ACA0, #5: An aside on PA + Th versus T(PA)

Suppose we bolt onto to first-order PA the axioms of Th, the arithmetization of a natural theory of a new truth-predicate T, in a way that – prima facie – shouldn’t upset even a deflationist/minimalist about truth. As we’d expect, the composite theory PA + Th proves each biconditional T(“φ”) ↔ φ for sentences φ. And as we’d also expect — though it isn’t entirely trivial to demonstrate it — PA + Th is conservative over PA: it proves no new arithmetical truths.

But now let’s reflect. Bolting the arithmetical truth-theory onto PA ‘externally’ leaves us, in particular, with just the same induction axioms as we started off with. However, you might say, why stop there? It would seem that the line of argument that I sketched a few posts ago for being generous with induction will apply again, and will motivate extending the induction axioms from instances involving just the original L-predicates (L is the language of PA) to instances involving the new predicate T too. So let T(PA) be the theory we get taking PA plus Th plus the closures of all instances of the first-order induction schema for new predicates constructed from T as well. Then, the argument seems to be, T(PA) should be as compelling a theory as PA + Th. But as is well known, T(PA) is not conservative over PA (it proves Con(PA) for a start).

However, the inflationary argument for generosity with induction can be resisted. But how exactly? A deflationist might be tempted to say: ‘As a deflationist, I don’t accept that truth is a genuine property: more precisely, ‘T’ in the theory Th doesn’t express a genuine property, so we can’t use it in inductive arguments.’ But this isn’t in fact terribly helpful, unless augmented by an independent account of the initially murky notion of ‘not expressing a genuine property’. So let’s proceed more carefully.

Suppose someone with a taste for formalizing his knowledge – call him Kurt – accepts PA (this is the apparatus he uses in fixing his arithmetical beliefs). Suppose we now offer him the axioms Th as a partial characterization of the uninterpreted new predicate T. If we give Kurt any particular number which happens to be the Gödel number of a sentence φ then he will in principle be able to prove the corresponding theorem T(“φ”) ↔ φ. What he can’t do, since he doesn’t yet have induction axioms for T, is prove anything general about T to the effect that, for every n, if n = “φ” for some φ, then T(n) ↔ φ, or else T(n) ↔ ⊥. So, while still working from inside PA + Th, Kurt has no way of knowing whether T(n) has been defined for all numbers n. In this sense, then, Kurt doesn’t yet know whether T expresses a determinate property of numbers. So, for a start, he isn’t entitled to employ universal quantifier introduction applied to complex expressions involving T. Hence, he won’t be in a position to establish the quantified antedecent needed to make use of an instance of the induction scheme involving a predicate embedding T. In other words, Kurt is not entitled to make any use of the extended instances of induction allowed in T(PA). In sum, a suitably cautious Kurt – so far – is in no position to inductively inflate PA + Th.

And now there’s an added wrinkle. For we can see in retrospect that talk of the theory PA + Th in fact glosses over an issue that matters. In bolting the axioms Th onto the theory PA, were we intending these new axioms to interact merely with sufficient logical rules governing the elimination of quantifiers and the use of conditionals to enable the extraction of the information packaged in those axioms? Or were we intending that the whole weight of first-order logic can be brought to bear on axioms from either pool – so that we can trivially prove, for example, ∀x(T(x) ∨ ¬T(x))? The issue doesn’t arise for practical purposes. But now the question has been raised, we see that the second alternative overgenerates in enabling us to deduce more than we are entitled to in just being given the axioms Th as (partially) characterizing the new predicate T. A cautious Kurt should only use quantifier elimination and the conditional rules on Th.

Note, it is not being suggested that Kurt reject the instances of the induction schema that embed the predicate T as false. How can he? As the argument for inductive generosity reminds us, if the antecedents of such an instance are true, the consequent has to be true too. Rather, as we said, the point is that Kurt so far doesn’t find himself entitled to get to the starting line for using such an instance.

Of course, Kurt can now start to ‘think outside the box’. He can stand back from PA, think about his practice, commit himself explicitly to the thought that every sentence of L is either true or false, reflect this this thought using an arithmetized truth-predicate which he takes to be fully defined, so induction must apply to it — and so he comes to endorse T(PA). We certainly don’t want to ban Kurt from such reflections or suppose that he must make some mistake if he takes on these further thoughts, and so comes to be able to demonstrate Con(PA), at least to his satisfaction. The point to emphasize is only that they are further thoughts, not commitments already implicity accepted in rationally endorsing PA in the first place. (Cf. Isaacson’s Thesis.)

We’re off!


Loads of apples
We’re off! The first logic lecture of the year done. No disasters. The data projector behaved. The last slide popped onto the screen with one minute to go. I talked mostly good sense. Bits of explanation had a beginning, middle and end — even sometimes in that order. Phew. After all these years, the first lecture is still nerve-racking. Piece of cake from now on.

But the lecture room doesn’t look much like that picture (of a scene that should really gladden the heart of Steve Jobs). I noticed just one laptop — odd, as I think a lot of our students have them, but they just haven’t yet got into the habit of bringing them en masse to classes here. But give it another year or two …

Giving it to ’em with both barrels

There’s a kind of stupidity that is, to put not too fine a point on it, morally offensive. I don’t mean common-or-garden dimness and/or a propensity for making daft mistakes (we’ve all been there!). I mean the “only I’m right and the rest of the world is wrong”, “there’s a conspiracy of mathematicians to cover over Cantor’s mistakes”, “Gödel proves that mathematics is self-contradictory” kind of stupidity that plagues the net and that is wilfully almost impervious to all reasoned response. So what do we do about the landslides of garbage that you find on newsgroups like sci.logic?

Well, let’s ignore a lot of it. But on the other hand, such newsgroups do get thousands of visitors a day and someone ought to stand up for the good name of logic! So I do think it is worth occasionally dropping by during an idle couple of minutes over a cup of coffee and giving the idiot du jour a blast or two with both barrels. A combination of argued refutation and more or less brutal mockery does (eventually) make many of the pretentious buffoons shut up. And if this means that, from time to time, even just a couple of students casually browsing past don’t get taken in by superficially plausible nonsense, then why not?

Some do crosswords, some do sudoku: as an alternative way of procrastinating, I can recommend a few bouts of idiot-bashing as mildly amusing fun. Though you can in fact learn a bit from doing it, and learn more from reading the contributions of others who are batting for the sanity-and-reason team. Give it a try!

Of making many books (again)

Michael Potter has just told me that the Schilpp volume on Michael Dummett is out. (Of course, the Library of Living Philosophers hasn’t been edited by Schilpp for nearly twenty years, but that’s how we still refer to the books, isn’t it?) So that’s another Amazon order winging its way to me. Nearly a thousand pages too. Gulp.

Suddenly there is bustle ….

After weeks when the faculty has been almost deserted apart from the admin staff, suddenly there is bustle in the philosophy grad. centre outside my room, and the place is filling up again. It is one of the main delights of being in Cambridge that we have such bright graduate students, and I really rather miss the logicians when they are not around. Sadly, it’s not obvious where the successors of the current crop are going to come from, as none of this year’s M.Phil. intake seems inclined to go in the direction of the serious stuff. But Michael (Potter) and I will try to get a convert or two …

Anyway, this term’s teaching for me ought to be a lot of fun. First year logic lectures (trying to enthuse even the symbol-phobic to work through my intro book); third year Gödel’s Theorems lectures (chatting about themes in my Gödel book); the Math Logic Reading Group (this year is model theory year: we are kicking off with Manzano’s book as revision, then aim to do Hodges’s Shorter Model Theory); Michael and my Logic Seminar (phil. logic this term — Davidson and Dummett revisited); and a handful of modal logic lectures. I can have no complaints about that!

ACA0, #4: And what about ACA?

I’ve been trying to get my head around the significance of (i) the relation between ACA0 and ACA (the theory you get by keeping arithmetic comprehension, but allowing induction for any second-order wff) and (ii) the relation between ACA and T(PA) (the theory you get by adding to first-order PA a Tarski-like truth-theory AND allowing the new truth-predicate to appear in induction axioms). The technicalities hereabouts are reasonably clear: but — as I say — their philosophical significance is not. So watch this space.

We might be tempted by the following argument for being generous with induction in moving from (i) ACA0 to (ii) ACA, or indeed in moving from (i’) PA + T, the result of taking PA and bolting on a truth-theory “from outside” while leaving the induction axioms untouched, to (ii) T(PA):

Instances of induction are conditionals, telling us that from F(0) and (Ax)(Fx -> Fx’) we can infer (Ax)Fx. So we can derive(Ax)Fx. when we have already established the corresponding premiss (i) F(0) and can also establish (ii) (Ax)(Fx -> Fx’). But if we can already establish (i) and (ii) then trivially we can (iii) case by case derive each and every one of $F(0), F(1), F(2), … However, there are no ‘stray’ numbers which aren’t denoted by some numeral; so that means (iv) that we can show of each and every number that F holds of it. What more can it possibly take for F to express a genuine property that indeed holds for every number, so that (v) (Ax)Fx is true? In sum, it seems that we can’t possibly overshoot by allowing instances of the Induction Schema for any open wff of the language we are working with. The only usable instances from our generous range of axioms will be those where we can in fact establish the antecedents (i) and (ii) of the relevant conditionals — and in those cases, we can’t go wrong in accepting the consequents (v) too.

This argument for inflating ACA0 to ACA, or PA + T to T(PA) — and thereby getting to be able to prove new arithmetical sentences like Con(PA) — is surely too easy! But saying exactly why isn’t so easy …

As I was saying, of making many books there is indeed no end!

This time a logic recommendation. I got in the post today a copy of the recently published paperback version of José Ferreirós, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics. History of mathematics is not at all my thing, but I got it as it seems very well regarded, and I wanted to have more sense of how we got to where we are with set theory. Having dipped into it over a couple of leisurely coffees in a cafe this afternoon, it seems very approachably written and intriguing. It is still jolly expensive; but at least make sure that your library has a copy.

Of making many books there is no end …

… and much study is a weariness of the flesh. Still, some study is a lot less wearisome, and some books a heck of a lot more fun, than the alternatives. So let me recommend Kate Belsey’s Why Shakespeare? as a good read, and really illuminating. I think she’s wrong about Bridget Jones’s big knickers, but then disagreeing is fun and illuminating too. However, you’ll have to see how Bridget gets into the story for yourself.

Atheists, quaffing wine

I mentioned two edited collections of articles in recent posts, Louise M. Antony’s Philosophers without Gods and Barry Smith’s Questions of Taste. Having now read them both, let me just very warmly recommend the first, but suggest that you might perhaps buy a moderately decent bottle of Rosso di Montalcino instead of forking out for the second … (in part because you’ll find a summary of some of the papers for free here).

The atheism volume is a serious affair, put together — it seems to me — with considerable tact and finesse, the papers (nearly all worth reading) arranged to draw in but then get under the guard of the sort of intelligent religious reader who might simply put up the shutters against (say) Dawkins’s more direct attack. Let’s hope that there is a cheap paperback in every student bookshop soon.

As to ‘the philosophy of wine’, there is indeed a nice piece by Barry himself but also some mildly daft provocations (Roger Scruton). Let me add a provocation of my own. A certain kind of English obsession with a somehow absolute category of “fine wine” seems to me an odd distortion — a thought prompted more than once in the past at college feasts, where indeed wonderful Bordeaux was served with frankly awful bland food, so the wine had to stand by itself and provide all the interest, making complexity a great virtue, while the food was little more than mere padding. I suspect an Italian, say, would find that imbalance, making the wine stand on its own, most peculiar — and think of excellence in wine as something much more relative to an occasion and a kind of meal and even the company.

Remembrance of photos past, #2

Putting a tracker here has certainly been very good for deflating any fantasies about the number of people who might read this blog or about why they arrive here. You dream that are oodles of logic enthusiasts out in the world. But ah no, people arrive having googled for “Tesco discrimination”, “cheap universities”, “donkey philosophy”, “JK Rowling eat your heart out” … and now “photos of Monica Vitti”. Heaven knows what you all make of it!

But so as not to disappoint at least the last contingent of surfers, here is Vitti again, with Alain Delon in L’Eclisse. An earlier time-slice of me used to think they were the epitome of cool; and in fact, I rather think I still do …

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