be driven towards by our reasoning, but which is highly
counterintuitive, nevertheless. There are, amongst these, a large
variety of paradoxes of a logical nature which have teased even
professional logicians, in some cases for several millennia. But what
are now sometimes isolated as "the logical paradoxes" are a much less
heterogeneous collection: they are a group of antinomies centered on
the notion of self-reference, some of which were known in Classical
times, but most of which became particularly prominent in the early
decades of last century. Quine distinguished amongst paradoxes such
antinomies. He did so by first isolating the "veridical" and
"falsidical" paradoxes, which, although puzzling riddles, turned out
to be plainly true, or plainly false, after some inspection. In
addition, however, there were paradoxes which "produce a
self-contradiction by accepted ways of reasoning," and which, Quine
thought, established "that some tacit and trusted pattern of reasoning
must be made explicit, and henceforward be avoided or revised." We
will first look, more broadly, and historically, at several of the
main conundrums of a logical nature which have proved difficult, some
since antiquity, before concentrating later on the more recent
troubles with paradoxes of self-reference. They will all be called
"logical paradoxes."
1. Classical Logical Paradoxes
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" (compare 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, that is 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. (For recent work in this area, see, for instance,
Williamson 1994, and Keefe 2001).
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
recognize 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, that is 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.
2. Moving to Modern Times
Between the classical times of Aristotle and the late nineteenth
century when Cantor worked, there was a period in the middle ages when
paradoxes of a logical kind were considered intensively. That was
during the fourteenth century. Notable individuals were Paul of
Venice, living towards the end of that century, and John Buridan, born
just before it. As models of the care, and clarity which is required
to extricate oneself from the above kind of difficulties with problem
propositions each of these writers will surely stand forever. As an
illustration, Buridan discusses "No change is instantaneous" in the
following way (Scott 1966, p178):
I prove it, because every change is either in an indivisible
instant or it is in a divisible time. But none is in an indivisible
instant, since an indivisible instant cannot be given in time, as is
always supposed. Hence every change is in divisible time, and every
such must be called temporal and not instantaneous.
The opposite is argued, because at least the creation of our
intellective soul is instantaneous. For since it is indivisible, it
must be made altogether at once, not one part after another. And such
creation we call instantaneous. Therefore.
Buridan also discusses "You know the one approaching," which resembles
Eubulides' Hooded Man (Scott 1966, p178):
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.
These two cases are "sophisms" in Buridan's book on such, Sophismata,
and amongst these, in chapter 8, are the "insolubles," which are the
ones involving some form of self-reference. Broadly speaking, that is
to say, Buridan made a distinction similar to that mentioned before,
between general paradoxes of a logical nature, and "the logical
paradoxes." Thus in his chapter 8 Buridan discusses Eubulides' Liar
Paradox in several forms, for instance as it arises with "Every
proposition is false" in the following circumstances (Scott 1966,
p191): "I posit the case that all true propositions should be
destroyed and false ones remain. And then Socrates utters only this
proposition: 'Every proposition is false'."
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 + δx) – f(x) / δx,
where δx is a very small quantity. In the elementary case where f(x) =
x2, for instance, we get
(x + δx)2 – x2 / δx,
and the calculation goes first to
2xδx + δx2/ δx,
and then to 2x + δx, with δx being subsequently set to zero to get the
exact derivative 2x. Berkeley objected that only if δx 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 δx to be zero. If it
took δx 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 δx 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.
These issues, it will be remembered, centred on the problem of
non-recognition, and in various ways two central cases of this have
been given close attention since the end of the nineteenth century. A
great deal of other relevant discussion has also gone on, but these
two cases are perhaps the most important, historically (see, for
example, Linsky 1967). First must be mentioned Frege's interest in the
difficulty of inferring someone believes something about the Evening
Star so long as they believe that thing about the Morning Star. In
fact the Morning Star is the same as the Evening Star, we now realise,
but this was not always recognised, and indeed it is now realised that
even the term "star" is a misnomer, both objects being the planet
Venus. Still someone ignorant of the astronomical identity, it may be
thought, might accept "The Evening Star is in the sky," but reject
"The Morning Star is in the sky." Quine produced another much
discussed case of a similar sort, concerning Bernard J. Ortcutt, a
respectable man with grey hair, once seen at the beach. In one
location he was taken to be not a spy, in another place he was taken
to be a spy, as one might say; but is that quite the best way the
situation should be described? Maybe one who does not recognise him
can have beliefs about the man at the beach without thereby having
those beliefs about the respectable man with grey hair — or even
Bernard J. Ortcutt. Certainly Quine thought so, which has not only
caused a large scale controversy in itself; it has also led to, or
been part of much broader discussions about identity in similar, but
non-personal, intensional notions, like modality. Thus, as Quine
pointed out, it would not seem to be necessary that the number of the
planets is greater than 4, although it is necessary that 9 is greater
than 4, and 9 is the number of the planets. A branch of formal logic,
Intensional Logic, has been developed to enable a more precise
analysis of these kinds of issue.
3. Some Recent Logical Paradoxes
It was developments in other parts of mathematics which were integral
to the discovery of the next logical paradoxes to be considered. These
were developments in the theory of real numbers, as was mentioned
before, but also in Set Theory, and Arithmetic. Arithmetic is now
taken to be concerned with a "denumerable" number of objects — the
natural numbers — while real numbers are "non-denumerable." Sets of
both infinite sizes can be formed, it is now thought, which is the
basis on which Cantor was to give his precise answer "two to the aleph
zero" to the question of how many points there are on a line.
The tradition up to the middle of the nineteenth century did not look
at these matters in this kind of way. For the natural numbers arise in
connection with counting, for instance counting the cows in a field.
If there are a number of cows in the field then there is a set of
them: sets are collections of such individuals. But with the beef in
the field we do not normally talk in these terms: "beef" is a mass
noun not a count noun, and so it does not individuate things, merely
name some stuff, and, as a result, a number can be associated with the
beef in the field only given some arbitrary unit, like a pound, or a
kilogram. When there is just some F then there isn't a number of F's,
although there might be a number of, say, pieces of F. It is the same
with continua like space and time, which we can divide into yards, or
seconds, or indeed any finite quantity, and that is perhaps the main
fact which supports Aristotle's view that any division of such a
continuum is merely potential rather than actual, and inevitably
finite both in the unit used and in the number of them in a whole.
But continua from Cantor onwards have been seen as composed of
non-finite individuals. And not only that is the change. For also the
number of individuals in some set of individuals — whether cows, or
the non-finite elements in beef — has been taken to be possibly
non-finite, with a whole containing those individuals being then still
available: the infinite set of them. We now commonly have the idea
that there may be infinite sets first of finite entities, which will
then be "countable" or "denumerable," but also there will be sets of
non-finite, infinitesimal entities, which will be "uncountable," or
"non-denumerable."
It is important to appreciate the grip that these new ideas had on the
late nineteenth century generation of mathematicians and logicians,
since it came to seem, as a result of these sorts of changes, that
everything in mathematics was going to be explainable in terms of
sets: Set Theory looked like it would become the entire foundation for
mathematics. Only once one has appreciated this expectation, which the
vanguard of theorists uniformly had, can one realise the very severe
jolt to that society which came with the discovery of Russell's
Paradox, and several others at much the same time, around the turn of
the century. For Russell's Paradox showed that not everything could be
a set.
If we write "x is F" as "Fx"—as came to be common in this same
period—then the set of F's is written
{x|Fx},
and to say a is F, that is Fa, would then seem to be to say that a
belonged to this set, that is
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).
In the foundations of Arithmetic which Frege described in his major
logical works Begriffschrift, and Grundgesetze, this principle is a
major axiom (Kneale and Kneale 1962, Ch 8), but Russell found it was
logically impossible, since if one takes for "Fy" the specific
predicate "y does not belong to y," that is "¬ y ∈ y" then it requires
(∃>z)(y)(y ∈ z iff ¬ y ∈ y),
wherefrom, given the above meanings of "(∃>z)" and "(y)", we get the
contradiction
z ∈ z iff ¬ z ∈ z,
that is z is a member of itself if and only if it is not a member of
itself. As a result of this paradox which Russell discovered, the
theory of sets was considerably altered, and limits were put on
Frege's axiom, so that, for instance, either it defined merely subsets
of known sets (Zermelo's theory), or allowed one to discriminate sets
from other entities — usually called "proper classes" (von Neumann's
theory). In the latter case those things which are not members of
themselves form a proper class but not a set, and proper classes
cannot be members of anything.
But there were other reasons why it came to be realised that sets
could not always be formed, following the discovery of Burali-Forti's
and Cantor's Paradoxes. Burali-Forti's Paradox is about certain sets
called "ordinals," because of their connection with the ordinals of
ordinary language, that is "first," "second," "third," etc. The sets
which are ordinals are so ordered that each one is a member of all the
following ones, and so, with no limit envisaged to the sets which
could be formed, it seemed possible to prove that any succession of
such ordinals would themselves be members of a further ordinal – which
would have to be distinct from each of them. The trouble came in
considering the totality of all ordinals, since that would mean that
there would have to be a further distinct ordinal not in this
totality, and yet it was supposed to be the totality of all ordinals.
A very similar contradiction is reached in Cantor's Paradox.
For, for finite sets of finite entities it is easy to prove Cantor's
Theorem, namely that the number of members of a set is strictly less
than the number of its subsets. If one forms a set of the subsets of a
given set then one produces the "power set" of the original set, so
another way of stating Cantor's Theorem is to say that the number of
members of a set is strictly less than the number of members of its
power set. Cantor extended this theorem to his infinite sets as well –
although there was at least one such set he realised it obviously
could not apply to, namely the set of everything, sometimes called the
universal set. For the set of its subsets clearly could not have a
greater number than the number of things in the universal set itself,
since that contained everything. This was Cantor's Paradox, and his
resolution of it was to say that such an infinity was "inconsistent,"
since it could not be consistently numbered. He thought, however, that
only the size of infinite sets had to be limited, assuming that lesser
infinities could be consistently numbered, and nominating, for a
start, "aleph zero" as the number, or more properly the "power" of the
natural numbers (Hallett 1984, p175). In fact an earlier paradox about
the natural numbers had suggested that even they could not be
consistently numbered: for they could be put into 1 to 1 correlation
with the even numbers, for one thing, and yet there were surely more
of them, since they included the odd numbers as well. This paradox
Cantor took to be avoided by his definition of the power of a set
(N.B. not the power set of a set): his definition merely required two
sets to be put into 1 to 1 correlation in order for them to have the
same power. Thus all infinite sequences of natural numbers have the
same power, aleph zero.
But the number of points in a line was not aleph zero, it was two to
the aleph zero, and Cantor produced several proofs that these were not
the same. The most famous was his diagonal argument which seems to
show that there must be orders of infinity, and specifically that the
non-denumerably infinite is distinct from the denumerably infinite.
For belief in real numbers is equivalent to belief in certain infinite
sets: real numbers are commonly understood simply in terms of
possibly-non-terminating decimals, but this definition can be derived
from the more theoretical ones (Suppes 1972, p189). But can the
decimals between, say, 0 and 1 be listed? Listing them would make them
countable in the special sense of this which has been adopted, which
amongst other things does not require there to be a last item counted.
The natural numbers are countable in this sense, as before, and any
list, it seems, can be indexed by the ordinal numbers. Suppose,
however, that we had a list in which the n-th member was of this form:
an = 0.an1an2an3an4…,
where ani is a digit between 0 and 9 inclusive. Then that list would
not contain the "diagonal" decimal am defined by
amn = 9 – ann,
since for n = m this equation is false, if only whole digits are
involved. This seems to show that the totality of decimals in any
continuous interval cannot be listed, which implies that there are at
least two separate orders of infinity.
Of course, if there were no infinite sets then there would be no
infinite numbers, countable or uncountable, and so an Aristotelian
would not accept the result of this proof as a fact. Discrete things
might, at the most, be potentially denumerable, for him. But the
difficulty with the result extends even to those who accept that there
are infinite sets, because of another paradox, Skolem's Paradox, which
shows that all theories of a certain sort must have a countable model,
that is must be true in some countable domain of objects. But Set
Theory is one such theory, and in it, supposedly, there must be
non-countable sets. In fact a denumerable model for Set Theory has
recently been specified, by Lavine (Lavine 1994), so how can Cantor's
diagonal proof be accommodated? Commonly it is accommodated by saying
that, within the denumerable model of Set Theory, non-denumerability
is represented merely by the absence of a function which can do the
indexing of a set, that is produce a correlation between the set and
the ordinal numbers. But if that is the case, then maybe the
difficulty of listing the real numbers in an interval is comparable.
Certainly given a list of real numbers with a functional way of
indexing them, then diagonalisation enables us to construct another
real number. But maybe there still might be a denumerable number of
all the real numbers in an interval without any possibility of finding
a function which lists them, in which case we would have no diagonal
means of producing another. We seem to need a further proof that being
denumerable in size means being listable by means of a function.
4. Paradoxes of Self-Reference
The possibility that Cantor's diagonal procedure is a paradox in its
own right is not usually entertained, although a direct application of
it does yield an acknowledged paradox: Richard's Paradox. Consider for
a start all finite sequences of the twenty six letters of the English
alphabet, the ten digits, a comma, a full stop, a dash and a blank
space. Order these expressions according, first, to the number of
symbols, and then lexicographically within each such set. We then have
a way of identifying the n-th member of this collection. Now some of
these expressions are English phrases, and some of those phrases will
define real numbers. Let E be the sub-collection which does this, and
suppose we can again identify the n-th place in this, for each natural
number n. Then the following phrase, as Richard pointed out, would
seem to define a real number which is not defined in the collection:
"The real number whose whole part is zero, and whose n-th decimal
place is p plus 1 if the n-th decimal of the real number defined by
the n-th member of E is p, and p is neither 8 or 9, and is simply one
if this n-th decimal is eight or nine." But this expression is a
finite sequence of the previously described kind.
One significant fact about this paradox is that it is a semantical
paradox, since it is concerned not just with the ordered collection of
expressions (which is a syntactic matter), but also their meaning,
that is whether they refer to real numbers. It is this which possibly
makes it unclear whether there is a specifiable list of expressions of
the required kind, since while the total list of expressions can
certainly be straightforwardly ordered, whether some expression
defines a real number is maybe not such a clear cut matter. Indeed, it
might be concluded, just from the very fact that a paradox ensues, as
above, that whether some English phrase defines a real number is not
always entirely settleable. In Borel's terms, it cannot be decided
effectively (Martin-Löf 1970, p44). Another very similar semantical
paradox with this same aspect is Berry's Paradox, about "the least
integer not nameable in fewer than nineteen syllables." The problem
here is that that very phrase has less than nineteen syllables in it,
and yet, if it names an integer, that integer would have to be not
nameable in less than nineteen syllables. So is there a definite set
of English expressions which name integers not nameable in less than
nineteen syllables?
If some sort of fuzziness was the case then there would be a
considerable difference between such paradoxes and the previous
paradoxes in logical theory like Russell's, Burali-Forti's and
Cantor's, for instance. Indeed it has been common since Ramsey's
discussion of these matters, in the 1920s, to divide the major logical
paradoxes into two: the semantic or linguistic on the one hand, and
the syntactic or mathematical on the other. Mackie disagreed with
Ramsey to a certain extent, although he was prepared to say (Mackie
1973, p262):
The semantical paradoxes…can thus be solved in a philosophical
sense by demonstrating the lack of content of the key items, the fact
that various questions and sentences, construed in the intended way,
raise no substantial issue. But these are comments appropriate only to
linguistic items; one would expect that this method would apply only
to the semantic paradoxes, and not to "syntactic" ones like Russell's
class paradox, which are believed to involve only (formal) logical and
mathematical elements.
Russell himself opposed the distinction, formulating his famous
"Vicious Circle Principle" which, he held, all the paradoxes of
self-reference violated. Specifically he held that statements about
all the members of certain collections were nonsense (compare Haack
1978, p141):
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.
But this, seemingly, would rule out specifying, for instance, a man as
the one with the highest batting average in his team, since he is then
defined in terms of a total of which he is a member. It effectively
imposes a ban on all forms of self-reference, and so Russell's uniform
solution to the paradoxes is usually thought to be too drastic. Some
might say "this may be using a cannon against a fly, but at least it
stops the fly!"; but it also devastates too much else in the vicinity.
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.
Let us use the term "heterological" for the property of being
non-self-applicable, so we can say that "large" and "French" are
heterological, for instance, and we can write as a general definition
"x" is heterological if and only if "x" is not x.
But clearly, substituting "heterological" for "x" produces a
contradiction. Does this contradiction mean there is no such concept
as heterologicality, just as there is no such set as the Russell set?
Goldstein has recently argued that this is so (Goldstein 2000, p67),
following a tradition Mackie calls "the logical proof approach"
(Mackie 1973, p254f), to which Ryle was a notable contributor (Ryle
1950-1). The point is made even more plausible given the very detailed
logical analysis which Copi provided (Copi 1973, p301).
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,
Then ¬H"H" entails in turn
(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,
for instance, whether this has any sense, or involves a substantive
issue, as Mackie would have it (see also Parsons 1984). But there is a
well known further paradox which seems to block this dismissal. For if
we allow, as well as "true" and "false" also "meaningless," then it
might well seem that The Strengthened Liar arises, which, in this
case, could be expressed
What I am now saying is false, or meaningless.
If I am saying nothing meaningful here, then seemingly what I say is
true, which seems to imply that it does have meaning, after all.
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,
or
Some sentence in this book is false,
if that sentence is the only sentence in a book, say in its preface.
It also arises with the following pair of sentences taken together:
The following sentence is false,The previous sentence is true;
and in a case of Buridan's,
What Plato is saying is false,What Socrates is saying is true,
if Socrates says the first, while Plato says the second. There are
many other variations, some of which we shall look at later.
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.
He believed that
Ts iff p,
holds, quite generally, if "s" is some phrase naming, or referring, to
the sentence "p" — for instance, as above, that same sentence in
quotation marks, or a number in some system of numbering, which was
the way Gödel handled such matters. Tarski's analysis of truth
involved denying that there could be "semantic closure" that is the
presence in a language of the semantic concepts relating to
expressions in that language (Tarski 1956, p402):
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.
This conclusion, which requires that any consistent language be
incomplete, Tarski derived directly by considering The Liar, since
"This is false" seems to provide a self-referential "s" for which
s = "¬Ts",
hence, substituting in the following example of the T-scheme
T"¬Ts" iff ¬Ts,
we get
Ts iff ¬Ts.
To block this conclusion Tarski held that the self-reference seemingly
available in the identity
s = "¬Ts"
was just not consistently available, and specifically that, if one
used the sentence "this is false" then the referent of "this" should
not be that very sentence itself – on pain of the evident
contradiction. Using "this is false" coherently meant speaking about
an object language, but in another, higher, language – the
meta-language. Of course the semantical concepts applicable in this
meta-language likewise could not be sensibly defined within it, so
generally there was supposed to be a whole hierarchy of languages.
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 (compare Haack 1978, p144), where Jones says
All of Nixon's utterances about Watergate are false,
and Nixon says
All of Jones' utterances about Watergate are true.
If, following Tarski, we were to try to assign levels of language to
this pair of utterances, then how could we do it? It would seem that
Jones' utterance would have to be in a language higher, in Tarski's
hierarchy, than any of Nixon's; yet, contrariwise, Nixon's would have
to be higher than any of Jones'.
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,
and
(T) Any sentence is true if and only if what it says is the case.
Tarski, in these terms, took claim (S) to be incorrect. But one also
might claim that (T) is incorrect, maybe because there are sentences
without a truth value, being meaningless, or lacking in content in
some other way, as is held by the theorists mentioned before. A third
general diagnosis claims that both (S) and (T) are correct, and indeed
incompatible, but proceeds to some "rational reconstruction" of them
so that the incompatibility is removed. Fourthly it is possible to
argue that (S) and (T) are correct, but really compatible. Martin sees
this happening as a result of some possible ambiguity in the terms
used in the two principles.
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, that
is 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 mentions specifically Fodor's "Language of Thought" in his
work; Asher and Kamp themselves show that modes of argument similar to
Thomason's can be used even to show that Montague's Intensional
Semantics has the same problems. Asher and Kamp go on to explain the
general method which achieves these results (Asher and Kamp 1989,
p87):
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.
The only ray of hope Asher and Kamp can offer is (Asher and Kamp 1989,
p94): "Only the familiar systems of epistemic and doxastic logic, in
which knowledge and belief are treated as sentential operators, and
which do not treat propositions as objects of reference and
quantification, seem solidly protected from this difficulty." But see
on these, for instance, Mackie 1973, p276f, although also Slater 1986.
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, that is 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,
then our abilities as humans might not seem to be too great. For many
people would argue:
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!
So, unless we can extricate ourselves from this impasse, as well as
the many others we have looked at, we would not seem to be too bright.
Or does this sort of argument show that there is, indeed, no escape?
Some people, of course, might want to follow Tarski, and run from
"natural language" in the face of these conclusions. For Gödel had no
reason to conclude, from his theorems, that the formal systems he was
concerned with were inconsistent. However, his formal arguments differ
crucially from that just given, since there is no proof within his
systems that "provability" entails "truth." There is no doubt that
what we have been dealing with are real paradoxes!
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 (compare
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." (Quine 1966, p7)
The particular law which "paraconsistent" logicians mainly doubt is
"ex impossibile quodlibet", that is "from an impossibility anything
follows," or
(p&¬p) ⊢ q.
It is thought that, if this traditional rule were removed from logic
then, at least, any true contradictions we find, e.g. anything of the
form "p&¬p" which we deduce from some paradox of self-reference, will
not have the wholesale repercussions that it otherwise would have in
traditional logic. Objectors to paraconsistency might say that the
premise of this rule could not arise, so its "explosive" repercussions
would never eventuate. But there is the broader, philosophical
question, as well, about whether a switch to a different logic does
not just change the subject, leaving the original problems unattacked.
That depends on how one views "deviant logics." There are reasons to
believe that deviant logics are not rivals of traditional logic, but
merely supplementary to, or extensions of it (Haack, 1974, Pt 1, Ch1).
For if one drops the above rule then hasn't one merely produced a new
kind of negation? Are "p" and "¬p" still contradictory, if they can,
somehow, both be true? And if "p" and "¬p" are not contradictory, then
what is contradictory to "p", and couldn't we formulate the previous
paradoxes in terms of it? It seems we may have just turned our backs
on the real difficulty.
5. A Contemporary Twist
There have been developments, in the last few years, which have shown
that the previous emphasis on paradoxes involving self-reference was
to some extent misleading. For a family of paradoxes, with similar
levels of intractability, have been discovered, which are not
reflexive in this way.
It was mentioned before that a form of the Liar paradox could be
derived in connection with the pair of statements
What Plato is saying is false,
What Socrates is saying is true,
when Socrates says the former, and Plato the latter. For, if what
Socrates is saying is true, then, according to the former, what Plato
is saying is false, but then, according to the latter, what Socrates
is saying is false. On the other hand, if what Socrates is saying is
false then, according to the former, what Plato is saying is true, and
then, according to the latter, what Socrates is saying is true. Such a
paradox is called a "liar chain"; they can be of any length; and with
them we are already out of the really strict "self-reference" family,
although, by passing along through the chain what Socrates is saying,
it will eventually come back to reflect on itself.
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):
(Si) For all k>i, Sk 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, Sn is true for some n. Then Sn+1 is false, and all
subsequent statements; but the latter fact makes Sn+1 true; giving a
contradiction. Hence for no n is Sn 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.
Goldstein has formulated a comparable process, which he thinks
improves upon Gaifman in certain details, and which ends up labelling
certain sentences "FA", meaning that the sentence has made a "failed
attempt" at making a statement (Goldstein 2000, p57). But the major
question with such approaches, as before, is how they deal with The
Strengthened Liar. Surely there remain major problems with
This sentence is false, or has a GAP,
and
This sentence makes a false statement, or is a FA.
6. References and Further Reading
* Asher, N. and Kamp, H. 1986, "The Knower's Paradox and
Representational Theories of Attitudes," in J. Halpern (ed.)
Theoretical Aspects of Reasoning about Knowledge, San Mateo CA, Morgan
Kaufmann.
* Asher, N. and Kamp, H. 1989, "Self-Reference, Attitudes and
Paradox" in G. Chierchia, B.H. Partee, and R. Turner (eds.)
Properties, Types and Meaning 1.
* Beall, J.C., 2001, "Is Yablo's Paradox Non-Circular?," Analysis 61.3.
* Copi, I.M., 1973, Symbolic Logic 4th ed. Macmillan, New York.
* Gaifman, H. 1992, "Pointers to Truth," The Journal of
Philosophy, 89, 223-61.
* Goldstein, L. 2000, "A Unified Solution to Some Paradoxes,"
Proceedings of the Aristotelian Society, 100, pp53-74.
* Grattan-Guinness, I. (ed.) 1980, From the Calculus to Set
Theory, 1630-1910, Duckworth, London.
* Haack, S. 1974, Deviant Logic, C.U.P., Cambridge.
* Haack, S. 1978, Philosophy of Logics, C.U.P., Cambridge.
* Hallett, M. 1984, Cantorian Set Theory and Limitation of Size,
Clarendon Press, Oxford.
* Keefe, R. 2001, Theories of Vagueness, C.U.P. Cambridge.
* Kneale, W. 1972, "Propositions and Truth in Natural Languages,"
Mind, 81, pp225-243.
* Kneale, W. and Kneale M. 1962, The Development of Logic,
Clarendon Press, Oxford.
* Lavine, S. 1994, Understanding the Infinite, Harvard University
Press, Cambridge MA.
* Linsky, L. 1967, Referring, Routledge and Kegan Paul, London.
* Mackie, J.L. 1973, Truth, Probability and Paradox, Clarendon
Press, Oxford.
* Martin, R.L. (ed.) 1984, Recent Essays on Truth and the Liar
Paradox, Clarendon Press, Oxford.
* Martin-Löf, P. 1970, Notes on Constructive Mathematics, Almqvist
and Wiksell, Stockholm.
* Montague, R. 1963, "Syntactic Treatments of Modality, with
Corollaries on Reflection Principles and Finite Axiomatisability,"
Acta Philosophica Fennica, 16, pp153-167.
* Owen, G.E.L. 1957-8, "Zeno and the Mathematicians," Proceedings
of the Aristotelian Society, 58, 199-222.
* Parsons, C. 1984, "The Liar Paradox" in R.L.Martin (ed.) Recent
Essays on Truth and the Liar Paradox, Clarendon Press, Oxford.
* Penrose, R. 1989, The Emperor's New Mind, O.U.P., Oxford.
* Priest, G.G. 1979, "The Logic of Paradox," Journal of
Philosophical Logic, 8, pp219-241.
* Priest, G.G. 1994, "The Structure of the Paradoxes of
Self-Reference," Mind, 103, pp25- 34.
* Quine, W.V.O. 1966, The Ways of Paradox, Random House, New York.
* Ryle, G. 1950-1, "Heterologicality," Analysis, 11, pp61-69.
* Sainsbury, M. 1995, Paradoxes, 2nd ed., C.U.P. Cambridge.
* Salmon, W.C. (ed.) 1970, Zeno's Paradoxes, Bobbs-Merrill, Indianapolis.
* Scott, T.K. 1966, John Buridan: Sophisms on Meaning and Truth,
Appleton-Century- Crofts, New York.
* Slater, B.H. 1986, "Prior's Analytic," Analysis, 46, pp76-81.
* Sorensen, R. 1998, "Yablo's Paradox and Kindred Infinite Liars,"
Mind, 107, 137-55.
* Suppes, P. 1972, Axiomatic Set Theory, Dover, New York.
* Tarski, A. 1956, Logic, Semantics, Metamathematics: Papers from
1923 to 1938, trans. J.H. Woodger, O.U.P. Oxford.
* Thomason, R. 1977, "Indirect Discourse is not Quotational," The
Monist, 60, pp340-354.
* Thomason, R. 1980, "A Note on Syntactical Treatments of
Modality," Synthese, 44, pp391-395
* Thomason, R. 1986, "Paradoxes and Semantic Representation," in
J.Halpern (ed.) Theoretical Aspects of Reasoning about Knowledge, San
Mateo CA, Morgan Kaufmann.
* Williamson, T. 1994, Vagueness, London, Routledge.
* Yablo, S. 1993, "Paradox without Self-Reference," Analysis, 53, 251-52.
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