The four main paradoxes attributed to Eubulides, who lived in the fourth century BC, were 'The Liar', 'The Hooded Man', 'The Heap', and 'The Horned Man' (c.f. Kneale and Kneale 1962, p114).

The Horned Man is a version of the 'When did you stop beating your wife?' puzzle. This is not a simple question, and needs a carefully phrased reply, to avoid the inevitable come-back to 'I have not'. How is one to understand this denial, as saying you continue to beat your wife, or that you once did but do so no longer, or that you never have, and never will? It is a question of what the 'not', or negation means, in this case. If 'stopped beating' means 'beat before, but no longer', then 'not stopped beating' covers both 'did not beat before' and 'continues to beat'. And in that case 'I haven't' is an entirely correct answer to the question, if you in fact did not beat your wife. However, your audience might still need to be taken slowly through the alternatives before they clearly see this. Likewise with the Horned Man, which arises if someone wants to say, for instance, 'what you have not lost you still have'. In that case they will maybe have to accept the unwelcome conclusion 'I still have horns', if they admit 'I have not lost any horns'. Here, if 'lost' means 'had, but do not still have' then 'not lost' would cover the alternative 'did not have in the first place' as well as 'do still have' -- in which case what you have not lost you do not necessarily still have.

The Heap is nowadays commonly referred to as the Sorites Paradox, and concerns the possibility that the borderline between a predicate and its negation need not be finely drawn. We would all say that a man with no hairs on his head was bald, and that a man with, say, 10,000 hairs on his head was hirsute, i.e. not bald, but what about a man with only 1,000 hairs on his head, which are, say, evenly spread? It is not too clear what we should say, although maybe some would still want to say positively 'bald', while others would want to say 'not bald'. The learned treatment of this issue, in recent years, has been very extensive, with 'the lazy solution' not being the only one favoured, by any means. The lazy solution says that any lack of certainty about what to say is merely a matter of us not having yet decided upon, or even having the need to make up our mind about, a 'precisification' of the concept of baldness. There are objectors to this 'epistemic' way of seeing the matter, some of whom would prefer to think, for instance (see, e.g. Sainsbury 1995), that there was something essentially 'fuzzy' about baldness, so it is a 'vague predicate' by the nature of things, instead of just through lack of effort, or need.

The Hooded Man is about the concept of knowledge, and in other versions has again been much studied in recent years, as we shall see. In its original version the problem is this: maybe you would be prepared to say that you know your brother, yet surely someone might come in, who was in fact your brother, but with his head covered, so you did not know who it was. One aspect of this paradox is that the verb 'know' is ambiguous, and in fact is translated by two separate terms in several other languages than English -- French, for instance, has 'connâitre' and 'savoir'. There is the sense of 'being acquainted with', in other words, and the sense of 'knowing a fact about something'. Perhaps these two senses are inter-related, but distinguishing them provides one way out of the Hooded Man. For we can distinguish being acquainted with your brother from knowing that someone is your brother. Although you do not know it, you are certainly acquainted with the hooded man, since he is your brother, and you are acquainted with your brother. But that does not entail that you know that the hooded man is your brother, indeed, evidently you do not. We could also say, in that case, that you did not recognise your brother, for the notion of recognition is close to that of knowledge. And that points to another aspect of the problem, and another way of resolving the paradox -- showing, in addition, that there needn't just be one solution, or way out. Thus you might well be able to recognise your brother, but that does not require you can always do so, it merely means you can do better at this than those people who cannot do so. If we re-phrase the case: 'you can recognise your brother, but you did not recognise him when he had his head covered', then there is not really a paradox.

The last of Eubulides' paradoxes mentioned above was The Liar, which is perhaps the most famous paradox in the 'self-reference' family. The basic idea had several variations, even in antiquity. There was, for instance, The Cretan, where Epimenides, a Cretan, says that all Cretans are liars, and The Crocodile, where a crocodile has stolen someone's child, and says to him 'I will return her to you if you guess correctly whether I will do so or not' -- to which the father says 'You will not return my child'! Indeed a whole host of complications of The Liar have been constructed, especially in the last century, as we shall see. Now in The Cretan there is no real antinomy -- it may simply be false that all Cretans are liars; but if someone says just 'I am lying', the situation is different. For if it is true that he is lying then seemingly what he says is false; but if it is false that he is lying then what he is saying may seem to be true. A pedant might say that 'lying' was strictly not telling an untruth, but telling merely what one believes to be an untruth. In that case there is not the same difficulty with the person's remark being true: maybe he is indeed lying, although he does not believe it. The pedant, however, misses the point that his verbal nicety can be circumvented, and the paradox re-constructed in another, indeed many other forms. We shall look in more detail later at the paradox here, in some of its more complicated versions.

Before leaving the ancients, however, we can look at Zeno's Paradoxes, which not only have a logical interest in their own right, but also have a very close bearing on some paradoxes which appeared later, to do with infinity, and infinitesimals. Zeno's Paradoxes are primarily about the possibility of motion, but more generally they are about the possibility of specifying the units, or atomic parts, of which either space or time, or indeed any continuum may be thought to be composed.

For, Zeno argued (see, for instance, Owen 1957, and Salmon 1970), if there were such units then they would either have a size, or not have a size. But if they had a size we would have the paradox of The Stadium, while if they had no size we would have the paradox of The Arrow. Thus if runners A and B are approaching one another both at unit speed, then, supposing the units have a finite size, after one time unit they will have each moved one space unit relative to the stadium. But they will have moved two space units relative to each other, which implies that there was a time unit in between when they were just one space unit apart. So the time unit must be divisible after all. On the other hand, if the units of division have no size, then, at any given time, an arrow in flight must occupy a space just equal to itself -- for it cannot move within that time. But if so then it is at rest, and the arrow never moves.

That would seem to mean that space and time are divided without limit. But Zeno argued that if space and time were in themselves divided without limit then we would have the paradox of Achilles and the Tortoise. A runner, before he gets to the end of his race would have to get to the half-way point, but then also to the half-way point beyond that, i.e. the three-quarter-way point, and so on. There would be no limit to the sequence of points he would have to get to, and so there would always be a bit more to be run, and he could never get to the end. Likewise in a competitive race, even, say, between the super-speedy Achilles and a tortoise: Achilles would not be able to catch the tortoise up -- so long as the tortoise was given a start. For Achilles has first to get to the tortoise's original position, but by then the tortoise will be, however fractionally, further on. Now Achilles must always reach the tortoise's previous position before catching him up. Hence he never catches it up.

Aristotle had a way of resolving Zeno's Paradoxes which convinced most people until more recent times. Aristotle's resolution of Zeno's Paradoxes involved distinguishing between space and time being in themselves divided into parts without limit, and simply being divisible (by ourselves, for instance) without limit. No continuous magnitude, Aristotle thought, is actually composed of parts, since, although it may be divisible into parts without limit, the continuum is given before any such resulting division into parts. In particular, Aristotle denied that there could be any non-finite parts, and so is often called a 'Finitist': non-finite 'parts' cannot be parts of space or time, he thought, since no magnitude can be composed of what has no extension. This view came to be challenged later, since it means that an arrow can only be 'at rest' if it is at the same place at two separate times -- for Aristotle both rest and motion can only be defined over a finite increment of time. But later the notion of an instantaneous velocity came to be accepted, and that includes the case where the velocity is zero.

The puzzle about non-finite parts may remind one of the question which occupied many scholastic theologians in the Middle Ages: how many angels can sit upon a pin? And it is perhaps no accident that the theorist who gave the currently received answer to the general question of how many things without any extension make up a whole which has such an extension was a fervent believer in God. Certainly Aristotle's Finitism only stayed generally persuasive until the latter end of the nineteenth century, when the theorist in question, Cantor, specified the number of non-finite points in a continuum to most learned people's satisfaction.

I posit the case that you see your father coming from a distance, in such a way that you do not discern whether it is your father or another.

Then it is proved, because you do indeed know your father, and he is the one approaching; hence, you know the one approaching. Likewise, you know him who is known by you, but the one approaching is known by you; hence, you know the one approaching. I prove the minor, because your father is known by you and your father is the one approaching; hence, the one approaching is known by you.

The opposite is argued, because you do not know him of whom, if you are asked who he is, you will answer truly 'I do not know'. But concerning the one approaching you say this; hence etc.

Extended discussion of such cases may seem somewhat academic, but between Buridan's period, and more recent times, one notable figure started to bring out something of the larger importance of these issues. Indeed, quite generally, sophisms about the nature of change and continuity, about knowledge and its objects, and the ones about the notion of self-reference, amongst many others, have attracted a great deal of very professional attention, once their significance was realised, with techniques of analysis drawn from developments in formal logic and linguistic studies being added to the careful and clear expression, and modes of argument found in the best writers before. The pace of change started to quicken in the later nineteenth century, but the one earlier thinker who will also be mentioned here is Bishop Berkeley, who was active in the early eighteenth. For a history of this period, in connection with the issues which concerned Berkeley, see, for instance, Grattan-Guinness 1980. Berkeley's argument was with Newton about the foundations of the calculus; he took, amongst other things, a sceptical line about the possibility of instantaneous velocities.

It will be remembered that in the calculation of a derivative the following fraction is considered:

     f(x + x) - f(x) / x,

where x is a very small quantity.  In the elementary case where f(x) = x², for instance, we get

     (x + x)² - x² / x,

and the calculation goes first to

     2xx  + x²/ x,

and then to 2x + x, with x being subsequently set to zero to get the exact derivative 2x.  Berkeley objected that only if x was not zero could one first divide through by it, and so one was in no position, with the result of that operation, to then take x to be zero.  If it took x to be zero Newton's calculus, it seemed, required the impossible notion of an instantaneous velocity, which, of course, Aristotle had denied in connection with his analysis of Zeno's Paradoxes.  The point was appreciated to some extent elsewhere.  For the association between the derivative and motion, initiated by Newton's use of the term 'fluxion', was largely confined to England, and on the Continent, Leibniz' cotemporaneous development of the calculus had more hold.  And that involved the idea that the increment x was never zero, but merely remained a still finite 'infinitesimal'. 

One way of putting Aristotle's Finitism is to say that he believed that infinities, such as the possible successive divisions of a line, were only 'potential', not 'actual' -- an actual infinite division would end up with non-extensional, and so non-finite points. Leibniz, however, had no problem with the notion of an actual infinite division of a line -- or with the idea that the result could be a finite quantity. However, while Leibniz introduced finite infinitesimals instead of fluxions, this idea was also questioned as not sufficiently rigorous, and both ideas lost ground to definitions of derivatives in terms of limits, by Cauchy and Weierstrass in the nineteenth century. Leibniz' notion of finite infinitesimals in fact has been given a more rigorous definition since that time, by Abraham Robinson, and other proponents of 'non-standard analysis', but it was on the previous, nineteenth century theory of real numbers that Cantor worked, before he came to formulate his theory of infinite numbers. Leibniz would not have thought it too sensible to ask how many of his infinitesimals made up the line, but Cantor made much more precise the answer 'infinitely many'.

It is necessary to get some idea about the theory of real numbers before we can understand the next logical paradoxes which emerged in this tradition: Russell's Paradox, Burali-Forti's Paradox, Cantor's Paradox, and Skolem's Paradox. We will look at those in the next section, which will then lead us into twentieth century developments in the area of self-reference. But before all that it should be mentioned how recent discussions of knowledge and its objects, for instance, has become very professionalised, since developed discussion of issues to do with Eubulides' Hooded Man has been just as dominant in this period.

{x|Fx},

and to say a is F, i.e. Fa, would then seem to be to say that a belonged to this set, i.e.

a {x|Fx},

where the symbol "" represents "is a member of".

It therefore seems plausible to enunciate this as a general principle,

for all y: y {x|Fx} if and only if Fy,

which is symbolised in contemporary logic,

(y)(y {x|Fx} iff Fy).

But if the result held for all predicates 'F', then we could say, for any 'F'

there is a z such that: (y)(y z iff Fy),

which is now formalised

(z)(y)(y z iff Fy).

Whatever involves all of a collection must not be one of a collection, or, conversely, if, provided a certain collection had a total it would have members only definable in terms of that total, then the said collection has no total.

A more recent theorist to oppose Ramsey's distinction has been Priest. In fact he has tried to prove that all the main paradoxes of self-reference have a common structure using a further insight of Russell's, which he calls 'Russell's Schema' (Priest 1994, p27). This pre-dates Russell's attachment to the Vicious Circle Principle, but Priest has shown that, when adapted and applied to all the main paradoxes, it matches the reasoning which leads to the contradiction in each one of them. This approach, however, presumes that semantical notions, like definability, designation, truth, and knowledge can be construed in terms of mathematical sets, which seems to be really the very supposition which Ramsey disputed.

Grelling's Paradox also makes this supposition questionable. It is a self-referential, semantical paradox resembling, to some extent, Russell's Paradox, and concerns the property which an adjective has if it does not apply to itself. Thus

'large' is not large,
'multi-syllabled' is multi-syllabled,
'English' is English,
'French' is not French.

'x' is heterological if and only if 'x' is not x.

Copi first introduces the definition

Hs =_df (F)(sDesF&(P)(sDesP iff P=F) & ¬Fs),

in which '¬' abbreviates 'not', and 'Des' refers to the relation between a verbal expression and the property it designates. Thus 'sDesF' reads: s designates F. Copi's proof of the contradiction then goes in the following way. First, H'H' entails in turn

(F)('H'DesF&(P)('H'DesP iff P=F)&¬F'H') - by substitution in the definition,

'H'DesF&(P)('H'DesP iff P=F)&¬F'H' - by taking the case thus said to exist,

('H'DesH iff H=F)&¬F'H' - by substitution in the 'for all P',

H=F&¬F'H' – by assuming 'H' designates H,

(F)¬('H'DesF&(P)('H'DesP iff P=F)&¬F'H') - since '¬(F)' is equivalent to '(F)¬',

¬(H'DesH&(P)('H'DesP iff P=H)&¬H'H') - substituting 'H' for 'F',

¬((P)('H'DesP iff P=H)&¬H'H') - assuming 'H' designates H, H'H' - assuming (P)('H'DesP iff P=H).

To get the contradiction

H'H' iff ¬H'H',

therefore, one has to be assured that there is one and only one property which 'H' designates. And Copi gives no proof of this.

The Liar Paradox is a further self-referential, semantical paradox, perhaps the major one to come down from antiquity. And one may very well ask, with respect to

What I am now saying is false,

What I am now saying is false, or meaningless.

Let us, therefore, look at some other notable ways of trying to escape even the Unstrengthened Liar. The Unstrengthened Liar comes in a whole host of variations, for instance:

This very sentence is false,

Some sentence in this book is false,

The following sentence is false,

The previous sentence is true;

What Plato is saying is false,

What Socrates is saying is true,

The semantical concepts in these paradoxes are truth and falsity, and the first major contribution to our understanding of these, in the twentieth century, was by Tarski. Tarski took truth and falsity to be predicates of sentences, and discussed at length the following example of his famous 'T-scheme':

'snow is white' is true if and only if snow is white.

Ts iff p,

The main source of the difficulties met with seems to lie in the following: it has not always been kept in mind that the semantical concepts have a relative character, that they must always be related to a particular language. People have not been aware that the language about which we speak need by no means coincide with the language in which we speak. They have carried out the semantics of a language in that language itself and, generally speaking, they have proceeded as though there was only one language in the world. The analysis of the antinomies mentioned shows, on the contrary, that the semantical concepts simply have no place in the language to which they relate, that the language which contains its own semantics, and within which the usual laws of logic hold, must inevitably be inconsistent.

s = '¬Ts',

T'¬Ts' iff ¬Ts,

Ts iff ¬Ts.

s = '¬Ts'

It seems difficult to apply this kind of stratification of languages to the way we ordinarily speak, however. Indeed, to assert that truth can attach to indexical sentences, like 'What I am now saying is false', would seem to be flying in the face of a very clear truth (Kneale 1972, p234f). Consider, further, this variation of the Plato-Socrates case above (c.f. Haack 1978, p144), where Jones says

All of Nixon's utterances about Watergate are false,

All of Jones' utterances about Watergate are true.

Martin has produced a typology of solutions to the Liar which locates Tarski's way out as one amongst four possible, general diagnoses (Martin 1984, p4). The two principles which Martin takes to categorise the Liar we have just seen, namely

(S) There is a sentence which says of itself only that it is not true,

(T) Any sentence is true if and only if what it says is the case.

We can isolate a further, fifth option, although Martin does not consider it. That option is to hold that both (S) and (T) are incorrect, as is done by the tradition which holds that it is not sentences which are true or false. One cannot say, for instance, that the sentence 'that is white' is true, in itself, since what is spoken of might vary from one utterance of the sentence to another. Following the second world war, because of this sort of thing, it became more common to think of semantical notions as attached not to sentences and words, but to what such sentences and words mean (Kneale and Kneale 1962, p601f). On this understanding it is not specifically the sentence 'that is white' but what is expressed by this sentence, i.e. the statement or proposition made by it, which may be true. But it was shown by Thomason, following work by Montague, that the same sorts of problems can be generated even in this case. We can create self-referential paradoxes to do with statements and propositions which again cannot be obviously escaped (Thomason 1977, 1980, 1986). And the problems are not just confined to the semantics of truth and falsity, but also arise in just the same way with more general semantical notions like knowledge, belief, and provability. In recent years, the much larger extent of the problems to do with self-reference has, in this way, become increasingly apparent.

Asher and Kamp sum up (Asher and Kamp 1989, p87):

Thomason argues that the results of Montague (1963) apply not only to theories in which attitudinal concepts, such as knowledge and belief, are treated as predicates of sentences, but also to 'representational' theories of the attitudes, which analyse these concepts as relations to, or operations on (mental) representations. Such representational treatments of the attitudes have found many advocates; and it is probably true that some of their proponents have not been sufficiently alert to the pitfalls of self-reference even after those had been so clearly exposed in Montague (1963)... To such happy-go-lucky representationalists, Thomason (1980) is a stern warning of the obstacles that a precise elaboration of their proposals would encounter.

Thomason's argument is, at least on the face of it, straightforward. He reasons as follows: Suppose that a certain attitude, say belief, is treated as a property of 'proposition-like' objects - let us call them 'representations' - which are built up from atomic constituents in much the way that sentences are. Then, with enough arithmetic at our disposal, we can associate a Gödel number with each such object and we can mimic the relevant structural properties of and relations between such objects by explicitly defined arithmetical predicates of their Gödel numbers. This Gödelisation of representations can then be exploited to derive a contradiction in ways familiar from the work of Gödel, Tarski and Montague.

Gödel's famous theorems in this area are, of course, concerned with the notion of provability, and they show that if this notion is taken as a predicate of certain formulas, then in any standard formal system which has enough arithmetic to handle the Gödel numbers used to identify the formulas in the system, certain statements can be constructed which are true, but are not provable in the system, if it is consistent. What is also true, and even provable in such a system is that, if it is consistent then (a) a certain specific self-referential formula is not provable in the system, and (b) the consistency of the system is not provable in the system. This means the consistency of the system cannot be proved in the system unless it is inconsistent, and it is commonly believed that the appropriate systems are consistent. But if they are consistent then this result shows they are incomplete, i.e. there are truths which they cannot prove.

The paradoxical thing about Gödel's Theorems is that they seem to show that there are things we can ourselves prove, in the natural language we use to talk about formal systems, but which a formal system of proof cannot prove. And that fact has been fed into the very large debate about our differences from, even superiority over mechanisms (see e.g. Penrose 1989). But if we consider the way many people would argue about, for instance,

this very sentence is unprovable,

If that sentence is provable then it is true, since provability entails truth; but that makes it unprovable, which is a contradiction. Hence it must be unprovable. But by this process we seem to have proved that it is unprovable - another contradiction!

The intractability of the impasse here, and the failure of many great minds to make headway with it, has lead some theorists to believe that indeed there is no escape. Notable amongst these is Priest (c.f. Priest 1979), who believes we must now learn to accept that some contradictions can be true, and adjust our logic accordingly. This is very much in line with the expectation we initially noted Quine had, that maybe 'some tacit and trusted pattern of reasoning must be made explicit and henceforward be avoided or revised'. The particular law which 'paraconsistent' logicians mainly doubt is 'ex impossibile quodlibet', i.e. 'from an impossibility anything follows', or

(p&¬p) q.

It seems, however, that, if one creates what might be called 'infinite chains' then there is not even this attenuated form of self-reference (though see Beall, 2001). Yablo asked us to consider an infinite sequence of sentences of which the following is representative (Yablo 1993):

(S_i) For all k>i, S_k is untrue.

Sorensen's 'Queue Paradox' is similar, and can be obtained by replacing 'all' by 'some' here, and considering the series of thoughts of some students in an infinite queue (Sorensen 1998). Suppose that, in Yablo's case, S_n is true for some n. Then S_n+1 is false, and all subsequent statements; but the latter fact makes S_n+1 true; giving a contradiction. Hence for no n is S_n true. But that means that S₁ is true, S₂ is true, etc; in fact it means every statement is true, which is another contradiction. In Sorensen's case, if some student thinks 'some of the students behind me are now thinking an untruth' then this cannot be false, since then all the students behind her are thinking the truth - although that means that some student behind her is speaking an untruth, a contradiction. So no student is thinking an untruth. But if some student is consequently thinking a truth, then some student behind them is thinking an untruth, which we know to be impossible. Indeed every supposition seems impossible, and we are in the characteristic impasse.

Gaifman has worked up a way of dealing with such more complex paradoxes of the Liar sort, which can end up denying the sentences in such loops, chains, and infinite sequences have any truth value whatever. Using 'GAP' for 'recognised failure to assign a standard truth value' Gaifman formulates what he calls the 'closed loop rule' (Gaifman 1992, pp225, 230):

If, in the course of applying the evaluation procedure, a closed unevaluated loop forms and none of its members can be assigned a standard value by any of the rules, then all of its members are assigned GAP in a single evaluation step.

This sentence is false, or has a GAP,

This sentence makes a false statement, or is a FA.