Spandrels of truth

J.C. Beall interviewed by Richard Marshall.

JC Beall is a Harley-Davidson of logic and maths. Yet he wanders round ponds watching eastern bluebirds and dreaming of Tasmania whilst in his head fantastic theories emerge about truth, liars, vagueness, other worlds and the sinister threat of Pinocchio. Male nipples and spandrels are metaphors that he finds open up a whole road. His book Logic: The Basics, is designed so beginners can be spellbound by the beauty of logic whils’t his other books, Spandrels of Truth, Revenge of the Liar, and Liars and Heaps go into the very depths of philosophical paradox. And he’s co-edited other booksthat examine these labyrinths as well. He's as complex as Borges. Baddam!

3:AM: What made you become a philosopher? Were you always a philosophical type or did something change?

JC Beall: I grew up in a family in which (mostly Judeo-Xtn) theology was often a topic of conversation. The problem of evil, free will, cosmologies, many-vs-one, and so on were just natural things to think about -- regularly. But I also had a good mind for mathematics, logic, and science. What I do now -- viz., academic philosophy -- is a happy and, I think, natural combination of those features.

3:AM: You’re a huge figure in contemporary logic and philosophy of logic and maths. I was listening to Steven Strogatz, a top mathematician, who said that he’d never understood how anyone could not find maths intuitive until one day at University he was confronted with a problem that he couldn’t get. He said that it was only then that he finally understood why others found what he was doing really really hard and kind of scary. Has anything like that happened to you, whereby you experienced how difficult the logic and maths are to people like me, or is it a stretch for your imagination to understand my almost phobic attitude towards your subject matter?

JCB: Sure. Let me give a long-winded answer. I think that the analogy of eyesight is useful. As you look around you, make a concerted effort to see shapes and patterns. Within 15 feet or so, all of the shapes (e.g., chairs, or trees, or boxes, or windows, or etc.) are easy to see -- sharp, clear, etc. You can also see relations among those objects: the chair is beside the table; the tree is beyond the window; etc. But all still is sharp. As you let your gaze move beyond 15 feet to 30 feet, certain objects are still sharp, but perhaps not quite as easy to see as the ones closer to you. And now let your gaze go far beyond 50 feet, 100 feet, 150 feet, etc. Distinguishing among the objects is now hard -- except, perhaps, for those lucky few with absurdly good vision (or nice vision-enhancing gadgets).

In my experience, the mathematical-cum-logical realm is much the same. In fact, in my experience, the maths-cum-logic realm is much like the room I'm in or the lovely garden outside -- except that the maths-cum-logic realm lacks the color. But the parallel continues. Go back to the range at which objects look quite blurry -- hard to distinguish. Depending on what you're trying to see (but can't), frustration might compel you to an easy solution: transport yourself closer to the target objects! In the maths-cum-logic realm, such transportation is also available -- not by walking, but by learning. In the case of maths/logic, the learning is often a matter of mastering precise definitions, learning some facts -- so-called theorems -- and putting these things together in a way that moves you closer to what you're trying to see.

With the eyesight analogy in hand, I can now answer your question directly. I find it relatively easy to see a lot of the basic areas in elementary set theory (and related areas of algebra and topology). I'm not expert in those areas, but they come relatively easily. But what about the far reaches of some of these areas? Well, here is where Strogatz' experience is familiar to me. I am not afraid to look at the far reaches; I'm downright frustrated that my 'sight' (so to speak) just seems not up to the task. (And the frustration leads to practical questions: is it worth the time to 'move closer' via a lot of study? Probably not, but it would probably be quite fun. One of the biggest frustrations is the one shared by all of us: life is too damn short!)

Can I understand the 'almost phobic attitude' experienced by many? Yes, and I am genuinely saddened by the fact that the experience is so common. I truly believe that the basic ideas of logic (and, hence, maths) are ones that not only come naturally to most people but are also interesting to most people. The problem is that the material is often either presented in a way that makes the subject entirely boring (e.g., a bunch of dry old facts that seem to have little bearing on creative theorizing) or presented to be exceedingly difficult.

3:AM: One reason I was wondering about this is because you’ve written a very cool introduction to the subject, ‘Logic: The Basics’ where you seem optimistic that anyone will be comfortable with the subject. You make it sound creative and fun when you write, for example, ‘ Usually, when an idea initially seems too hard or abstract, a bit more thinking will eventually do the trick. My advice is that in times of initial difficulty you give the matter a bit more thought. Moreover, don’t just worry to master the given matters. Try hard to think about different logical options from those explicitly canvassed in the book… if a theory claims that the ‘right logic’ works this way, try to think about an alternative theory according to which the ‘right logic’ works that way.’

This suggests you think we can all do it if we just work harder. Is that right? And generally, not just you, but do you think the teachers of logic have thought hard enough about how to teach it to those who might not be as confident about their abilities in this area as themselves, being teachers who probably never or rarely felt de-skilled by the subject?

JCB: Thanks. Yes, I think that working a bit more on the problem -- e.g., going back and reviewing critical definitions, thinking of questions that are related to the problem at hand, etc. -- usually does the trick. (I do hope, however, that this isn't taken as a charge that those who are struggling are lazy!! I don't think that at all!)

On the pedagogy question: yes, I believe that many have thought hard about how to best teach logic to those who feel like they just don't have the requisite skills -- those who think themselves blind to the realm (to use the belabored analogy from above). Some folks think it best to construct models of (perhaps very boring) 'worlds', and then learn how to use formal languages to 'describe' such worlds. Once one is comfortable with formal languages (which 'talk about' our little constructed worlds), one then turns to the logical relations on such languages. For example, if all of the sentences in theory X are true about a world, does it 'have to be' that all of the sentences in theory Y are true about the given world? If so, we have a sort of logical relation holding among the sentences in X and Y, something approaching a notion of 'logical validity'.

This way of doing things is sometimes called 'model-theoretic'. One thinks first about the language-model (or language-world) relations – for example, such and so name 'Richard' denotes such and so object in such and so model, and such and so predicate 'writer' picks out such and so set of objects, etc. -- and then tries to explain logical relations on the language. This approach tends to be more intuitive for many people than a so-called proof-theoretic approach, which tries to ignore any hint of 'meaning' in the languages and defines logical relations in terms of, let me say, 'pure shape'.

But it's difficult to have a book -- or book-cum-computer program -- peel away already-existing phobias from those who, for whatever reason, think themselves ill-equipped to do logic, let alone enjoy it! This is where in-person teaching can make a huge difference, at least in my experience. But I'm no expert on teaching methods. All of this is armchair pedagogy -- though backed with some experience.)

3:AM: You’re written about truth, about the liar paradox and about vagueness and they all link up. But for the non-specialist some of this will be unfamiliar so I thought we’d start by discussing some of the ideas in your recent book first and then see how the ideas there help us untangle your approaches to other issues. ‘Spandrels Of Truth’ is a philosophy book written largely without the technical symbolic formulations and is about a theory of ‘transparent truth.’ So first of all, could you say what you mean by this term ‘transparent truth’? I really like your use of Aiehtela and Aiehtelanu to illustrate this version of deflationism. Could you say something about this?

JCB: Transparent truth theorists are deflationists about truth of a certain stripe. Common to all deflationists is the thought that the truth predicate wasn't brought into our language to name some important property. (Contrast the predicate 'tree', or 'standing', or many other predicates.) Instead, it was brought into our language as a bit of logical vocabulary -- to do some logical work: namely, the truth predicate affords us the effect of asserting infinitely long conjunctions (e.g., "A1 and A2 and A3 and A4 and … and …") even though we don't have infinitely long sentences in our language. (In particular, by saying "everything in theory X is true" you thereby say something that is equivalent to a would-be conjunction of all elements of theory X, even an infinite such 'conjunction' if X is infinite.) Transparent truth theorists think that applying the truth predicate to any sentence in our language results in a sentence equivalent to the original sentence -- that is, the two can be substituted for each other in any (sentential) context. (I ignore matters of so-called opaque contexts.)

One way to think about that is via a metaphor: God could uniquely (completely, accurately) specify our world without using 'true' at all. In other words, if you take the 'true'-free fragment of our language (the part of our language that doesn't contain 'true' or any equivalents), God could use that to completely describe our world -- to tell the entire truth about our world (without using 'true' or any equivalent). But we're in a different situation from God's. We don't have the time or resources to assert each sentence of long or complicated theories; and so we can use our device "is true" to have the same effect -- saying "everything in the given theory is true".

On the Aiehtela (-nu) story: this is just a way of making the point above. Instead of having "is true" in the language, we could instead have something like "is believed by Aiehtela", and then tell a story about Aiehtela: namely, that if Aiehtela's believing A implies A, and vice versa. (So, e.g., since grass is green, we infer that Aiehtela believes that grass is green. And if Aiehtela believes that NYC is south of LA, then NYC is south of LA. (It's not. Hence, Aiehtela doesn't believe it.)) Had we grown up with the story of Aiehtela instead of talk of "truth" etc., then we could use the "is believed by Aiehtela" predicate to do the logical, expressive work that we brought "true" into the language to do. Of course, had we grown up with such a story, you can very easily imagine that reasonable people might wonder about the "nature" of Aiehtela -- debating about the rich metaphysics of Aiehtela, etc. But just as such wondering would be unproductive, so too in the case of our actual expressive device (viz., "is true"): folks are ill-advised to think that there's more to truth (Aiehtela) than what's given in the logical behavior of the predicate.

A terminological note (perhaps only for those who already know a lot about deflationism etc.): Paul Horwich's minimalism about truth is not committed to the everywhere-intersubstitutability of T<A> and A for all sentences A of our language. This is why Horwich, though a pioneering figure in deflationary approaches to truth, is not a transparent-truth theorist (unlike, e.g., Field, me, perhaps Quineand Leeds, and other recent philosophers like Ole Hjortland, David Ripley, Lionel Shapiro, Elia Zardini, among others).

3:AM: So in your theory ‘truth’ just let’s us generalize. And you also accept principles of excluded middle and bivalence and see them as taking in each others washing. Is that right? You say that ‘an essential role of negation is to be exhaustive’ and this is important to you because you think that even though that’s the case, sometimes it fails to be exhaustive. So can you explain what this is getting at so that we start to see the motivation for the dialetheistic stuff?

JCB: Right. By "exhaustive", I just mean that negation cuts all of our sentences into the true and the not true. (If, contrary to my views, negation were also "exclusive", then it would cut the sentences in a non-overlapping way. But we have gluts, and so negation isn't exclusive in the given sense.)

A clarification is needed here, specifically on your point about negation "sometimes [failing] to be exhaustive". In the book ‘Spandrels of Truth’, I use a framework in which there are two different types of "worlds" -- the normal worlds and the abnormalworlds. The former can be thought of as logically possible (e.g., negation is always exhaustive there), and the latter logically impossible (e.g., negation may fail to be exhaustive there). So, on my view, negation is always exhaustive in our world and every logically possible one; but there are abnormal worlds where it fails. The dialetheic stuff comes into the picture because negation is exhaustive: with such exhaustive (perhaps exhausting!) behavior, contradictions emerge. For example, we have spandrels of truth like "This sentence is not true". Well, by Exhaustion, it's either true or not; but in either case, it will be both true and not. Hence, we have a glut -- though only at the 'semantic' level, nothing to worry about.

Of course, all of that is terribly complicated-sounding. In fact, the story involving different sorts of worlds, etc., is too complicated, or so I've come to think. This is why my latest work has gone back to a strong distinction between the logic of our language and the extra-logical work that wedo. I won't go into this too much here, but it is a fairly dramatic departure from the traditional path (certainly, the last 50 years or so) of non-classical-logic truth theories. In particular, I've come to think that much of what we took to be logical behavior (e.g., of the interaction between negation and disjunction and the logical consequences of their interactions) is in fact extra-logical behavior on our part. And while what I just said sounds cryptic and complicated, it ultimately seems (when filled out) to happily simplify much of the ugly complexities of some of my previous work -- but a detailed discussion of this would not be of interest to your readers, I suspect. (And to those precious few who might find the details interesting, I have papers on the matter that they can read.)

3:AM: So there are cases where negation fails to be exhaustive - vagueness and the semantic paradoxes which we’ll talk about in a minute – but just for now let's imagine that we have a case where negation fails – so then we get some options on the table don’t we? So maybe in these cases the proposition is both true and false, in a kind of glut, or maybe its neither true or false in a kind of gap, for example? Is this right? So what is your solution to this situation? Is it always the same solution or are there different ways of handling these cases depending on the case?

JCB: As above, it's misleading to say that there are places where negation fails to be exhaustive -- and I should've been clearer about that in the book. If we ignore other "worlds" (especially ignore the logically impossible ones), then there simply aren't any cases where negation fails to be exhaustive.

But you are right that if we reject that negation is (everywhere) exhaustive, then we have a new logical option -- actually, one that has long seemed attractive to many. In particular, if negation is not exhaustive, then (here setting aside some distinctions that Bas van Fraassen pointed out some time ago) we no longer have that every sentence is either true or false (or, for transparent-truth theorists, true or not true). In this case, perhaps we should say that a sentence like "this sentence is true" (which is not contradictory) is neither true nor false (i.e., one for which negation fails to exhaustively cut into one camp or the other); and this might be the right thing to say about certain cases of vagueness too. Maybe.

In fact, in my latest work, I'm very tempted to think that logic (hence, the logic of negation) is neutral on whether we have gluts or gaps. In particular, it's not a logicalmatter as to whether excluded middle or non-contradiction (by which, in this context, I mean "explosion" or "ex contradictione quodlibet" or "ex falso quodlibet") are valid; they simply don't hold in virtue of logic -- full stop. But whether a certain discourse or theory has "gaps" or "gluts" is a matter of extra-logicalmatters -- perhaps features of a given domain in question. But I shouldn't say too much more about this here, and these last remarks aren't my official doctrine (yet).

3:AM: So this is why you bring in the notion of spandrels isn’t it? Like male nipples, they are accidental byproducts and you think that the deflationary transparent notion of truth of your theory brings with it spandrels of language that cause headaches. Is that right? Why are they interesting and significant? Is your idea that studying them helps us understand language and thought?

JCB: The thought is a simple one. We have the truth predicate in our language not because it names some important property in the world, but simply to serve a practical need -- allowing us to express things that we couldn't otherwise express. (It is much like other logical vocabulary in this sense: it doesn't play any important explanatory role in our theories; it rather simply allows us to express our explanatory theories in a simple fashion and in a finite amount of time.) So, we bring in this see-through device -- our truth predicate -- to do this work.

But when you introduce something into an environment, there will be inevitable and often unintended byproducts of the introduced thing -- call these "spandrels" (to borrow a usage from Gould and Lewontin in biology). What are you going to do with the spandrels? Well, it depends. You could paint them; you could ignore them; you could spend the rest of your life thinking about what to do with them; etc. In the case of our see-through device -- our truth predicate -- the spandrels of the device (of truth) are odd sentences like "I am not true", which, given the rules of the device and logical framework of our language, are true if and only if they're not true. Contradiction. Now what?!

My answer, joining other so-called glut theorists (Asenjo, Mortensen, Priest, Routley, and more), is that we let them live their curious little lives. The difference between me and other glut theorists (at least one notable difference) is this: the paradox-driven contradictions don't call into question the very coherence of our semantic enterprise or any other sort of theoretical explanations. That's not what truth is doing anyway. Truth is just an expressive, logical notion -- not explanatory. We brought it in for expressive reasons alone. The question of the paradox-driven contradictions is not whether semantics is coherent or etc (a typical way to think of the matter, very nicely voiced by Vann McGeein the early 90s); the question is a practical one of utility. Do the contradictions get in the way of the device doing its job -- the expressive work for which we introduced the device?

If not, then there's no reason to worry about the contradictions. And here is where the tradition of paraconsistent logic finds its utility: we say that the logic of our language is paraconsistent, which nicely explains why the truth of some contradictions (viz., as I believe, only the spandrels of truth) doesn't imply the absurdity that all contradictions are true.

3:AM: You call your position a deflated dialetheisim. Graham Priest thinks it is a ‘straight version’ of his expansionist version. So where are you two differing and why do you find your version more satisfactory?

JCB: I'll stick to the ideas in the Spandrels book. The 'deflated dialetheism' has two parts: it takes truth to be transparent (deflated), as above; and it takes there to be gluts -- 'true contradictions'. But there's an important sense in which it is a deflated dialetheism: the gluts are one and all mere side-effects of our (in-principle dispensable but in-practice indispensable) 'semantic devices' -- e.g., truth, exemplification (or predication, if you like), etc. And since I think that this is the right account of truth (full stop), I think that anything less is thereby less than satisfactory (including Priest's version, though we agree on a lot of other matters). That's the long and short of it. But let me say a bit more by way of salient features of comparison and contrast.

The position I advance in the Spandrels book is in some ways more expansive than Priest's (and, I should say, that of Priest's cohort Routley/Sylvan) and in some (different) ways very 'straight'. The chief difference between Priest and me centers on truth: Priest thinks that truth is less than transparent, and in particular that truth doesn't commute with negation -- truth of negation is not equivalent to negation of truth.

(In other words, if T is our truth predicate, and <A> a name of sentence A, and ~ is negation, then a transparency theorist maintains that T<~A> and ~T<A> are everywhere intersubstitutable; but Priest thinks that they're not equivalent; he thinks that ~T<A>implies T<~A>but not vice versa.)

And this difference has one 'expansive' upshot. In particular, my theory -- as in Priest's and any other dialetheic theory -- contains a sentence of the form A&~A, and so also contains T<A&~A>. No problem. Further, in the logic that Priest and I both endorse (a logic that goes back to the brilliant Pittsburgh mathematician Florencio Asenjoin the mid-50s, and published in 1966, and later worked out quite extensively by Priest under the name 'LP'), if a contradiction A&~A is true, then so too is its negation ~(A&~A); and so my theory -- like Priest's -- contains the sentence T<~(A&~A)>. But, now, here's where my theory 'expands' in a way that Priest's doesn't: namely, since (on my view) truth commutes with negation, I'm not only committed to the original contradiction T(A&~A) and, in turn, T<(A&~A)&~(A&~A)>, but also to the 'new' one:

T<A&~A> & ~T<A&~A>.

Priest thinks that this sort of 'expansion' is too much, multiplying contradictions 'beyond necessity' (as Priest puts it somewhere). But there's no way around such 'expansion' on a transparency theory of truth. But I also don't see it as much of a problem. After all, on a transparency view, the truth predicate is just a see-through device; one is 'multiplying new contradictions' only in a very, very, very minimal sense -- they're one and all equivalent to each other. (Indeed, even the size of the theory doesn't increase, though this is for silly reasons having to do with the size of already-infinite theories.)

So, in one sense, my theory of truth, just in virtue of being a transparency theory, is more 'expansive' than Priest's: it contains some gluts that Priest's doesn't contain. But there's a more important, more interesting sense in which Priest's version of dialetheism -- not unlike one of the other Aussie pioneers of strong paraconsistency (viz. , Routley/Sylvan) -- is vastly more 'expansive'.

In particular, whereas I see gluts - 'true contradictions' - only as side-effects of our so-called semantic vocabulary (e.g., truth, exemplification, etc.), Priest sees them everywhere: gluts may emerge at any fragment of our language and reality. Indeed, Priest, following some pioneering work of Chris Mortensen, even thinks that arithmetic - arithmetic!!! - is inconsistent!

(Incidentally, I'm all for exploring so-called inconsistent theories of arithmetic; but I don't think that any of them are even possibly true theories of *arithmetic*.) And Priest is quite sanguine about gluts at all levels of the physical and metaphysical space -- though this is not to say that he actually believes that every level of reality is peppered with gluts. (But he -- unlike me -- doesn't see this as beyond possibility.)

3:AM: So it would be good perhaps to talk about some of the things that cause negation to fail. So you’ve written a load about the liar paradox and put a collection together of proposals about it in ‘Revenge of the Liar: New Essays on the Paradox’ . ‘This sentence is false’ is the kind of spandrelly assertion that gets us all puzzling. It’s a Godfather puzzle that works like the mafia: you think you’ve killed the problem, but it’s cousins pop up to get revenge. Can you explain this and how do you get rid of all family members?

JCB: I like the Godfather analogy. (That's a new image to associate with the Liar!) The thing that makes the Liar so tough is that it seems to many that it is irresolvable. Let me just give a rough idea of such thoughts. You might think of the 'basic liar' as working off of the following two principles:

Exhaustion: Every sentence is either true or false.

Exclusion: No sentence is both truth and false.

OK, what about the following sentence marked with "$"?

($) The sentence marked with "$" is false.

Now, the $-marked sentence is true if and only if the sentence marked "$" is false. By Exhaustion, the sentence is either true or false. But either way, the sentence is both true and false -- which conflicts with Exclusion. So, it seems that one or the other of these principles need to go. A very natural thought is to give up Exhaustion. After all, one wouldn't be crazy to think that a sentence like the $-marked sentence is a counterexample to Exhaustion -- that is, that it's neither true nor false. If one gives up Exhaustion, one thereby avoids the contradiction(s) arising from the $-marked sentence.

But now the problem of so-called revenge emerges. We can reformulate the true version of Exhaustion as something like

Exhaustion.1: Every sentence is either true or false or neither true nor false.

In turn, Exclusion gets tweaked too:

Exclusion.1: No sentence is both true and false; and if a sentence is neither true nor false, it's not true.

Consider a cousin of the $-marked sentence:

(#) The sentence marked with "#" is either false or neither true nor false.

And you can see the problem: exactly the same contradiction(s) that emerged with the $-marked sentence emerge with the #-marked sentence.

And the problem generalizes. In abstract, we have

Exhaustion: Every sentence is true or Other.

Exclusion: No sentence is both true and Other.

Now, say whatever you want about the category of Otherness -- this is supposed to hold the key to solving the Liar -- and you'll face the pattern often labelled "revenge". In particular, consider a sentence

(!) The sentence marked with "!" is Other.

This will generate the familiar Liar-like contradiction(s).

…And so the Liar seems to generate revenge-like patterns as you said. (Shoot down the one gangster, and another is knocking at your door -- to use your Godfather analogy.) That's the basic thought.

3:AM: And the other big puzzle, one that Graham Priest and Roy Sorensen think is even a bigger spandrel in the works than the liar, is vagueness. You collected different proposals about this issue in ‘Liars and Heaps: New Essays on Paradox’. So why are philosophers so impressed by the challenge of this particular paradox and how does your approach sort it out?

JCB: well, I agree that vagueness is harder than the semantic-cum-logical paradoxes. First, I've never thought that the Zermelo/Russell paradox (of a set of all sets that don't contain themselves) was ever a paradox for mathematics. Mathematicians can axiomatize away the paradox; and they did -- successfully, it seems. Is the given axiomatization (say, standard classical set theory) 'intuitively satisfying'? Well, I'm not sure; but I don't know why mathematics needs to be subject to our intuitions, which of course are built on rather ordinary cases. Kurt Godel, according to a brief report in a paper by Jon Myhill (mathematician), held the same sort of view: there was never a paradox for mathematics; the (e.g., Russell) paradox was a paradox for the (semantic) notion of 'exemplification' or 'predication'. And for *that* sort of paradox (which is similar if not exactly the same as the Liar), a paraconsistent solution has always struck me as very natural.

Vagueness is much harder. There's no real paradox there. After all, the conclusion of a sorites argument is not even remotely plausible (e.g., that every person in NYC is bald, etc.). But the 'pull' of some of the premises (to use nice terminology that Matti Eklund uses) of such arguments is very strong. And so something has to give.

My thoughts on vagueness continue to evolve, and I don't think that I have much of great interest to say on the matter. I have always found so-called paracomplete solutions (where negation fails to be exhaustive, excluded middle fails) to be very natural, and I continue to think in that direction -- despite having long ago rejected paracompleteness in general. At the moment, I am officially committed to some classical-logic solution or other -- perhaps somewhere in the Williamson or Sorensen vein (both extraordinary philosophers on the topic).

One thing that has never seemed right is to treat both the semantic-cum-logical paradoxes and vagueness-driven paradoxes in the same manner, at least if -- as I think -- the former involve gluts. (There are glut-theoretic philosophers who strongly disagree: David Ripley, Mark Colyvan, Zach Weber, Graham Priest, and Dominic Hyde-- to name a few! But let me set this aside.) One nice thing about paracomplete approaches to vagueness is that they do seem to allow a more natural unified solution to both sorts of paradox than one gets in the dialetheic case. (Perhaps one of the great pioneers along these lines is Vann McGee, and more recently the work of Hartry Field.)

3:AM: How does your theory track ordinary uses of truth and falsity (untruth)? Would you be interested if studies inx-phi showed that the folk did or didn’t adhere to your view of modest dialeithism?

JCB: Truth theories, especially those that take the logical structure of truth seriously, are invariably (let me say) 'rational reconstructions' of our (target) usage of 'true'. My theory is supposed to be in line with ordinary use of (at least one very important usage of) "true".

But to highlight which usage we're discussing, one needs to give the game away a bit. In particular, one needs to make clear that we're only talking about the usage of "true" for which the unrestricted rules of transparency (or rules in the vicinity) hold! Now, if folks immediately say "I don't recognize such a usage", then this would be surprising; but I've yet to encounter them -- and I don't know that the x-phi folk will either. But if I've simply made up such a usage, then this would be important to know -- and really disappointing, but still important to know.

Of course, the chief reason that people question whether we have a transparent truth predicate (or a truth predicate such that the familiar T-biconditionals hold for all sentences) are the paradoxes! And these are extraordinary cases, ones for which it would be hard to rely -- one way or the other -- on 'ordinary intuitions' (built from ordinary cases!). So, one way to proceed is to take the simplest picture: we have a transparent predicate (rather than a highly complicated one involving many restrictions to avoid paradox, etc.) and we want to figure out the simplest and most natural story about how we coherently have such a thing in our language (despite the inevitable paradoxes). And that's partly what I'm doing, and what I take other truth theorists to be doing.

3:AM: Hartry Field when discussing norms of rationality is revisionary. He holds that if people are committed to a metaphysical view of these norms they are in error and so meanings in those cases need replacing. But he also thinks that the revisions wouldn’t change everything about ordinary practice. It seems you’re saying something similar – we shouldn’t look for the metaphysical essence of truth because there isn’t one, they’re spandrels (like we shouldn’t look for the purpose of a male nipple) - but the error is not likely to be manifest often. Is that right?

JCB: Field is probably right about the topic you mention (norms of rationality); and I often find myself agreeing with him on a variety of topics (certainly truth, though not its logic). But I think that the case of norms of rationality is different from the topic of spandrels and the 'nature' of truth.

Field and I completely agree that you'll have better luck finding Hogwarts than you will an interesting or explanatory 'nature of truth'. I think that Field also agrees that the paradoxes may be seen as spandrels of the expressive device which is our truth predicate. Do such spandrels manifest themselves in nature? Nope. But this isn't to say that they -- the spandrels (viz., the paradoxes of truth) -- don't make a difference to our theory of reality. In particular, if you have a contradiction (e.g., "I'm true and not true") in your theory, and your theory is governed by a logic that can't handle contradictions in a sane fashion, then you’re done: your theory of reality will be absurd.

3:AM: How does all that you’ve discussed here link with logical pluralism? Why should logic be plural? Why can’t there be a unified logic handling all cases and contexts? I guess my thought is why you reject the approach of Williamson and Sorensen who seem very reluctant to accept the need for anything over and above a classical notion of logic?

JCB: I was lucky to have met my longtime friend and collaborator Greg Restall when I first moved to Australia in May 1998 (moved to Tasmania, which is still my absolute favorite place in the world, and to where I'd move back if the right opportunity emerged). Greg had ideas about logical pluralism, and we found it natural to work together -- and our papers and subsequent book were the result.

Let me take each of your questions about logical pluralism in turn.

How does all that we've discussed here link with logical pluralism? Well, pluralism, as Restall and I put it, is the view that there is more than one (deductive) consequence relation on our language; and, in particular, there is at least one paraconsistent such relation, in addition to the classical relation.

(NB: I'm a very weak pluralist, since I definitely reject that the classical consequence relation is a relation over our language. But let me skip these details, which can be seen in that book for those who are interested.)

If nothing else, this sort of pluralism motivates exploration of different logical systems and different hypotheses about what (e.g., non-classical) consequence relations our language may enjoy. If nothing else, logical pluralism fosters a good spirit of exploration.

Why should logic be plural? It's not that logic is plural (in the sense in question); it's rather that there are a plurality of logics on our language.

Why can't there be a unified logic handling all cases/contexts? We're happy to say that there is a unified logic in one sense: it is the intersection of all of the other logics (i.e., it's the relation you get by taking the stuff common to all of the relations). But this will be boringly weak.

Why isn't classical logic along the right (and sufficient) logic? Unfortunately, I've never found arguments that are telling in this context (again, setting aside my anti-classical arguments from the existence of gluts!). On one hand, I have been enormously impressed by the ideas that Williamson and Sorensen have come up with via their methodology -- stick to classical logic come what may. This approach has simply never seemed natural or right to me, but I continue to be amazed -- awe-struck at times -- by the richness of ideas that the approach has yielded, particularly in the hands of the two you mentioned, though others could be listed too. And actually, perhaps surprisingly, both Williamson and Sorensen could easily be logical pluralists along the lines that Restall and I sketch. Indeed, Restall himself is as classical -- and committed to the classical-logic approach -- as either Williamson or Sorensen. The pluralism is in many ways built for such folks -- giving them an entirely new set of tools to use.

3:AM: Why are dialetheists against Pinocchio? (I had to ask, it’s such a great title!)

JCB: We dialetheists are against scary-looking puppets of all sorts, particularly wooden ones. (I had to say it. It was such a great question!)

More seriously, Eldridge-Smith argued that the Pinocchio paradox (viz., Pinocchio says that his nose is growing) poses a serious problem for the sort of 'merely-semantic dialetheism' that I endorse (according to which the only gluts are those that arise in virtue of our semantic notions, not non-semantic notions like noses, etc.). I point out that if Pinocchio poses a problem, so too does the so-called Barber paradox, according to which there exists a barber who shaves all and only those who do not shave themselves. But for the Barber to be a problem, we'd need the actual existence of the Barber -- who (needless to say) doesn't in fact exist. So too with Pinocchio. (Actually, it's slightly more complicated that all this, but I'll skip the details.)

3:AM: That last paper was notable for its brevity. Your last book was going to be long but ended up short (relative to intentions at the beginning of the project). Do you think brevity is a virtue in philosophical argument and thought, and should be encouraged? I can think of many advantages – referees would be able to sail through stuff at speed, publishing stuff would be cheaper, time spent writing them would be shorter, they’d be less boring and so on. Blogging seems to reflect this, and you are a blogger. I guess this is a general question about the culture of philosophy now – is it changing because of technology and so on? Is this good?

JCB: Yes, brevity in philosophical writing should be encouraged -- widely and often (and then again, and again). Philosophy is hard. Publishing the result of one's philosophical work should be as streamlined as possible. When I look at a paper, or a book, I want to know: a) what's the issue? b) what's the thesis? c) what's the argument? If these can't be answered clearly and concisely, I grow suspicious of whether the author is clear on the answers herself.

Ed Gettier(famous philosopher for a 3pp paper called "Is justified true belief knowledge?") was one of my philosophical teachers, and he required clear, concise and correct papers. (Debate can occur over correctness, of course; but the debate is not fruitful if we don't have clarity, and it is not efficient if we don't have conciseness.) I learned from him just how hard it is to write clearly and, especially, concisely in philosophical work. It is really hard; but I think it is one of the most valuable features of good philosophical work. If you can clearly say what needs to be said in a concise fashion, then why not do so?

Of course, if one doesn't have time to edit things down to a clean, concise work, then so it goes. (I admit that my terribly long answers to your questions are the result of having no time to make them clean and concise. And I apologize to all readers who've made it this far!) And perhaps this touches on your blogging point. Blogs are there to allow for less than polished work, and I think that this is good. I myself don't have much time to read many philosophy blogs, but I think that they're a good thing, in general.

You say that I'm a blogger. I wish. A few years back I thought that a blog would be a great way to write down some of my many philosophical-cum-logical ramblings. (For years, I've carried a small notebook or some note cards or the like to jot down thoughts.) But I just did not have time *at all*, and it came to naught.

On your general culture question: I'm not sure whether the philosophical culture is changing in terms of pursuing brevity. I wish it would, because I truly believe that the resulting work would be a better product. This is one reason that I am very happy to support -- and excited about -- the new journal ‘Thought’, edited by Crispin Wright and John Divers and an exciting editorial board, and published by Wiley-Blackwell. This is in the tradition of ‘Analysis’ (a longstanding philosophy journal), and I hope, and now very much expect, that it will flourish in the same great tradition of ‘Analysis’: clear, concise, and -- let the debate begin -- correct papers in the max-10pp range (usually 3-5pp). But whether technology has anything to do with any of this is an interesting question -- on which I've got no guesses.

3:AM: Outside of this philosophical work, are there books, films, art and so on from outside philosophy and maths that have helped you?

JCB: Hmmm…. Certainly, there's a lot that has made life enjoyable -- music (classical, folk, rock, chants), films of all sorts (from silly to confrontational), good children's fiction and interesting sci-fi, and art. I also think that the beauty of nature -- I live between two lovely ponds close to a nice nature trail with lots of eastern bluebirds and more -- is very helpful to me. But I don't think that there's anything in non-philosophy or non-maths that has had direct bearing on my work. (But maybe I'm wrong.)

3:AM: And finally, for the rational truth seekers here at 3:AM: are there five books apart from your own, (which we’ll be dashing away to read straight after this) , that you could recommend that would help us further understand this cool philosophical area?

JCB: Hmmm…. Well, if you want some simple logical concepts and a sense of philosophical theorizing, I hope my ‘Logic: The Basics’ is useful. (Yes, you said "apart from your own", but this isn't gratuitous self-promotion; it's hard to point to entry-level ones that give a real taste for logical theorizing.) I think that Graham Priest's ‘Logic: A Short Intro’is similarly useful -- though not intended to go into things at even the elementary level of my Basics book. (And there are lots and lots of others that are slightly more advanced -- recent ones by Burgess, one by N. J. J. Smith, one by Restall, and the list goes on and on and on…) Unfortunately, the really cool philosophical ideas in logic require one to jump in and do some logic. And one can't really do that without starting at the beginning -- the sorts of books I mentioned.

But I can say this: almost any philosophical topic opens the door for logical theorizing! So, find any good book in philosophy -- be it in metaphysics, epistemology, mind, language, religion, science, art, love, whatever -- and you'll have some logical options to play with. Life is short. Ride safe, but enjoy it!

ABOUT THE INTERVIEWER
Richard Marshallis still biding his time.