One general comment, though. I note that A does define a model for a FOL language as having a set for quantifiers to range over,  but with a function (of the right arity) over that set as interpretation for each function symbol, and a relation (of the right arity) over that set as interpretation for each relation symbol. My attention might have flickered, but A seems happy to treat functions and relations as they come, not explicitly trading them in for set-theoretic surrogates (sets of ordered tuples). But then it is interesting to ask — if we treat functions and relations as they come, without going in for a set-theoretic story, then why not treat the quantifiers as they come, as running over some objects plural? That way we can interpret e.g. the first-order language of set theory (whose quantifiers run over more than set-many objects) without wriggling.

A does nicely downplay the unnecessary invocation of sets — though not consistently. E.g. he later falls into saying “the set of primitive recursive functions is the set of functions [defined thus and so]” when he could equally well have said, simply,  “the primitive recursive functions are the functions [defined thus and so]”. I’d go for consistently avoiding unnecessary set talk from the off — thus making it much easier for the beginner at serious logic to see when set theory starts doing some real work for us. Three cheers for sets: but in their proper place!

Avigad, MLC — 2: And what is PL all about?

After the chapter of preliminaries, there are two chapters on propositional logic (substantial chapters too, some fifty-five large format pages between them, and they range much more widely than the usual sort of introductions to PL in math logic books).

The general approach of MLC foregrounds syntax and proof theory. So these two chapters start with §2.1 quickly reviewing the syntax of the language of PL (with \land, \lor, \to, \bot as basic — so negation has to be defined by treating \neg A as A \to \bot). §2.2 presents a Hilbert-style axiomatic deductive system for minimal logic, which is augmented to give systems for intuitionist and classical PL. §2.3 says more about the provability relations for the three logics (initially defined in terms of the existence of a derivation in the relevant Hilbert-style system). §2.4 then introduces natural deduction systems for the same three logics, and outlines proofs that we can redefine the same provability relations as before in terms of the availability of natural deductions. §2.5 notes some validities in the three logics and §2.6 is on normal forms in classical logic. §2.7 then considers translations between logics, e.g. the Gödel-Gentzen double-negation translation between intuitionist and classical logic. Finally §2.8  takes a very brisk look at other sorts of deductive system, and issues about decision procedures.

As you’d expect, this is all technically just fine. Now, A says in his Preface that “readers who have had a prior introduction to logic will be able to navigate the material here more quickly and comfortably”. But I suspect this rather misleads: in fact some prior knowledge will be pretty essential if you are really going get much out the discussions here. To be sure, the point of the exercise isn’t (for example) to get the reader to be a whizz at knocking off complex Gentzen-style natural deduction proofs; but are there quite enough worked examples for the real newbie to get a good feel for the claimed naturalness of such proofs? Is a single illustration of a Fitch-style alternative helpful? I’m doubtful. And so on.

To continue, Chapter 3 is on semantics. We get the standard two-valued semantics for classical PL, along with soundness and completeness proofs, in §3.1. Then we get interpretations in Boolean algebras in §3.2. Next, §3.3 introduces Kripke semantics for intuitionistic (and minimal) logic — as I said, A is indeed casting his net significantly more widely that usual in introducing PL. §3.4 gives algebraic and topological interpretations for intuitionistic logic. And the chapter ends with §3.5, ‘Variations’, introducing what A calls generalised Beth semantics.

Again I really do wonder about the compression and speed of some of the episodes in this chapter; certainly, those new to logic could find them very hard going. Filters and ultrafilters in Boolean algebras are dealt with at speed; some more examples of Kripke semantics at work might have helped to fix ideas; Heyting semantics is again dealt with at speed. And §3.5 will (surely?) be found challenging.

Still, I think that for someone coming to MLC who already does have enough logical background (perhaps half-baked, perhaps rather fragmentary) and who is mathematically adept enough, these chapters — perhaps initially minus their last sections — should bring a range of technical material into a nicely organised story in a very helpful way, giving a good basis for pressing on through the book.

For me, the interest here in the early chapters of MLC is in thinking through what I might do differently and why. As just implied, without going into more detail here, I’d have gone rather more slowly at quite a few points (OK, you might think too slowly — these things are of course a judgment call). But perhaps as importantly, I’d have wanted to add some more explanatory/motivational chat.

For example, what’s so great about minimal logic? Why talk about it at all? We are told that minimal logic “has a slightly better computational interpretation” than intuitionistic logic. But so what? On the face of it doesn’t treat negation well, its Kripke semantics doesn’t take the absurdity constant seriously, and lacking disjunctive syllogism minimal logic isn’t a sane candidate for regimenting mathematical reasoning (platonistic or constructive). And if your beef with intuitionist logic are relevantist worries about its explosive character, then minimal logic is hardly any better (since from a contradiction, we can derive any and every negated wff).  So — a reader coming with a bit of logical background might reasonably wonder — why seriously bother with it? It would be worth saying something.

Another, even more basic, example. We are introduced to the language of propositional logic (with a fixed infinite supply of propositional letters). Glancing ahead, in Chapter 4 we meet a multiplicity of first-order languages (with their various proprietary and typically finite supplies of constants and/or relation symbols and/or function symbols). Why the asymmetry? Connectedly but more basically, what actually are the propositional variables doing in a propositional language? Different elementary textbooks say different things, so A can’t assume that everyone is approaching the discussions here with the same background of understanding of what PL is all about. He cheerfully says that the propositional variables “stand for” propositions. Which isn’t very helpful given that neither “stand for” nor “proposition” is at all clear!

Now, of course, you can get on with the technicalities of PL (and FOL) while blissfully ignoring the question of what exactly the point of it all is, what exactly the relation is between the technical games being played and the real modes of mathematical argumentation you are, let’s hope, intending to throw light on when doing mathematical logic. But I still think that an author should fess up about  their understanding of these rather contentious issues, as such an understanding which must shape their overall approach. Avigad notes in his Preface that we can use formal systems to understand patterns of mathematical inference  (though he then oddly says that “the role of a proof system” for PL “is to derive tautologies”, when you would expect something like “is to establish tautological entailments”); and perhaps more will come out as the book progresses about how he conceives of this use. But for myself, I’d have liked to have seen rather more upfront.

Avigad, MLC — 1: What are formulas?

I noted before that Jeremy Avigad’s new book Mathematical Logic and Computation has already been published by CUP on the Cambridge Core system, and the hardback is due any day now. The headline news is that this looks to me the most interesting and worthwhile advanced-student-orientated book that has been published recently.

I’m inspired, then, to blog about some of the discussions in the book that interest me for one reason or another, either because I might be inclined to do things differently, or because they are on topics that I’m not very familiar with, or (who knows?) maybe for some other reason. I’m not at all planning a judicious systematic review, then: these will be scattered comments shaped by the contingencies of my own interests!

Chapter 1 of MLC is on “Fundamentals”, aiming to “develop a foundation for reasoning about syntax”. So we get the usual kinds of definitions of inductively defined sets, structural recursion, definitions of trees, etc. and applications of the abstract machinery to defining the terms and formulas of FOL languages, proving unique parsing, etc.

This is done in a quite hard-core way (particularly on trees), and I think you’d ideally need to have already done a mid-level logic course to really get the point of various definitions and constructions. But A [Avigad, the Author, of course!] notes that he is here underwriting patterns of reasoning that are intuitively clear enough, so the reader can at this point skim, returning later to nail down various details on a need-to-know basis.

But there is one stand-out decision that is perhaps worth pausing over for a moment. Take the two expressions \forall xFx and \forall yFy. The choice of bound variable is of course arbitrary. It seems we have two choices here:

  1. Just live with the arbitrariness. Allow such expressions as distinct formulas, but prove that  formulas like these which are can be turned into each other by the renaming of bound variables (formulas which are \alpha-equivalent, as they say) are always interderivable, are logically equivalent too.
  2. Say that formulas proper are what we get by quotienting expressions by \alpha-equivalence, and lift our first-shot definitions of e.g. wellformedness for expressions of FOL to become definitions of wellformedness for the more abstract formulas proper of FOL.

Now, as A says, there is in the end not much difference between these two options; but he plumps for the second option, and for a reason. The thought is this. If we work at expression level, we will need a story about allowable substitutions of terms for variables that blocks unwanted variable-capture. And A suggests there are three ways of doing this, none of which is entirely free from trouble according to him.

  1. Distinguish free from bound occurrences of variables, define what it is for a term to be free for a variable, and only allow a term to be substituted when it is free to be substituted. Trouble: “involves inserting qualifications everywhere and checking that they are maintained.”
  2. Modify the definition of substitution so that bound variables first get renamed as needed — so that the result of substituting y + 1 for x in \exists y(y > x) is something like \exists z(z > y + 1). Trouble: “Even though we can fix a recipe for executing the renaming, the choice is somewhat arbitrary. Moreover, because of the renamings, statements we make about substitutions will generally hold only up to \alpha-equivalence, cluttering up our statements.”
  3. Maintain separate stocks of free and bound variables, so that the problem never arises. Trouble: “Requires us to rename a variable whenever we wish to apply a binder.”

But the supposed trouble counting against the third option is, by my lights, no trouble at all. In fact A is arguably quite misdescribing what is going on in that case.

Taking the Gentzen line, we distinguish constants with their fixed interpretations, parameters or temporary names whose interpretation can vary, and bound variables which are undetachable parts of a quantifier-former we might represent ‘\forall x \ldots\ x\ldots \ x\ldots’. And when we quantify Fa to get \forall xFx we are not “renaming a variable” (a trivial synactic change) but replacing the parameter which has one semantic role with an expression which is part of a composite expression with a quite different semantic role. There’s a good Fregean principle, use different bits of syntax to mark different semantic roles: and that’s what is happening here when we replace the ‘a’ by the ‘x’ and at the same time bind with the quantifier ‘\forall x’ (all in one go, so to speak).

So its seems to me that option 1c is markedly more attractive than A has it (it handles issues about substitution nicely, and meshes with the elegant story about semantics which has \forall xFx true on an interpretation when Fa is true however we extend that interpretation to give a referent to the temporary name a). The simplicity of 1c compared with option 2 in fact gets the deciding vote for me.

Avigad on Mathematical Logic and Computation

A heads up, as they say. Jeremy Avigad’s new book Mathematical Logic and Computation has now been published by CUP (or at least, an e-version is already available on the Cambridge Core system if you have access — with the hardback due soon). Here’s a link to the front matter of the book, which gives you the Table of Contents and the Preface. Between them, they give you a fair idea of the coverage of the book.

As you’d expect from this author, this book is very worth having, an excellent addition to the literature, with plenty more than enough divergences and side-steps from the more well-trodden paths through the material to be consistently interesting. Having quickly read a few chapters, and dipped into a few more, I’d say that the treatments of topics, though very clear, are often rather on the challenging side (Avigad’s Carnegie Mellon gets very high-flying students in this area!). For example, the chapters on FOL would probably be best tackled by someone who has already done a course based on something like Enderton’s classic text. But that’s not a complaint, just an indication of the level of approach.

When the physical version becomes available — so much easier to navigate! — I’m going to enjoy settling down to a careful read through, and maybe will comment in more detail here. Meanwhile, this is most certainly a book to make sure your library gets.

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