Can one distinguish between different types of self-referential paradoxes? Is it useful to do so?
In his essay 'The Ways of Paradox'1 Quine differentiates three different types of sentence commonly referred to with the title 'paradox'. Veridical paradoxes are sentences where we are initially surprised by an unexpected result, but are soon able to see from where the contradiction (quite rightly - hence the name) arises. An example of this sort is the barber paradox, where we are invited to consider a village where there is a barber who shaves all and only those men who do not shave themselves. We conjure up an image of a place where some of the male-population shave themselves, and thus do not need to be shaved again, but where no man has a beard, since the barber shaves all those who haven't done it themselves. The problem arises when we consider the barber - it seems likely that he would shave himself, being a barber and everything, but if he did then he would come under the category of people that the barber doesn't share (namely people that shave themselves), so he doesn't. But since he doesn't, he is one of the people that he should shave (someone who isn't going to get a shave any other way), and so he does, so he doesn't, so he does, ad infinitum. Quine suggests that, having reasoned through the example, we can see that the logic is valid, and are forced simply to deny that such a village, or at least such a barber, could exist.
The second sort of paradox, falsidical ones are those, such as fallacious proofs that 2 = 1, where we get a surprise, but are able to isolate a trick - and through examination to discover that the argument is invalid, and show just where the problem arises. For example, the mathematical examples tend to rely at some point on a division by 0 (which can have any result), and thus ultimately fail to convince anyone of their purported conclusion.
The third type are antinomies - and these are the sort that produce the most active debate. Antinomies include (many variations of) the liar paradox, Grelling's paradox, and Russell's paradox. The liar paradox is perhaps the most famous paradox, and certainly the simplest, and most startling. Thought to originate from Epimenides' paradox2, the liars paradox can simply be the sentence 'This sentence is false'. If the sentence is true, then it is false, but if it is false then the sentence 'tells it how it is', and is thus true, and so false, and so on. Grelling's paradox, another particularly neat one, is that of whether 'heterological' is heterological itself. Autological is an adjective which describes words which, effectively, 'are what they themselves describe', and heterological describes just those things which are not autological - so 'The adjective 'short' is short; the adjective 'English' is English; the adjective 'adjectival' is adjectival; the adjective 'polysyllabic' is polysyllabic' - whereas 'long', 'German', 'verb' and 'monosyllabic' are heterological. But if 'heterological' were heterological, then it would describe itself, and thus be autological, and thus not describe itself and so be heterological, and so would describe itself.......
As I have said, antinomies are the focus of the majority of the debate, and whilst it is obvious to see why falsidical paradoxes might be relatively uninteresting for the philosopher, warning only of potential dangers within a system that could be abused to confuse an audience, but ultimately do not produce major problems, or require adjustments of our current thinking to avoid them. It is not clear, however, whether a separation of the sort Quine outlines will be successful between veridic paradoxes and antinomies. Let us consider one of the more complex, but also very powerful antinomies - that of Russell's Paradox - which produces a startling result, and has caused philosophers of mathematics to significantly alter their system.
Russell's Paradox involves classes. According to Frege's position on classes (into whose nature he made significant investigation), there is a class for any group of objects that might be described. So if I talk about 'every red object' there is a class populated by red objects; if we discuss 'three-legged dogs' we discuss the members of the class of three-legged dogs; 'cows' are simply those things that belong to the class of 'cows'. Classes are not limited to medium-sized objects, either - all 'happy thoughts' together form a set, 'ways of skinning a cat' would make up another, 'the odd numbers' would be an (enumerably infinite) other. Classes might even be filled with other classes - classes containing a single member make up the class of 'singletons'; we could imagine the class of 'empty classes'; a class of 'all classes that can be described in only a single way' would be possible, though perhaps likely to be empty. The last two examples are particularly important: we are happy for there to be empty classes, such as that of square circles, so these will not cause problems. Russell's paradox, however, is that of the class of all classes which do not belong to themselves: this class will belong to itself only if it does not.
This clearly causes problems - and whilst there would potentially be other avenues to investigate, the solution which is most widely accepted is to deny Frege's conception of a class - we must accept that it will not necessarily be the case that any description which would seem to lay out membership conditions will describe an actual class, since some descriptions are capable of describing classes with members that they both do and do not have. Whilst Russell's theory spelt the end of Frege's substantial project to reduce practically everything to classes, there seem to be few alternatives but to significantly alter our conception of classes - either by denying that any arbitrary description refers to a class, or by allowing for the (seemingly far more serious) possibility of things which are both members and non-members of the same class.
Let us look at some of the possible solutions to various of these paradoxes, and examine whether they support any fruitful separations of types of paradox, or not. Initially, it might seem obvious that there is some similarity between two of the solutions I have so far discussed - namely those of the barber paradox and Russell's paradox: with both of these we simply argue that the paradox probes the impossibility of the person or class (respectively) described.
Both these 'solutions', however, are somewhat lacking, and a philosopher should not be satisfied simply to accept them and move on. Clearly, as well as accepting that these things are impossible, we would like some explanation of why this should be so, and, furthermore (though this might be answered by the explanation) we would like to know just what group of things we are to abandon. To look at the second request first, there seem a number of constraints to what group we are to outlaw: clearly we do not want to be too strict - such as ruling out any talk of imagined barbers, so as to prevent the possibility of this particular one, since obviously there is much fruitful discussion of barbers; we do not want to be too weak, and fail to rule out all the things we want to - it's not use simply getting rid of talk of barbers, since we obviously need simply alter the example to that of a shoemakers that makes shoes for all and only those people that do not make their own shoes. One thing much ignored in the literature, however, is what I would call a 'fruitfulness recommendation' - we should, I believe, hope that our solution does not rule out exactly the problem cases we are currently worried about, and no more or less, since to do so would simply be to invite accusations, which might likely be correct, that we have produced no solution at all, just reasserted our dislike of certain problem cases. I feel that the 'fruitfulness recommendation' (which, it should be noted, is not a strict or infallible constraint) fits well with the requirement for an explanation. As an example of what I mean by this recommendation, as we shall see, one of the potential solutions to the liar paradox denies acceptable usage not just to the sentence 'This sentence is false', but also to its pair 'This sentence is true'. I would suggest that, rather than being a fault of this solution, 'fruitfulness' might well be a virtue, and we should at least accept the possibility that in explaining away certain of our paradoxes, we might well have to disallow certain other sentences from out language, in just the way that Russell's paradox shows us interesting things about the constraints of set theory, rather than simply telling us to rule out very specific instances of unpleasant class-definitions.
I shall consider a number of responses, all but one of which are outlined in Sainsbury's 'Paradoxes'3. The first paradox, and the one which seems much neglected (and does not appear in Sainsbury's appraisal of such responses) is, put simply a negative one. This response just argues that when we come across a paradox we should simply take its content to be false. The explanation (and motivation) for such a solution might be the argument that for something to be true is for it to correctly describe the world. Since the world cannot be correctly described by a paradox, they must be false. So, since we find that there is no state of affairs according to which 'This sentence is false' might be how the world is, the sentence simply can't be classified as true, and need say no more than this - even if, in not being true it would seem that the sentence is false, such a position fails to lay out what set of facts would make the sentence true. To elaborate in logical terms - to be true is to be true in an interpretation, and, since with a paradox there can be no consistent interpretation on which the sentiment of the paradox is true, it is false.
The first objection to this claim might be that paradoxes do not simply discuss truth - in fact most of the paradoxes I have used as examples have been about other notions. This, however, might not be a problem for the solution at hand - since it seems likely that most of these paradoxes can be unpacked into a sentence involving truth. When asked 'Is 'heterological' heterological?' we might ask 'Is the claim that 'heterological' is heterological true?' in which case we could reply that no, there is no interpretation, so consistent set of facts, that would make this proposition true, so 'heterological' is not heterological. Unfortunately, such a response does not seem terribly likely to be successful. It seems likely that, whilst we might claim to be able to unpack every sentence into one involving truth, it seems likely that there may often be more than one way of doing this, and each will produce a different result. If asked 'Is the claim that 'heterological' is autological true?' we might find ourselves having to respond that, no, it's not, since there is no consistent set of facts that would make that true. So 'heterological' is not autological, so, presumably it is heterological. The only way to avoid such a conclusion is to deny that the last deduction is correct - to argue that simply because 'heterological' is not autological does not make it 'heterological' - but to deny the possibility of binary opposites is to deny the law of the excluded middle, and we find this results in a collapse to another of the (widely debated) responses - that of the 'gap' response.
The 'gap' response argues, quite simply, that the truth value of a paradox is neither true nor false, but somewhere in between ('in the gap'). We should quickly think of intuitionistic logics, and, indeed, 'gap' theorists proposed a three-valued logic, whereby the content of a paradox is neither true nor false. So 'heterological' is neither heterological nor not heterological (autological); the class of all classes not belonging to themselves is neither in itself nor not in itself. Intuitionistic logic has, however, has a rough ride at best, and the arguments against it are all but conclusive, if only because we hit upon huge other problems if, when we find something is not 'true', it still isn't necessarily 'not true'. Intuitionistic logic, I would suggest, is at the very least a last resort, if not completely out of the question - only the most extreme of circumstances could motivate us to reject one of the most basic rules of our thinking - that of the law of the excluded middle.
A better response in my mind, however, is that of Tarski's. Tarski argues that 'true' should not be seen as a simple, single predicate, but as a group of terms, which might be labelled true1, true2, true3, etc., and that such terms belong only to a meta-language. Rather than 'true' being a predicate in a language itself, he argues that 'true' can only be applied to a language from outside - as a meta-linguistic device pointing to some property of a given sentence in the target language. When we find ourselves 'nesting truths' we must give each instance a label relative to its position, and the (meta-)language it is in. So when we state 'This sentence is false', or rather 'This sentence is not true', we are actually saying 'The sentence 'This sentence is not true1' is not true2', and clearly this is simply false. Such a response is certainly attractive, though suggestions have been made that Tarski is incorrect to suggest that this is already a property of the word 'true', and that in fact the word should be extended in just the way Tarski suggests to enable this possibility. I'm not sure where I stand on this claim, but I think I lean to Tarski's side - we clearly do have a conception of how the word 'true' is to be used, and such a conception does not include such labelling4, yet I would argue that such a conception does not extend to the sort of sentences up for debate. Clearly (I would argue) 'true', as it is used in the liar paradox, is some sort meta-linguistic device, in that it is talking not just about the reference of the words, but how the words are used - the debate is simply whether the meta-linguistic device can be expressed in the language it is commenting on. It seems quite possible that our common usage of the word 'true' does not extend to such unorthodox sentences, and that if we are to use it in these cases we might have to extend it. If we are to do so, it might be wise to extend it in just the way that Tarski has suggested, and thus form a complex set of truth-predicates.
How does this connect to other sorts of paradox? Well, a traditional response to Russell's paradox is his 'ramification' - the separation of sets into hierarchies. It is argued that a set can only contain members lower in the hierarchy, so that any set can contain individuals (roughly speaking, actual objects - anything described by a noun), then all but the bottom row can contain 'singletons' - sets with a single individual only, and so it continues, until we have sets of sets of sets. This would seem to be a neat parallel of Tarski's hierarchy of truth-predicates, translated into terms of a membership relation.
And at that point, since I'm rambling, tired, the essay is already far too long, and I'm feeling like I'm getting RSI, I shall leave the debate.1
W. V. (1966), The Ways of the Paradox, (Harvard University Press) pp. 1-182
Which, being a Cretan saying 'All Cretans are liars', in fact isn't a paradox at all, since it could be the fact that some are - including Epimenides - whereas some are not, or simply that Epimenides sometimes lies and sometimes does not, as those whom we are accustomed to call liars frequently do.3Sainsbury, R. M. (1988), Paradoxes, (CUP) Ch. 5 4If simply by virtue of the fact that no device in natural language seems to mimic the sort of sub-scripting that logicians use to disambiguate different uses of a word - natural language does not object to ambiguities, and expects the context to make it clear in some way (though whether this is possible is unclear).