Peter Eldridge-Smith

I distinguish paradoxes and hypodoxes among the conundrums of time travel. I introduce ‘hypodoxes’ as a term for seemingly consistent conundrums that seem to be related to various paradoxes, as the Truth-teller is related to the Liar. In this article, I briefly compare paradoxes and hypodoxes of time travel with Liar paradoxes and Truth-teller hypodoxes. I also discuss Lewis’ treatment of time travel paradoxes, which I characterise as a Laissez Faire theory of time travel. Time travel paradoxes are impossible according to Laissez Faire theories, while it seems hypodoxes are possible.

The Pinocchio paradox, devised by Veronique Eldridge-Smith in February 2001, is a counter-example to solutions to the Liar that restrict the use or definition of semantic predicates. Pinocchio’s nose grows if and only if what he is stating is false, and Pinocchio says ‘My nose is growing’. In this statement, ‘is growing’ has its normal meaning and is not a semantic predicate. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, it is not growing. ‘Because’ stands here for a non-semantic relation; it might be supposed to be causal or of some other nature, but it is not semantic. The paradox is discussed in relation to Tarski’s and Kripke’s theories of truth. Although the paradox is not necessarily a counter-example to a theory of a truth predicate, it is a problem for a theory of truth of the kind preserved by validity.

The Pinocchio paradox poses one dialetheia too many for semantic dialetheists (Eldridge-Smith 2011). However, Beall (2011) thinks that the Pinocchio scenario is merely an impossible story, like that of the village barber who shaves just those villagers who do not shave themselves. Meanwhile, Beall maintains that Liar paradoxes generate dialetheia. The Barber scenario is self-contradictory, yet the Pinocchio scenario requires a principle of truth for a contradiction. In this and other respects the Pinocchio paradox is a version of the Liar, unlike the Barber. One wonders why some Liars would be impossible if others generate dialetheias

... and a reading knowledge of formal logical symbolism is essential too. (Philosophers often use bits of logical symbolism to clarify their arguments.) Because the artificial and simply formal languages of logic give us highly illuminating objects of comparison when we come thinking about how natural languages work. (Relevant to topics in ‘philosophical logic’ and the philosophy of language.) But mainly because it us the point of entry into the study of one of the major intellectual achievements by philosophers of the 20th – i.e. the development of mathematical logic. (Not least in Cambridge: Russell and Whitehead, Ramsey, Turing.)

Many of our concepts are introduced to us via, and seem only to be constrained by, roughand-ready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (from numbers to numbers) – i.e. one whose value for any given input can be determined by a step-by-step calculating procedure, where each step is fully determined by some antecedently given finite set of calculating rules. We are told that we are to abstract from practical considerations of how many steps will be needed and how much ink will be spilt in the process, so long as everything remains finite. We are also told that each step is to be ‘small’ and the rules governing it must be ‘simple’, available to a cognitively limited calculating agent: for we want an algorithmic procedure, step-by-minimal-step, to be idiot-proof. For a classic elucidation of this kind, see e.g. Rogers (1967, pp. 1–5). Church’s Thesis, in one form, then claims this informally explicated concept in fact has a perfectly precise extension, the set of recursive functions. Church’s Thesis can be supported in a quasi-empirical way, by the failure of our searches for counterexamples. It can be supported too in a more principled way, by the observation that different appealing ways of sharpening up the informal chararacterization of algorithmic computability end up specifying the same set of recursive functions. But such considerations fall short of a demonstration of the Thesis. So is there a different argumentative strategy we could use, one that could lead to a proof? Sometimes it is claimed that there just can’t be, because you can never really prove results involving an informal concept like algorithmic computability..

Semantic dialetheists astutely dodge Explosion, the logical contagion of everything being true if a single contradiction is true. A dialetheia is contained in their semantics, and sustained by a paraconsistent logic. Graham Priest has shown that this is a solution to the Liar paradox. I use the Pinocchio paradox, devised by Veronique Eldridge-Smith, as a counter-example. The Pinocchio paradox turns on the truth of Pinocchio, whose nose grows if and only if what he is saying is not true, saying ‘My nose is growing’. It is not just a matter of interpretation whether Pinocchio’s nose is and is not growing.

odel’s Theorems (CUP, heavily corrected fourth printing 2009: henceforth IGT ). Surely that’s more than enough to be going on with? Ah, but there’s the snag. It is more than enough. In the writing, as is the way with these things, the book grew far beyond the scope of the lecture notes from which it started. And while I hope the result is still pretty accessible to someone prepared to put in the time and effort, there is – to be frank – a lot more in the book than is really needed by philosophers meeting the incompleteness theorems for the first time, or indeed by mathematicians wanting a brisk introduction. You might reasonably want to get your heads around only those technical basics which are actually necessary for understanding how the theorems are proved and for appreciating philosophical discussions about incompleteness. So you need a cut-down version of the book – an introduction to the Introduction! Well, isn’t that what lectures are for? Indeed. But there’s another snag. I haven’t got many lectures to play with. So either (A) I crack on at a very fast pace (hard-core mathmo style), cover those basics, but perhaps leave too many people puzzled and alarmed. Or (B) I do relaxed talk’n’chalk, highlighting the really Big Ideas, making sure everyone is grasping them as we go along, but inevitably omit important stuff and leave quite a gap between what happens in the lectures and what happens in the book. What to do? I’m going for plan (B). But then I ought to do something to fill that gap between lectures and book. Hence these notes. The idea, then, is to give relaxed lectures, highlighting Big Ideas, not worrying too much about depth or fine-detail (nor even about getting through all of the day’s intended menu of topics). These notes then expand things just enough, and give pointers to relevant chunks of IGT. Though I hope these notes will be to a fair extent be stand-alone, and tell a brief but coherent story read by themselves: so occasionally I’ll copy a paragraph or two from the book, rather than just refer to them..

Preface 1 The First Theorem revisited 1.1 Notational preliminaries 1.2 Definitional preliminaries 1.3 A general version of G¨ odel’s First Theorem 1.4 Giving the First Theorem bite 1.5 Generic G¨ odel sentences and arithmetic truth 1.6 Canonical and standard G¨ odel sentences 2 The Second Theorem revisited 2.1 Definitional preliminaries 2.2 Towards G¨ odel’s Second Theorem 2.3 A general version of G¨ odel’s Second Theorem 2.4 Giving the Second Theorem bite 2.5 Comparisons 2.6 Further results about provability predicates 2.7 Back to the First Theorem 2.8 Introducing Rosserized provability predicates..

In a reading group, we’ve been working through the first three parts of Field’s Saving Truth from Paradox, by the end of which he has presented his core proposals. At this point, we’ve now rather lost the will to continue – for this is an astonishingly badly written book, which makes ridiculous demands on the patience of even a sympathetic reader. It so happened that it fell to me to introduce the last two chapters in Part III, Ch. 17 in which Field rounds out his key technical construction, and Ch. 18, ‘What Has Been Done’. I talked mainly about the latter. Here, for what they are worth, are some very quickly written reflections on what has and hasn’t been achieved

Unlike his other major typescripts, the Big Typescript is divided into titled chapters, themselves divided into titled sections. But within a section we still get a collection of remarks typically without connecting tissue and lacking any transparently significant ordering or helpful signposting. So we still encounter the usual difficulties in trying to think our way through into what Wittgenstein might be wanting to say. Some enthusiasts like to try to persuade us that the aphoristic style is really of the essence. But I’ve always wondered how true that is. So I here propose an experiment. Let’s take §108 of the ‘Big Typescript’ (the first of the sections in the part of the book titled ‘The Foundations of Mathematics’). There are four and a bit pages of remarks. I’m going to try to render them into rather more conventional prose. In the section that follows, then, I have embedded almost every sentence Wittgenstein wrote in his §108 – or rather, of course, I’ve embedded almost every sentence from the well-regarded translation. I have, however, often changed the order in which remarks appear, omitted some ‘and’s and ‘but’s and the like, changed some punctuation, demoted some passages to footnotes, but not – I hope – in any way that radically changes the significance of any remark, or distorts what is going on. And I’ve added a lot of signposting. I’ll comment, necessarily very briefly, on the results of the exercise starting in §3 below. So here we go . .

There are two paradoxes of satisfaction, and they are of different kinds. The classic satisfaction paradox is a version of Grelling’s: does ‘does not satisfy itself’ satisfy itself? The Unsatisfied paradox finds a predicate, P, such that Px if and only if x does not satisfy that predicate: paradox results for any x. The two are intuitively different as their predicates have different paradoxical extensions. Analysis reduces each paradoxical argument to differing rule sets, wherein their respective pathologies lie. Having different pathologies, they are paradoxes of different kinds. Furthermore, each of these satisfaction paradoxes has an analogue with the same pathology in set theory. Therefore, these analogues are respectively of the same two kinds. This level of abstraction is significant in that it tracks two related but different pathologies. Thus, not all paradoxes of semantics and set theory share the same pathology: there are at least two kinds of paradox cutting across the semantic and set-theoretic distinction

At some step in proving the Liar Paradox in natural language, a sentence is derived that seems overdetermined with respect to its semantic value. This is complemented by Tarski’s Theorem that a formal language cannot consistently contain a naive truth predicate given the laws of logic used in proving the Liar paradox. I argue that proofs of the Eubulidean Liar either use a principle of truth with non-canonical names in a fallacious way or make a fallacious use of substitution of identicals. Tarski never committed the first fallacy and may have himself considered it fallacious. Nevertheless, I clarify that it is fallacious. I then argue substitution of identicals needs to be restricted within the scope of the truth predicate. A logic for truth implementing this restriction is a monotonic extension of a classical first order logic, or indeed a formalizable fragment of natural language. Proofs of Tarski’s Indefinability of Truth theorem are invalid in this logic. This approach generalizes to invalidate proofs of Liar-like paradoxes, particularly the predicate form of the Knower paradox. Consequently, such a logic can be further extended in a way that avoids Montague’s theorem for such a system. Yet, the semantic status of a Liar sentence is not fully resolved. It is no longer overdetermined; it is now underdetermined.

In the section ‘Further reading’, I listed a book that arrived on my desk just as I was sending IGT off to the press, namely Church’s Thesis after 70 Years edited by Adam Olszewski et al. On the basis of a quick glance, I warned that the twenty two essays in the book did seem to be of ‘variable quality’. But actually, things turn out to be a bit worse than that: the collection really isn’t very good at all! After I sent my book to press, I gave a paper-by-paper review on my blog, at http://logicmatters.blogspot.com. It is probably more fun to chase up the reviews ‘in the wild’, so to speak, starting from the entry for May 14, 2007. But here they are wrapped up into a single document, only marginally tidied. Some of the points made here should help further explain and support the general line on the Thesis taken in..

It seems that the Truth-teller is either true or false, but there is no accepted principle determining which it is. From this point of view, the Truth-teller is a hypodox. A hypodox is a conundrum like a paradox, but consistent. Sometimes, accepting an additional principle will convert a hypodox into a paradox. Conversely, in some cases, retracting or restricting a principle will convert a paradox to a hypodox. This last point suggests a new method of avoiding inconsistency. This article provides a significant example. The Liar paradox can be defused to a hypodox by relatively minimally restricting three principles: the T-schema, substitution of identicals and universal instantiation. These restrictions are not arbitrary. For each, I identify the source of a contradiction given some presumptions. Then I propose each restriction as a reasonable way to deal with that source of contradiction.

The concept of hypodox is dual to the concept of paradox. Whereas a paradox is incompatibly overdetermined, a hypodox is underdetermined. Indeed, many particular paradoxes have dual hypodoxes. So, naively the dual of Russell’s Paradox is whether the set of all sets that are members of themselves is self-membered. The dual of the Liar Paradox is the Truth-teller, and a hypodoxical dual of the Heterological paradox is whether ‘autological’ is autological. I provide some analysis of the duality and I search for modal hypodoxes as duals to known paradoxes. I also try to continue this search further by mapping the relations between some paradoxes and hypodoxes in duality squares. This investigation increases the known extension of the concept of hypodox, and the duality relation between paradoxes and hypodoxes assists in finding some of these.

Where to begin? I’ll take three books from my shelves. First, now nearly forty years old, a little book of television lectures by the great physicist Richard Feynman, The Character of Physical Law. He talks about the laws of motion, the inverse square law of gravitation, conservation laws, symmetry principles and the various ways these all hang together. Feynman obviously takes it that it is a prime aim of science to discover such laws. But what are laws? He writes – and this is about his one and only shot at a characterization at the level of abstraction that we might think of as philosophical –.

Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for a rather bumpy read. There is another route Parsons could have taken: he could have reprinted the relevant papers with postscripts, and then top-and-tailed the collection with a preface and added concluding reflections. It must sound very ungrateful, but I rather suspect that that might have worked better. Much of the book is about arithmetic. But Parsons has woven into the discussion claims about mathematics more generally and about set theory in particular. We might well have a basic worry about this structure: for do defensible claims about the ontology and epistemology of arithmetic have to be generalizable to apply to more infinitary mathematics? For example, suppose you are attracted to a Hellman-like modal structuralist account of arithmetic: then should you think it a problem if you suspect that such an account can’t readily be extended to cope e.g. with set theory (since you boggle at the idea of possible worlds free of abstracta but with enough structure to somehow model ZFC)? Parsons himself seems to waver over such questions. So for present purposes, I’ll focus just on arithmetic, and in what follows I’ll revisit two of the most familiar Parsonian themes, his views on structuralism as an account of the ontology of arithmetic, and his exploration of the role of intuition in grounding arithmetical knowledge