## 1. Paradoxes of Self-Reference


### 1.1 Semantic Paradoxes



Paradoxes of self-reference have been known since antiquity. The
discovery of the liar paradox is often credited to Eubulides the
Megarian who lived in the 4th century BC. The liar paradox belongs to
the category of *semantic* paradoxes, since it is based on the
semantic notion of truth. Other well-known semantic paradoxes include
Grelling's paradox, Berry's paradox, and Richard's
paradox.



*Grelling's paradox* involves a predicate defined as
follows. Say a predicate is *heterological* if it is not true
of itself, that is, if it does not itself have the property it
expresses. Then the predicate "German" is heterological,
since it is not itself a German word, but the predicate
"deutsch" is not heterological. The question that leads to
the paradox is now:



Is "heterological" heterological?




It is easy to see that we obtain a contradiction independently of
whether we answer "yes" or "no" to this
question (the argument runs more or less like in the liar paradox).
Grelling's paradox is self-referential, since the definition of
the predicate heterological refers to *all* predicates,
including the predicate heterological itself. Definitions such as this
which depends on a set of entities, at least one of which is the
entity being defined, are called *impredicative*.



*Berry's paradox* is another paradox based on an
impredicative definition, or rather, an impredicative description.
Some phrases of the English language are descriptions of natural
numbers, for example, "the sum of five and seven" is a
description of the number 12. Berry's paradox arises when trying
to determine the denotation of the following description: 



the least number that cannot be referred to by a description
containing less than 100 symbols.




The contradiction is that this description containing 93 symbols
denotes a number which, by definition, cannot be denoted by any
description containing less than 100 symbols. The description is of
course impredicative, since it implicitly refers to *all*
descriptions, including itself.



*Richard's paradox* considers phrases of the English
language defining real numbers rather than natural numbers. For
example, "the ratio between the circumference and diameter of a
circle" is a phrase defining the number \(\pi\). Assume an
enumeration of all such phrases is given (e.g. by putting them into
lexicographical order). Now consider the phrase:



the real number whose \(n\)th decimal place is 1 whenever the
\(n\)th decimal place of the number denoted by the \(n\)th
phrase is 0; otherwise 0.




This phrase defines a real number, so it must be among the enumerated
phrases, say number \(k\) in this enumeration. But, at the same
time, by definition, it differs from the number denoted by the
\(k\)th phrase in the \(k\)th decimal place. Thus we have a
contradiction. The defining phrase is obviously impredicative. The
particular construction employed in this paradox is called
*diagonalisation*. Diagonalisation is a general construction
and proof method originally invented by Georg Cantor (1891) to prove
the uncountability of the power set of the natural numbers. It was
also used as a basis for Cantor's paradox, one of the
set-theoretic paradoxes to be considered next.


### 1.2 Set-Theoretic Paradoxes



The best-known set-theoretic paradoxes are Russell's paradox and
Cantor's paradox. *Russell's paradox* arises from
considering the *Russell set* \(R\) of all sets that are
not members of themselves, that is, the set defined defined by
\(R = \{ x \mid x \not\in x \}\). The
contradiction is then derived by asking whether \(R\) is a member
of itself, that is, whether \(R \in R\) holds. If
\(R \in R\) then \(R\) is a member of itself,
and thus \(R \not\in R\), by definition of \(R\).
If, on the other hand, \(R \not\in R\) then \(R\)
is not a member of itself, and thus \(R \in R\),
again by definition of \(R\).



*Cantor's paradox* is based on an application of
Cantor's theorem. *Cantor's theorem* states that
given any finite or infinite set \(S\), the power set of
\(S\) has strictly larger cardinality (greater size) than
\(S\). The theorem is proved by a form of diagonalisation, the
same idea underlying Richard's paradox. Cantor's paradox
considers the set of all sets. Let us call this set the *universal
set* and denote it by \(U\). The power set of \(U\) is
denoted \(\wp(U)\). Since \(U\) contains *all*
sets it will in particular contain all elements of
\(\wp(U)\). Thus \(\wp(U)\) must be a subset of
\(U\) and must thus have a cardinality (size) which is less than
or equal to the cardinality of \(U\). However, this immediately
contradicts Cantor's theorem. 



The *Hypergame paradox* is a more recent addition to the list
of set-theoretic paradoxes, invented by Zwicker (1987). Let us call a
two-player game *well-founded* if it is bound to terminate in a
finite number of moves. Tournament chess is an example of a
well-founded game. We now define *hypergame* to be the game in
which player 1 in the first move chooses a well-founded game to be
played, and player 2 subsequently makes the first move in the chosen
game. All remaining moves are then moves of the chosen game. Hypergame
must be a well-founded game, since any play will last exactly one move
more than some given well-founded game. However, if hypergame is
well-founded then it must be one of the games that can be chosen in
the first move of hypergame, that is, player 1 can choose hypergame in
the first move. This allows player 2 to choose hypergame in the
subsequent move, and the two players can continue choosing hypergame
*ad infinitum*. Thus hypergame cannot be well-founded,
contradicting our previous conclusion.


### 1.3 Epistemic paradoxes



The most well-know epistemic paradox is the *paradox of the
knower*. This paradox has many equivalent formulations, one of
them based on the sentence "This sentence is not known by
anyone." Let us call this sentence the *knower sentence*,
abbreviated \(KS. KS\) is obviously quite similar to the
liar sentence, except the central concept involved is knowledge rather
than truth. The reasoning leading to a contradiction from \(KS\)
is a bit more complex than in the liar paradox. First \(KS\) is
shown to be true by the following piece of reasoning:



Assume to obtain a contradiction that \(KS\) is not true. Then
what \(KS\) expresses cannot be the case, that is, \(KS\)
must be known by someone. Since everything known is true (this is part
of the definition of the concept of knowledge), \(KS\) is true,
contradicting our assumption. This concludes the proof that
\(KS\) is true.




The piece of reasoning just carried out to prove the truth of
\(KS\) should be available to any agent (person) with sufficient
reasoning capabilities. That is, an agent should be able to prove the
truth of \(KS\), and thus come to know that \(KS\) holds.
However, if \(KS\) is known by someone, then what it expresses is
not the case, and thus it cannot be true. This is a contradiction, and
thus we have a paradox. The role of self-reference in this paradox is
obvious, as it is based on a sentence, \(KS\), referring directly
to itself.



The paradox of the knower is just one of many epistemic paradoxes
involving self-reference. See the entry on
 epistemic paradoxes
 for further information on the class of epistemic paradoxes. A quite
recent epistemic paradox, cast in the setting of beliefs and
assumptions in a two-player game, is the Brandenburger-Keisler paradox
(Brandenburger & Keisler, 2006), described in detail in the entry
on
 epistemic foundations of game theory.
 For a detailed discussion and history of the paradoxes of
self-reference in general, see the entry on
 paradoxes and contemporary logic.
 


### 1.4 Common Structures in the Paradoxes



The paradoxes above are all quite similar in structure. In the case of
the paradoxes of Grelling and Russell, this can be seen as follows.
Define the *extension* of a predicate to be the set of objects
it is true of. For a predicate \(P\) we denote its extension by
ext\((P)\). Grelling's paradox involves the predicate
heterological, which is true of all those predicates that are not true
of themselves. Thus the extension of the predicate heterological is
the set \(\{ P \mid P \not\in\) ext\((P) \}\). Compare
this to the Russell set \(R\) given by \(\{ x \mid x \not\in x \}\). The only significant difference between these
two sets is that the first is defined on predicates whereas the second
is defined on sets. The proofs of contradictions based on these two
sets also share the same structure, as seen below (where
"het" abbreviates "heterological"):



\[\begin{align\*}
\tag{Grelling}
\het \in \ext(\het) &\Leftrightarrow \het \in \{ P \mid P \not\in \ext(P)\} \\
 &\Leftrightarrow \het \not\in \ext(\het). \\
 & \\
\tag{Russell}
R \in R &\Leftrightarrow R \in \{ x \mid x \not\in x\} \\
 &\Leftrightarrow R \not\in R. 
\end{align\*}\]


We have here two paradoxes of an almost identical structure belonging
to two distinct classes of paradoxes: one is semantic and the other
set-theoretic. What this teaches us is that even if paradoxes seem
different by involving different subject matters, they might be almost
identical in their underlying structure. Thus in many cases it makes
most sense to study the paradoxes of self-reference under one, rather
than study, say, the semantic and set-theoretic paradoxes separately.




Russell and Cantor's paradoxes are also more similar than they
appear at first. Cantor's paradox is based on an application of
Cantor's theorem to the universal set \(U\) (cf. Section
1.2 above). Below we give the proof of Cantor's theorem for an
arbitrary set \(S\).




We need to prove that \(\wp(S)\) has greater cardinality than
\(S\). Assume to obtain a contradiction that this is not the
case. Then there must exist a map \(f\) from \(S\) onto
\(\wp(S)\). Now consider the set \(C = \{ x \in S \mid x \not\in f(x)
\}\). Since \(f\) is onto \(\wp(S)\), there must exist a set \(c \in
S\) such that \(f(c)=C\). However, we now obtain a contradiction,
since the following holds: 



\[\begin{align\*}
c \in f(c) &\Leftrightarrow c \in \{ x \in S \mid x \not\in f(x) \} \\
 &\Leftrightarrow c \not\in f(c).
\end{align\*}\]



Note the similarity between this sequence of equivalences and the
corresponding sequences of equivalences derived for Russell's
and Grelling's paradoxes above. Now consider the special case of
Cantor's theorem where \(S\) is the universal set. Then we
can simply choose \(f\) to be the identity function, since
\(\wp(U)\) must necessarily be a subset of the universal set
\(U\). But then \(C\) becomes the Russell set! Thus
Cantor's paradox is nothing more than a slight variant of
Russell's paradox; the core argument leading to the
contradiction is the same in both. 



Priest (1994) gives even firmer evidence to the similarity between the
paradoxes of self-reference by showing that they all fit into what he
originally called the *Qualified Russell's Schema*, now
termed the *Inclosure Schema*. The idea behind it goes back to
Russell himself (1905) who also considered the paradoxes of
self-reference to have a common underlying structure. Given two
predicates predicates \(P\) and \(Q\), and a possibly partial function
\(\delta\), the Inclosure Schema consists of the following two
conditions:


1. \(w = \{ x \mid P(x) \}\) exists and
\(Q(w)\) holds;
2. if \(y\) is a subset of \(w\) such that
\(Q(y)\) holds then:
	1. \(\delta(y) \not\in y\),
	2. \(\delta(y) \in w\).


 If these conditions are satisfied we have the following
contradiction: Since \(w\) is trivially a subset of \(w\) and since
\(Q(w)\) holds by condition 1, we have both \(\delta(w) \not\in w\)
and \(\delta(w) \in w\), by 2a and 2b, respectively. Thus any triple
\((P,Q,\delta)\) satisfying the Inclosure Schema will produce a
paradox. Priest shows how most of the well-known paradoxes of
self-reference fit into the schema. Below we will consider only a few
of these paradoxes, starting with Russell's paradox. In this
case we define the triple \((P,Q,\delta)\) as follows:


* \(P(x)\) is the predicate "\(x \not\in x\)".
* \(Q(y)\) is the universal predicate true of any
object.
* \(\delta\) is the identity function.



Then \(w\) in the Inclosure Schema becomes the Russell set and
the contradiction obtained from the schema becomes Russell's
paradox.



In the case of Richard's paradox we define the triple by:


* \(P(x)\) is the predicate "\(x\) is a
real definable by a phrase in English."
* \(Q(y)\) is the predicate "\(y\) is a
denumerable set of reals definable by a phrase in English."
* \(\delta\) is the function that maps any denumerable set \(y\)
of reals to the real \(z\) whose \(n\)th decimal place is 1
whenever the \(n\)th decimal of the \(n\)th real in
\(y\) is 0; otherwise 0. (Any enumeration of the elements in
\(y\) will do.)



Here \(w = \{ x \mid P(x) \}\) becomes the
set of all reals definable by phrases in English. For any denumerable
subset \(y\) of \(w, \delta(y)\) is a real that by
construction will differ from all reals in \(y\) (it differs from
the \(n\)th real in \(y\) on the \(n\)th decimal
place). Letting \(y\) equal \(w\) we thus get
\(\delta(w) \not\in w\). However, at the same time
\(\delta(w)\) is definable by a phrase in English, so
\(\delta(w) \in w\), and we have a contradiction.
This contradiction is Richard's paradox. 



The liar paradox also fits Russell's schema, albeit in a
slightly less direct way:


* \(P(x)\) is the predicate "\(x\) is true."
* \(Q(y)\) is the predicate "\(y\) is definable."
* \(\delta(y)\) is the sentence "this sentence does not belong
to the set \(y\)."



Here \(w = \{ x \mid P(x) \}\) becomes the
set of true sentences, and \(\delta(w)\) becomes a version of
the liar sentence: "this sentence does not belong to the set of
true sentences". 



From the above it can be concluded that all, or at least most,
paradoxes of self-reference share a common underlying 
structure--independent of whether they are semantic, set-theoretic or
epistemic. Priest (1994) argues that they should then also share a
common solution. Priest calls this the *principle of uniform
solution*: "same kind of paradox, same kind of
solution." Whether the Inclosure Schema can in full generality
count as a necessary and sufficient condition for self-referential
paradoxicality is however disputable (Slater, 2002; Abad, 2008;
Badici, 2008; Zhong, 2012, and others), hence not all authors agree on
the principle of uniform solution either. 



The
 Sorites paradox
 is a paradox that on the surface does not involve self-reference at
all. However, Priest (2010b, 2013) argues that it still fits the
inclosure schema and can hence be seen as a paradox of self-reference,
or at least a paradox that should have the same kind of solution as
the paradoxes of self-reference. This has led Colyvan (2009), Priest
(2010) and Weber (2010b) to all advance a dialetheic approach to
solving the Sorites paradox. This approach to the Sorites paradox has
been attacked by Beall (2014a, 2014b) and defended by Weber et al.
(2014).


### 1.5 Paradoxes without Negation



Most paradoxes considered so far involve negation in an essential way,
e.g. sentences saying of themselves that they are *not* true or
knowable. The central role of negation will become even clearer when
we formalise the paradoxes of self-reference in Section 2 below.
*Curry's paradox* is a similar paradox of self-reference
that however does not directly involve negation. A semantic variant of
Curry's paradox comes from the following *Curry sentence*
\(C\): "If this sentence is true then \(F\)",
where \(F\) can be any statement, for instance an obviously false
one. Suppose the Curry sentence \(C\) is true. Then it expresses
a true fact, that is, if \(C\) is true then \(F\). However,
we already assumed \(C\) to be true, so we can infer \(F\),
using Modus Ponens. We have now proved that if we assume \(C\) to
be true, then \(F\) follows. This is exactly what the Curry
sentence itself expresses. In other words, we have proved that the
Curry sentence itself is true! But then we also have that \(F\)
is true, and this is a paradox, since \(F\) can be any statement,
including things that are obviously false (Smullyan (2006) let
\(F\) be the sentence "Santa Claus exists", thereby
using Curry's paradox to prove the existence of Santa Claus). In
a classical logical setting where the implication \(C \rightarrow F\)
is equivalent to \(\neg C \vee F\), Curry's paradox still
implicitly involves negation, but Curry's paradox is still
independently interesting since it goes through with fewer assumptions
about the underlying logic than the liar paradox. See the entry on
 Curry's paradox
 for more details. 


### 1.6 Paradoxes without Self-Reference



In 1985, Yablo succeeded in constructing a semantic paradox that does
not involve self-reference in the strict sense. Instead, it consists
of an infinite chain of sentences, each sentence expressing the
untruth of all the subsequent ones. More precisely, for each natural
number \(i\) we define \(S\_i\) to be the
sentence "for all \(j\gt i,
S\_j\) is not true". We can then derive
a contradiction as follows:




First we prove that none of the sentences
\(S\_i\) can be true. Assume to obtain a
contradiction that \(S\_i\) is true for some
\(i\). Then it is true that "for all
\(j\gt i, S\_j\) is not
true". Thus none of the sentences
\(S\_j\) for \(j\gt i\) are true.
In particular, \(S\_{i+1}\) is not true.
\(S\_{i+1}\) is the sentence "for all
\(j\gt i+1, S\_j\) is not
true". Since this sentence is not true, there must be some
\(k\gt i+1\) for which \(S\_k\)
is true. However, this contradicts that none of the sentences
\(S\_j\) with \(j\gt i\) are
true.



We have now proved that none of the sentences
\(S\_i\) are true. Then, in particular, we have
that for all \(j\gt 0\), S\(\_j\) is not true. This
is exactly what is expressed by S\(\_0\), so S\(\_0\) must
be true. This is again a contradiction.




Yablo calls this paradox the \(\omega\)*-liar*, but others usually
refer to it as *Yablo's paradox*. Note that none of the
sentences \(S\_i\) refer to themselves (not
even indirectly), but only to the ones that occur later in the
sequence. Yablo's paradox is semantic, but as shown by Yablo
(2006), similar set-theoretic paradoxes involving no self-reference
can be formulated in certain set theories. 



Yablo's paradox demonstrates that we can have logical paradoxes
without self-reference--only a certain kind of
non-wellfoundedness is needed to obtain a contradiction. There are
obviously structural differences between the ordinary paradoxes of
self-reference and Yablo's paradox: The ordinary paradoxes of
self-reference involve a cyclic structure of reference, whereas
Yablo's paradox involve an acyclic, but non-wellfounded,
structure of reference. More precisely, the referential structure in
self-referential paradoxes such as the liar is a reflexive relation on
a singleton set (a cycle), whereas the referential structure in
Yablo's paradox is isomorphic to the usual less-than ordering on
the natural numbers, which is a strict total order (contains no
cycles). Even though there is this difference, Yablo's paradox
still share most properties with the ordinary paradoxes of
self-reference. When solving paradoxes we might thus choose to
consider them all under one, and refer to them as *paradoxes of
non-wellfoundedness*. In the following we will however stick to
the term *paradoxes of self-reference*, even though most of
what we say will apply to Yablo's paradox and related paradoxes
of non-wellfoundedness as well.



Given the insight that not only cyclic structures of reference can
lead to paradox, but also certain types of non-wellfounded structures,
it becomes interesting to study further these structures of reference
and their potential in characterising the necessary and sufficient
conditions for paradoxicality. This line of work was initiated by
Gaifman (1988, 1992, 2000), and later pursued by Cook (2004), Walicki
(2009) and others.



Significant amounts of newer work on self-reference has gone into
trying to make a complete graph-theoretical characterisation of which
structures of reference admit paradoxes, including Rabern and Macauley
(2013), Cook (2014) and Dyrkolbotn and Walicki (2014). A complete
characterisation is still an open problem (Rabern, Rabern and
Macauley, 2013), but it seems to be a relatively widespread conjecture
that all paradoxical graphs of reference are either cyclic or contain
a Yablo-like structure. If this conjecture turns out to be true, it
would mean that in terms of structure of reference, all paradoxes of
reference are either liar-like or Yablo-like. The precise
characterisation of what it means to be a "Yablo-like
structure" is still discussed.



Even though the structure of reference involved in Yablo's
paradox does not contain any cycles (each sentence only refers to
later sentences in the sequence), it is still being discussed whether
the paradox is self-referential or not (Cook, 2014; Halbach and Zhang,
2017). Yablo (1993) himself argues that it is non-self-referential,
whereas Priest (1997) argues that it is self-referential. Butler
(2017) claims that even if Priest is correct, there will be other
Yablo-like paradoxes that are not self-referential in the sense of
Priest. In the analysis of Yablo's paradox, it is essential to
note that it involves an *infinite* sequence of sentences,
where each sentence refers to *infinitely* many other
sentences. To formalise it in a setting of propositional logic, it is
hence necessary to use infinitary propositional logic. Any finitary
variant of Yablo's sequence--where every sentence only
refers to finitely many later sentences in the sequence--must
necessarily be consistent (non-paradoxical) due to the compactness
theorem in propositional logic (every finite subset of sentences in
the sequence induces a well-founded reference relation and the
sentences can hence consistently be assigned truth-values bottom-up).
In finitary first- and second-order arithmetic, one can instead
attempt to formalise Yablo's paradox by a unary predicate
\(S(x)\) where, for every natural numbers \(i\),
\(S(\inumeral)\) represents the formalisation of the \(i\)th
sentence \(S\_i\) of the Yablo sequence (where
\(\inumeral\) is the numeral representing \(i)\). How and
whether the Yablo paradox can truthfully be represented this way, and
how it relates to compactness of the underlying logic, has been
investigated by Picollo (2013). 



Yablo's paradox has also inspired the creation of similar
paradoxes involving non-wellfounded, acyclic structures of reference
in other areas than truth, e.g. the "Yabloesque" variant
of the Brandenburger-Keisler paradox of epistemic game theory by
Baskent (2016), a variant concerning provability by Cieslinski and
Urbaniak (2013), and a variant in the context of Godel's
incompleteness theorems by Leach-Krouse (2014).


## 2. Why the Paradoxes Matter



After having presented a number of paradoxes of self-reference and
discussed some of their underlying similarities, we will now turn to a
discussion of their *significance*. The significance of a
paradox is its indication of a flaw or deficiency in our understanding
of the central concepts involved in it. In case of the semantic
paradoxes, it seems that it is our understanding of fundamental
semantic concepts such as *truth* (in the liar paradox and
Grelling's paradox) and *definability* (in Berry's
and Richard's paradoxes) that are deficient. In case of the
set-theoretic paradoxes, it is our understanding of the concept of a
*set*. If we fully understood these concepts, we should be able
to deal with them without being led to contradictions. 



To illustrate this, consider the case of Zeno's classical
paradox on *Achilles and the Tortoise* (see the entry
 Zeno's paradoxes
 for details). In this paradox we seem able to prove that the tortoise
can win a race against the 10 times faster Achilles if given an
arbitrarily small head start. Zeno used this paradox as an argument
against the possibility of motion. It has later turned out that the
paradox rests on an inadequate understanding of infinity. More
precisely, it rests on an implicit assumption that any infinite series
of positive reals must have an infinite sum. The later developments of
the mathematics of infinite series has shown that this assumption is
invalid, and thus the paradox dissolves. The original acceptance of
Zeno's argument as a paradox was a symptom that the concept of
infinity was not sufficiently well understood at the time. In analogy,
it seems reasonable to expect that the existence of semantic and
set-theoretic paradoxes is a symptom that the involved semantic and
set-theoretic concepts are not yet sufficiently well understood. 



Another possible answer could be that it is our understanding of the
very concept of "contradiction" that is flawed. The
reasoning involved in the paradoxes of self-reference all end up with
some contradiction, a sentence concluded to be both true and false. We
consider this an impossibility, hence the paradox, but maybe we
don't have to? *Dialetheism* is the view that there can
be "true contradictions", meaning that it is not
impossible for a sentence to be both true and false. If adopting the
view of dialetheism, all the paradoxes of self-reference dissolve and
instead become existence proofs of certain "dialetheia":
sentences being both true and false. Priest (1987) is a strong
advocate of dialetheism, and uses his principle of uniform solution
(see Section 1.4 above) to defend the dialetheist solution. See the
entries on
 dialetheism
 and
 paraconsistent logic
 for more information.



Currently, no commonly agreed upon solution to the paradoxes of
self-reference exists. They continue to pose foundational problems in
semantics and set theory. No claim can be made to a solid foundation
for these subjects until a satisfactory solution to the paradoxes has
been provided. Problems surface when it comes to formalising semantics
(the concept of truth) and set theory. If formalising the intuitive,
"naive" understanding of these subjects, inconsistent
systems linger as the paradoxes will be formalisable in these
systems.


### 2.1 Consequences of the Semantic Paradoxes



The liar paradox is a significant barrier to the construction of
formal theories of truth as it produces inconsistencies in these
potential theories. A substantial amount of research in self-reference
concentrates on formal theories of truth and ways to circumvent the
liar paradox. There are two articles that have influenced the work on
formal theories of truth and the liar paradox more than any other:
Alfred Tarski's "The Concept of Truth in Formalised
Languages" (1935) and Saul Kripke's "Outline of a
Theory of Truth" (1975). Below we first introduce some of the
ideas and results of Tarski's article. Kripke's article is
discussed in Section 3.2.



Tarski gives a number of conditions that, as he puts it, any
*adequate definition of truth* must satisfy. The central of
these conditions is what is now most often referred to as *Schema
T* (or the *T-schema* or *Convention T* or the
*Tarski biconditionals*):



\[
\tag{Schema \(T\)}
\phi \leftrightarrow T \langle \phi \rangle,\text{ for all sentences } \phi.
\]


Here \(T\) is the predicate intended to express truth and
\(\langle \phi \rangle\) is a *name* for the sentence \(\phi\). Applying the
predicate \(T\) to the name \(\langle \phi \rangle\) gives the expression
\(T\langle \phi \rangle\) intended to represent the phrase "\(\phi\)
is true". Schema \(T\) thus expresses that for every
sentence \(\phi , \phi\) holds if and only if the sentence "\(\phi\)
is true" holds. The \(T\)-schema is usually regarded as a
set of sentences within a formal theory. It is customary to use
first-order arithmetic, that is, first-order predicate logic extended
with a set of standard axioms for arithmetic like PA (Peano
Arithmetic) or Robinson's Q. What is being said in the following
will apply to any such first-order formalisation of arithmetic. In
this setting, \(\langle \phi \rangle\) above denotes the *Godel code*
of \(\phi\), and \(T\langle \phi \rangle\) is short for
\(T(\langle \phi \rangle)\). The reader not familiar with Godel
codings (also known as Godel numberings) can just think of the
mapping \(\langle \cdot \rangle\) as a naming device or quotation mechanism for
formulae--just like quotation marks in natural language. Often
used notational variants for \(\langle \phi \rangle\) are \(\ulcorner \phi \urcorner\)
and '\(\phi\)'.



Tarski showed that the liar paradox is formalisable in any formal
theory containing his schema T, and thus any such theory must be
inconsistent. This result is often referred to as *Tarski's
theorem on the undefinability of truth*. The result is basically a
formalisation of the liar paradox within first-order arithmetic
extended with the \(T\)-schema. In order to construct such a
formalisation it is necessary to be able to formulate self-referential
sentences (like the liar sentence) within first-order arithmetic. This
ability is provided by the diagonal lemma. 



**Diagonal lemma.**
  

Let \(S\) be a theory extending first-order arithmetic. For every
formula \(\phi(x)\) there is a sentence \(\psi\) such that
\(S \vdash \psi \leftrightarrow \phi \langle \psi \rangle\).




Here the notation \(S \vdash \alpha\) means that \(\alpha\) is
provable in the theory S, and \(\phi \langle \psi \rangle\) is short for
\(\phi(\langle \psi \rangle)\). Assume a formula \(\phi(x)\) is given that
is intended to express some property of sentences - truth, for
instance. Then the diagonal lemma gives the existence of a sentence
\(\psi\) satisfying the biimplication \(\psi \leftrightarrow \phi \langle \psi \rangle\).
The sentence \(\phi \langle \psi \rangle\) can be thought of as expressing that
the sentence \(\psi\) has the property expressed by \(\phi(x)\).
The biimplication thus expresses that \(\psi\) is equivalent to the
sentence expressing that \(\psi\) has property \(\phi\). One can therefore
think of \(\psi\) as a sentence *expressing of itself* that it has
property \(\phi\). In the case of truth, it would be a sentence
expressing of itself that it is true. The sentence \(\psi\) is of course
not self-referential in a strict sense, but mathematically it behaves
like one. It is therefore possible to use sentences generated by the
diagonal lemma to formalise paradoxes based on self-referential
sentences, like the liar. The diagonal lemma is sometimes called the
*fixed-point lemma*, since the equivalence \(\psi \leftrightarrow \phi \langle \psi \rangle\) can be seen as expressing that \(\psi\) is a
fixed-point of \(\phi(x)\).



A theory in first-order predicate logic is called
*inconsistent* if a logical contradiction is provable in it.
Tarski's theorem (on the undefinability of truth) may now be
stated and proved.




**Tarski's theorem.**
  

Any theory extending first-order arithmetic and containing schema
\(T\) is inconsistent.



*Proof.* Assume the existence of a consistent formal theory
\(S\) extending first-order arithmetic and containing schema
\(T\). We need to show that this assumption leads to a
contradiction. The proof mimics the liar paradox. Apply the diagonal
lemma to obtain a sentence \(\lambda\) satisfying \(\lambda \leftrightarrow \neg T \langle \lambda \rangle\) in S. The sentence \(\lambda\) expresses
of itself that it is not true, so \(\lambda\) corresponds to the liar
sentence. Instantiating schema \(T\) with the sentence \(\lambda\)
gives us \(\lambda \leftrightarrow T\langle \lambda \rangle\). We now have that
both \(\lambda \leftrightarrow \neg T\langle \lambda \rangle\) and \(\lambda \leftrightarrow T\langle \lambda \rangle\) hold in \(S\) (are provable in
\(S)\), and thus \(T\langle \lambda \rangle \leftrightarrow \neg T\langle \lambda \rangle\) must hold in \(S\). This
contradicts \(S\) being consistent. \(\Box\)




Note that the contradiction \(T\langle \lambda \rangle \leftrightarrow \neg T\langle \lambda \rangle\) above expresses: The liar sentence is true
if and only if it is not. Compare this to the informal liar presented
in the beginning of the article. Tarski's theorem shows that, in
the setting of first-order arithmetic, it is not possible to give what
Tarski considers to be an "adequate theory of truth". The
central question then becomes: How may the formal setting or the
requirements for an adequate theory of truth be modified to regain
consistency--that is, to prevent the liar paradox from
trivialising the system? There are many different answers to this
question, as there are many different ways to regain consistency. In
Section 3 we will review the most influential approaches.


### 2.2 Consequences of the Set-Theoretic Paradoxes



The set-theoretic paradoxes constitute a significant challenge to the
foundations of mathematics. They show that it is impossible to have a
concept of set satisfying the *unrestricted comprehension
principle* (also called *full comprehension* or
*unrestricted abstraction*):



**Unrestricted comprehension:**
  

\(\forall u (u \in \{ x \mid \phi(x)\} \leftrightarrow \phi(u))\), for
all formulae \(\phi(x)\).




In an informal setting, the formulae \(\phi(x)\) could be
allowed to be arbitrary predicates. In a more formal setting they
would be formulae of e.g. a suitable first-order language. The
unrestricted comprehension principle says that for any property
(expressed by \(\phi)\) there is the *set* of those entities that
satisfy the property. This sounds as a very reasonable principle, and
it more or less captures the intuitive concept of a set. Indeed, it is
the concept of set originally brought forward by the father of set
theory, Georg Cantor (1895), himself. Unfortunately, the principle is
not sound, as it gives rise to Russell's paradox. Consider the
property of non-self-membership. This can be expressed by the formula
\(x \not\in x\). If we let \(\phi(x)\) be the
formula \(x \not\in x\) then the set \(\{ x \mid 
\phi(x) \}\) becomes the Russell set \(R\), and we obtain
the following instance of the unrestricted comprehension
principle:



\[
\forall u (u \in R \leftrightarrow u \not\in u).
\]


Analogous to the argument in Russell's paradox a contradiction
is now obtained by instantiating \(u\) with \(R\):



\[
R \in R \leftrightarrow R \not\in R.
\]


This contradiction expresses that the Russell set is a member of
itself if and only if it is not. What has hereby been proven is the
following.



**Theorem (Inconsistency of Naive Set Theory).**
  

Any theory containing the unrestricted comprehension principle is
inconsistent.




Compare this theorem with Tarski's theorem. Tarski's
theorem expresses that if we formalise the intuitively most obvious
principle concerning truth we end up with an inconsistent theory. The
theorem above expresses that the same thing happens when formalising
the intuitively most obvious principle concerning set existence and
membership.



Given the inconsistency of unrestricted comprehension, the objective
becomes to find a way to restrict either the comprehension principle
itself or the underlying logical principles to regain a consistent
theory, that is, a set theory that will not be trivialised by
Russell's paradox. Many alternative set theories excluding the
unrestricted comprehension principle have been developed during the
last century, among them the type theory of Russell and Whitehead,
Simple Type theory (ST), Godel-Bernays set theory (GB),
Zermelo-Fraenkel set theory (ZF), and Quine's New Foundations
(NF). These are all believed to be consistent, although no simple
proofs of their consistency are known. At least they all escape the
known paradoxes of self-reference. We will return to a discussion of
this in Section 3.


### 2.3 Consequences of the Epistemic Paradoxes



The epistemic paradoxes constitute a threat to the construction of
formal theories of knowledge, as the paradoxes become formalisable in
many such theories. Suppose we wish to construct a formal theory of
*knowability* within an extension of first-order arithmetic.
The reason for choosing to formalise knowability rather than knowledge
is that knowledge is always relative to a certain agent at a certain
point in time, whereas knowability is a universal concept like truth.
We could have chosen to work directly with knowledge instead, but it
would require more work and make the presentation unnecessarily
complicated. To formalise knowability we introduce a special predicate
\(K\), and use sentences of the form \(K\langle \phi \rangle\) to
express that \(\phi\) is knowable. Analogous to the cases of truth and
set membership, there must be certain logical principles that
\(K\) needs to satisfy in order for our formal theory to qualify
as an adequate theory of knowability. First of all, all knowable
sentences must be true. This property can be formalised by the
following logical principle:


* **A1.**
\(K\langle \phi \rangle \rightarrow \phi\), for all sentences
\(\phi\).



Of course this principle must itself be knowable, that is, we get the
following logical principle:


* **A2.**

\(K\langle K\langle \phi \rangle \rightarrow \phi \rangle\), for
all sentences \(\phi\).



In addition, all theorems of first-order arithmetic ought to be
knowable:


* **A3.**

\(K\langle \phi \rangle\), for all sentences \(\phi\) of first-order arithmetic.



Furthermore, knowability must be closed under logical
consequences:


* **A4.**

\(K\langle \phi \rightarrow \psi \rangle \rightarrow 
(K\langle \phi \rangle \rightarrow K\langle \psi \rangle)\),   for
all sentences \(\phi\).



Now, the principles A1-A4 is all it takes to formalise the
paradox of the knower. More precisely, we have the following theorem
due to Montague (1963).




**Montague's theorem.**
  

Any formal theory extending first-order arithmetic and containing
axiom schemas A1-A4 is inconsistent.



*Proof.* Assume the existence of a consistent formal theory
\(S\) extending first-order arithmetic and containing axiom
schemas A1-A4. We need to show that this assumption leads to a
contradiction. The proof mimics the paradox of the knower. Apply the
diagonal lemma to obtain a sentence \(\lambda\) satisfying \(\lambda \leftrightarrow \neg K \langle \lambda \rangle\) in \(S\). The sentence
\(\lambda\) expresses of itself that it is not knowable, so \(\lambda\)
roughly corresponds to the knower sentence, \(KS\). The first
piece of argumentation used in the paradox of the knower led to the
conclusion that \(KS\) is indeed true. This piece of
argumentation is mimicked by the following piece of formal reasoning
within \(S\):




|  |  |  |
| --- | --- | --- |
| 1. | \(\lambda \rightarrow \neg K\langle \lambda \rangle\) | by choice of \(\lambda\) |
| 2. | \(\neg K\langle \lambda \rangle \rightarrow \lambda\) | by choice of \(\lambda\) |
| 3. | \(K\langle \lambda \rangle \rightarrow \lambda\) | axiom A1 |
| 4. | \((K\langle \lambda \rangle \rightarrow \lambda) \rightarrow\)

\( ((\lambda \rightarrow \neg K\langle \lambda \rangle) \rightarrow \neg K\langle \lambda \rangle)\) | propositional tautology |
| 5. | \((\lambda \rightarrow \neg K\langle \lambda \rangle)
\rightarrow \neg K\langle \lambda \rangle\) | modus ponens on 4,3  |
| 6. | \(\neg K\langle \lambda \rangle\) | modus ponens on 5,1 |
| 7. | \(\lambda\) | modus ponens on 2,6  |



This proof shows that \(\lambda\), our formal version of \(KS\), is
provable in \(S\). The proof corresponds to the informal argument
that \(KS\) is true. As argued in the paradox of the knower, any
agent with sufficient reasoning capabilities will be able to prove the
truth of \(KS\), and thus come to know that \(KS\) holds.
Thus \(KS\) must be knowable. What this means in the present
formal framework is that we can also prove the knowability of \(\lambda\)
in \(S\):




|  |  |  |
| --- | --- | --- |
| 8. | \(K\langle \lambda \rightarrow \neg K\langle \lambda \rangle \rangle\) | by A3 and choice of \(\lambda\) |
| 9. | \(K\langle \neg\)K\(\langle \lambda \rangle \rightarrow \lambda \rangle\) | by A3 and choice of \(\lambda\) |
| 10. | \(K\langle K \langle \lambda \rangle \rightarrow \lambda \rangle\) | axiom A2 |
| 11. | \(K\langle(K\langle \lambda \rangle \rightarrow \lambda) \rightarrow\)

\( ((\lambda \rightarrow \neg K\langle \lambda \rangle) \rightarrow \neg K\langle \lambda \rangle)\rangle\) | axiom A3 on propositional
tautology |
| 12. | \(K\langle(\lambda \rightarrow \neg K\langle \lambda \rangle)
\rightarrow \neg K\langle \lambda \rangle \rangle\) | axiom A4 on 11,10  |
| 13. | \(K\langle \neg K\langle \lambda \rangle \rangle\) | axiom A4 on 12,8 |
| 14. | \(K\langle \lambda \rangle\) | axiom A4 on 9,13 |



This completes the proof of the knowability of \(\lambda\), corresponding
to the informal argument that \(KS\) is known by some agent. Note
the similarity between the two pieces of proof in lines 1-7 and
8-14. The only difference is that in the latter all formulae are
preceded by an extra K. This is because lines 8-14 express the
same line of reasoning as lines 1-7 with the only difference
that the latter is a proof of the *knowability* of \(\lambda\)
rather than the *truth* of \(\lambda\). Having concluded that
\(\lambda\) is both true and knowable, we now immediately obtain a
contradiction as in the paradox of the knower:




|  |  |  |
| --- | --- | --- |
| 15. | \(\neg K\langle \lambda \rangle\) | modus ponens on 1,7 |
| 16. | \(K\langle \lambda \rangle \wedge \neg K\langle \lambda \rangle\)
 | conjunction of 14 and 15 |



This completes the proof of Montague's theorem. \(\Box\)




The proof above is simpler than the original proof of Montague (1963)
and directly mimicks the reasoning underlying the paradox of the
knower. Montague's theorem shows that in the setting of
first-order arithmetic we cannot have a theory of knowledge or
knowability satisfying even the basic principles A1-A4.
Montague's theorem is a generalisation of Tarski's
theorem. If a predicate symbol \(K\) satisfies Tarski's
schema \(T\) then it is easy to see that it will also satisfy
axiom schemas A1-A4. Thus axiom schemas A1-A4 constitute a
weakening of the \(T\)-schema, and Montague's theorem shows
that even this much weaker version of the \(T\)-schema is
sufficient to produce inconsistency.



Formalising knowledge as a predicate in a first-order logic is
referred to as the *syntactic treatment* of knowledge.
Alternatively, one can choose to formalise knowledge as a modal
operator in a suitable modal logic. This is referred to as the
*semantic treatment* of knowledge. In the semantic treatment of
knowledge one generally avoids problems of self-reference, and thus
inconsistency, but it is at the expense of the expressive power of the
formalism (the problems of self-reference are avoided by propositional
modal logic not admitting anything equivalent to the diagonal lemma
for constructing self-referential formulas).


### 2.4 Consequences Concerning Provability and Computability



The central argument given in the proof of Tarski's theorem is
closely related to the central argument in Godel's first
incompleteness theorem (Godel, 1931). Godel's theorem
can be given the following formulation.



**Godel's first incompleteness theorem.**
  

If first-order arithmetic is \(\omega\)-consistent then it is incomplete.




A theory is called \(\omega\)-*consistent* if the following holds
for every formula \(\phi(x)\) containing \(x\) as its only
free variable: If \(\vdash \neg \phi(n)\) for every natural
number \(n\) then \(\not\vdash \exists x\phi(x).
\omega\)-consistency is a stronger condition than ordinary consistency,
so any \(\omega\)-consistent theory will also be consistent. A theory is
*incomplete* if it contains a formula which can neither be
proved nor disproved.




*Proof sketch of Godel's first incompleteness
theorem*. Assume first-order arithmetic is both \(\omega\)-consistent
and complete. We need to show that this leads to a contradiction.
Godel constructs a formula *Bew* (for
"Beweis") in formal arithmetic satisfying, for all \(\phi\)
and all \(n\),


* (1)

\(\vdash\)*Bew*\((n, \langle \phi \rangle)\) iff \(n\) is
the Godel code of a proof of \(\phi\).



Assuming the theory to be \(\omega\)-consistent and complete we can prove
that for all sentences \(\phi\),


* (2)

\(\vdash \exists x\)*Bew*\((x, \langle \phi \rangle)\)
iff \(\vdash \phi\).



The proof of (2) runs like this. First we prove the implication from
left to right. If \(\vdash \exists x\)*Bew*\((x,
\langle \phi \rangle)\) then there is some \(n\) such that
\(\not\vdash \neg\)*Bew*\((n, \langle \phi \rangle)\), by
\(\omega\)-consistency. By completeness we get
\(\vdash\)*Bew*\((n, \langle \phi \rangle)\) for this \(n\). By
(1) above we get that \(n\) denotes a proof of \(\phi\). That is,
\(\phi\) is provable, so we have \(\vdash \phi\). To prove the implication
from right to left, note that if \(\vdash \phi\) then there must be an
\(n\) such that \(\vdash\) *Bew*\((n, \langle \phi \rangle)\),
by (1). From this we get
\(\vdash \exists x\)*Bew*\((x, \langle \phi \rangle)\), as
required. This concludes the proof of (2). Now, when in a complete
theory we have (2), we must also have:


* (3)

\(\vdash \exists x\)*Bew*\((x, \langle \phi \rangle)
\leftrightarrow \phi\), for all sentences \(\phi\).



If we let \(\exists x\)*Bew*\((x, \langle \phi \rangle)\) be
abbreviated \(T\langle \phi \rangle\) then (3) becomes:



\[
\vdash T\langle \phi \rangle \leftrightarrow \phi, \text{ for all sentence } \phi.
\]


This is the \(T\)-schema! Thus if we assume first-order
arithmetic to be \(\omega\)-consistent and complete then schema
\(T\) turns out to be interpretable in it. Now, Tarski's
theorem above shows that there exists no such consistent theory, and
we thus have a contradiction. \(\Box\)




In the proof above we reduced Godel's incompleteness
theorem to an application of Tarski's theorem in order to show
the close link between the two (this version of the proof is due to
Bolander, 2002). Godel was well aware of this link, and indeed it
seems that Godel even proved Tarski's theorem before Tarski
himself did (Feferman, 1984). Godel's theorem can be
interpreted as demonstrating a limitation in what can be achieved by
purely formal procedures. It says that if first-order arithmetic is
\(\omega\)-consistent (which it is believed to be), then there must be
arithmetical sentences that can neither be proved nor disproved by the
formal procedures of first-order arithmetic. One might at first expect
this limitation to be resolvable by the inclusion of additional
axioms, but Godel showed that the incompleteness result still
holds when first-order arithmetic is extended with an arbitrary finite
set of axiom schemas (or, more generally, an arbitrary recursive set
of axioms). Thus we obtain a general limitation result saying that
there cannot exist a formal proof procedure by which any given
arithmetical sentence can be proved to hold or not to hold. For more
details on Godel's incompleteness theorem, see the entry on
 Kurt Godel.



The limitation result of Godel's theorem is closely related
to another limitation result known as the *undecidability of the
halting problem*. This is a result stating that there are
limitations to what can be computed. We will present this result in
the following. The result is based on the notion of a *Turing
machine*, which is a generic model of a computer program running
on a computer having unbounded memory. Thus any program running on any
computer can be thought of as a Turing machine (see the entry on
 Turing machines
 for more details). When running a Turing machine, it will either
terminate after a finite number of computation steps, or will continue
running forever. In case it terminates after a finite number of
computation steps we say that it *halts*. The *halting
problem* is the problem of finding a Turing machine that can
decide whether other Turing machines halt or not. We say that a Turing
machine \(H\) *decides the halting problem* if the
following holds:



\(H\) takes as input a pair \((\langle A\rangle ,x)\)
consisting of the Godel code \(\langle A\rangle\) of a Turing
machine \(A\) and an arbitrary string \(x. H\)
returns the answer "yes" if the Turing machine \(A\)
halts when given input \(x\), and "no" otherwise.




Thus if a Turing machine \(H\) decides the halting problem, we
can use it to determine for an arbitrary Turing machine \(A\) and
arbitrary input \(x\) whether \(A\) will halt on input
\(x\) or not. The *undecidability of the halting problem*
is the following result, due to Turing (1937), stating that no such
machine can exist: 




**Theorem (Undecidability of the Halting Problem).**
  

There exists no Turing machine deciding the halting problem.



*Proof.* Assume the existence of a Turing machine \(H\)
deciding the halting problem. We need to show that this assumption
leads to a contradiction. The proof mimics Grelling's paradox.
We call a Turing machine \(A\) *heterological* if
\(A\) doesn't halt on input \(\langle A\rangle\), that is, if
\(A\) doesn't halt when given its own Godel code as
input. Using \(H\), we can construct a Turing machine \(G\)
that halts if and only if it is given the Godel code of a
heterological Turing machine as input. \(G\) works as
follows:



\(G\) takes as input the Godel code of a Turing machine
\(A\). Then it runs \(H\) on input
\((\langle A\rangle ,\langle A\rangle)\). If \(H\) on input
\((\langle A\rangle ,\langle A\rangle)\) returns "no" we
know that \(A\) is heterological, and \(G\) is halted. If,
on the other hand, \(H\) on input
\((\langle A\rangle ,\langle A\rangle)\) returns "yes" then
\(G\) is forced into an infinite loop (that is, is forced to
never halt).




In analogy to Grelling's paradox we can now ask whether
\(G\) is a heterological Turing machine or not. This leads to the
following sequence of equivalences: 



\[\begin{align\*}
G \text{ is heterological } 
 &\Leftrightarrow 
 G \text{ doesn't halt on input } \langle G\rangle \\
 &\Leftrightarrow 
 H \text{ returns "no" on input } (\langle G\rangle ,\langle G\rangle) \\
 &\Leftrightarrow
 G \text{ halts on input } \langle G\rangle \\
 &\Leftrightarrow
 G \text{ is not heterological.}
\end{align\*}\]


This gives the required contradiction. \(\Box\)




From the two theorems above we see that in the areas of provability
and computability the paradoxes of self-reference turn into limitation
results: there are limits to what can be proven and what can be
computed. This is actually quite similar to what happened in the areas
of semantics, set theory and epistemology: The paradoxes of
self-reference turned into theorems showing that there are limits to
which properties we can consistently assume a truth predicate to have
(Tarski's theorem), a set theory to have (inconsistency of naive
set theory), and a knowledge predicate to have (Montague's
theorem). It is hard to accept these limitation results, because most
of them conflict with our intuitions and expectations. The central
role played by self-reference in all of them makes them even harder to
accept, and definitely more puzzling. However, we are forced to accept
them, and forced to accept the fact that in these areas we cannot have
all we might (otherwise) reasonably ask for. 


## 3. Solving the paradoxes



The present section takes a look at how to solve--or rather,
circumvent--the paradoxes. To solve or circumvent a paradox one
has to weaken some of the assumptions leading to the contradiction. It
is very difficult to choose which assumptions to weaken, since each of
the explicitly stated assumptions underlying a paradox appears to be
"obviously true"--otherwise it would not qualify as
a paradox. Below we will take a look at the most influential
approaches to solving the paradoxes.



So far the presentation has been structured according to type of
paradox, that is, the semantic, set-theoretic and epistemic paradoxes
have been dealt with separately. However, it has also been
demonstrated that these three types of paradoxes are similar in
underlying structure, and it has been argued that a solution to one
should be a solutions to all (the principle of uniform solution).
Therefore, in the following the presentation will be structured not
according to type of paradox but according to type of solution. Each
type of solution considered in the following can be applied to any of
the paradoxes of self-reference, although in most cases the
constructions involved were originally developed with only one type of
paradox in mind.


### 3.1 Building Explicit Hierarchies



Building hierarchies is a method to circumvent both the set-theoretic,
semantic and epistemic paradoxes. Russell's original solution to
his paradox--as well as Tarski's original solution to his
*undefinability of truth* problem--was to build
hierarchies. In Russell's case, this led to *type
theory*. In Tarski's case, it led to what is now known as
*Tarski's hierarchy of languages*. In both cases, the
idea is to stratify the universe of discourse (sets, sentences) into
levels. In type theory, these levels are called *types*. The
fundamental idea of type theory is to introduce the constraint that
any set of a given type may only contain elements of lower types (that
is, may only contain sets which are located lower in the
stratification). This effectively blocks Russell's paradox,
since no set can then be a member of itself.



In Tarski's case, the stratification is obtained in the
following way. Assume one wants to equip a language \(L\_0\) with a
truth predicate. From Tarski's theorem (Section 2.1) it is known
that this truth predicate cannot be part of the language \(L\_0\)
itself--at least not as long as we want the truth predicate to
satisfy schema \(T\). Instead one builds a hierarchy of languages,
\(L\_0, L\_1, L\_2,\ldots\), where each language \(L\_{i+1}\) has a truth
predicate \(T\_{i+1}\) that only applies to the sentences of \(L\_j,
j\le i\). In this hierarchy, \(L\_0\) is called the *object
language* and, for all \(i, L\_{i+1}\) is called the
*meta-language* of \(L\_i\). This hierarchy effectively blocks
the liar paradox, since now a sentence can only express the truth or
untruth of sentences at lower levels, and thus a sentence such as the
liar that expresses its *own* untruth cannot be formed.



Russell's type theory can be regarded as a solution to
Russell's paradox, since type theory demonstrates how to
"repair" set theory such that the paradox disappears.
Similarly, Tarski's hierarchy can be regarded as a solution to
the liar paradox. It is the same stratification idea that underlies
both of Russell's and Tarski's solutions. The point to
note is that Russell's paradox and the liar paradox depend
crucially on circular notions (*self*-membership and
*self*-reference). By making a stratification in which an
object may only contain or refer to objects at lower levels,
circularity disappears. In the case of the epistemic paradoxes, a
similar stratification could be obtained by making an explicit
distinction between first-order knowledge (knowledge about the
external world), second-order knowledge (knowledge about first-order
knowledge), third-order knowledge (knowledge about second-order
knowledge), and so on. This stratification actually comes for free in
the semantic treatment of knowledge, where knowledge is formalised as
a modal operator.



Building explicit hierarchies is sufficient to avoid circularity, and
thus sufficient to block the standard paradoxes of self-reference.
However, there also exist paradoxes such as Yablo's that do not
rely on circularity and self-reference. Such paradoxes can also be
blocked by a hierarchy approach, but it is necessary to further
require the hierarchy to be well-founded, that is, to have a lowest
level. Otherwise, the paradoxes of non-wellfoundedness can still be
formulated. For instance, Yablo's paradox may be formalised in a
*descending* hierarchy of languages. A descending hierarchy of
languages consists of languages \(L\_0, L\_{-1}, L\_{-2},\ldots\) where
each language \(L\_{-i}\) has a truth predicate that only applies to
the sentences of the languages \(L\_{-j}, j\gt i\). Similarly, a
set-theoretic paradox of non-wellfoundedness may be formulated in a
type theory allowing negative types. The conclusion drawn is that a
stratification of the universe is not itself sufficient to avoid all
paradoxes--the stratification also has to be well-founded. 



Building an explicit (well-founded) hierarchy to solve the paradoxes
is today by most considered an overly drastic and heavy-handed
approach. The hierarchies introduce a number of complicating
technicalities not present in a "flat universe", and even
though the paradoxes do indeed disappear, so do all non-paradoxical
occurrences of self-reference. Kripke (1975) gives the following
illustrative example taken from ordinary discourse. Let \(N\) be
the following statement, made by Nixon,


* \((N)\)

All of Jones' utterances about Watergate are true,



and let \(J\) be the following statement, made by Jones,


* \((J)\)

Most of Nixon's utterances about Watergate are false.



In a Tarskian language hierarchy, the sentence \(N\) would have
to be on a higher level than all of Jones' utterances, and,
conversely, the sentence \(J\) would have to be on a higher level
than all of Nixon's utterances. Since \(N\) is an utterance
of Nixon, and \(J\) is an utterance of Jones, \(N\) would
have to be on a higher level than \(J\), and \(J\) on a
higher lever than \(N\). This is obviously not possible, so in a
hierarchy like the Tarskian, these sentences cannot even be
formulated. The sentences \(N\) and \(J\) are indeed both
indirectly self-referential, since \(N\) makes reference to a
totality including \(J\), and \(J\) makes reference to a
totality including \(N\). Nevertheless, in most cases \(N\)
and \(J\) are harmless, and do not produce a paradox. A paradox
is only produced in the special case where all of Jones'
utterances except possibly \(J\) are true, and exactly half of
Nixon's utterances are false, disregarding \(N\). Kripke
uses the fact that \(N\) and \(J\) are only problematic in a
certain special case as an argument against an approach that
altogether excludes the possibility of formulating \(N\) and
\(J\). 



Another argument against the hierarchy approach is that explicit
stratification is not part of ordinary discourse, and thus it might be
considered somewhat *ad hoc* to introduce it into formal
settings with the sole purpose of circumventing the paradoxes. Not
having an explicit stratification in ordinary discourse obviously
doesn't imply the non-existence of an underlying, implicit,
stratification, but at least it's not explicitly represented in
our syntax. 



The arguments given above are among the reasons the work of Russell
and Tarski has not been considered to furnish the final solutions to
the paradoxes. Many alternative solutions have been proposed. One
might for instance try to look for *implicit* hierarchies
rather than *explicit* hierarchies. An implicit hierarchy is a
hierarchy not explicitly reflected in the syntax of the language. In
the following section we will consider some of the solutions to the
paradoxes obtained by such implicit stratifications. 


### 3.2 Building Implicit Hierarchies



Tarski's hierarchy approach to the semantic paradoxes dominated
the field until 1975, when Kripke published his famous and highly
influential paper, "Outline of a Theory of Truth". This
paper has greatly shaped most later approaches to theories of truth
and the semantic paradoxes. It should be noted, however, that ideas
quite similar to Kripke's were developed simultaneously and
independently by Martin and Woodruff (1975), and that a parallel
approach in a set-theoretic setting was developed independently by
Gilmore (1974).


#### 3.2.1 Kripke's Theory of Truth



Kripke's ideas are based on an analysis of the problems involved
in Tarski's hierarchy approach. Kripke lists a number of
arguments against having a language hierarchy in which each sentence
lives at a fixed level, determined by its syntactic form. He proposes
an alternative solution which still uses the idea of having levels,
but where the levels are not becoming an explicit part of the syntax.
Rather, the levels become stages in an iterative construction of a
truth predicate. To explain Kripke's construction, some
additional technical machinery is required.



At each stage in Kripke's construction, the truth predicate is
only *partially defined*, that is, it only applies to some of
the sentences of the language. To deal with such partially defined
predicates, a *three-valued logic* is employed, that is, a
logic which operates with a third value, *undefined*, in
addition to the truth values *true* and *false*. Often
the third value is denoted "\(u\)" or
"\(\bot\)" (bottom). A partially defined predicate only
receives one of the classical truth values, *true* or
*false*, when it is applied to one of the terms for which the
predicate has been defined, and otherwise it receives the value
*undefined*. There are several different three-valued logics
available, differing in how they treat the third value. Here only one
of them is briefly described, called *Kleene's strong
three-valued logic*. More detailed information on this and related
logics can be found in the entry on
 many-valued logic.



In Kleene's strong three-valued logic, the value \(\bot\)
(*undefined*) can be interpreted as "not yet
defined". So one could think of formulae with the value \(\bot\)
as formulae that actually have a classical truth value (*true*
or *false*), but which has just not been determined yet. This
interpretation of *undefined* is reflected in the truth tables
for the logic, given below. The upper truth table is for disjunction,
the lower for negation:




| \(\vee\) | true | false | \(\bot\) |
| --- | --- | --- | --- |
| true | true | true | true |
| false | true | false | \(\bot\) |
| \(\bot\) | true | \(\bot\) | \(\bot\) |




|  |  |
| --- | --- |
| \(\neg\) |  |
| true | false |
| false | true |
| \(\bot\) | \(\bot\) |



These truth tables define the three-valued logic completely, as \(\vee\)
and \(\neg\) are taken to form an adequate set of connectives and
existential and universal quantification are treated as infinite
disjunction and conjunction, respectively. 



To handle partially defined truth predicates, it is necessary to
introduce the notion of partial models. In a *partial model*
for a first-order language, each \(n\)-place predicate symbol
\(P\) is interpreted by a pair \((U,V)\) of
disjoint \(n\)-place relations on the domain of the model.
\(U\) is called the *extension* of \(P\) and
\(V\) its *anti-extension*. In the model, \(P\) is
true of the objects in \(U\), false of the objects in \(V\),
and undefined otherwise. In this way, any atomic sentence receives one
of the truth values *true*, *false* or
*undefined* in the model. Non-atomic formulae are given truth
values in the model by using Kleene's strong three-valued logic
for evaluating the connectives. 



Given the definition of a partial model, a *partially interpreted
language* is a pair \((L,M)\) where \(L\) is a
first-order language and \(M\) is a partial model of \(L\).
Kripke recursively defines a sequence of partially interpreted
languages \(L\_0, L\_1, L\_2,\ldots\), only differing in their
interpretation of the truth predicate \(T\). The first language,
\(L\_0\), is taken to be an arbitrary language in which
both the extension and anti-extension of \(T\) are the empty set.
Thus in \(L\_0\), the truth predicate is completely
undefined. For any \(\alpha\), the language
\(L\_{\alpha +1}\) is like \(L\_{\alpha}\)
except that \(T\) is interpreted by the extension/anti-extension
pair \((U,V)\) given by:


* \(U\) is the set of Godel codes \(\langle \phi \rangle\) of
sentences \(\phi\) true in \(L\_{\alpha}\).
* \(V\) is the set of Godel codes \(\langle \phi \rangle\) of
sentences \(\phi\) false in \(L\_{\alpha}\).



This definition immediately gives that for all \(\alpha\),


* (4)

\(\phi\) is true (false) in \(L\_{\alpha} \Leftrightarrow T\langle \phi
\rangle\) is true (false) in \(L\_{\alpha +1}\).



What has been constructed is a sequence \(L\_0,
L\_1, L\_2,\ldots\) of partially
interpreted languages such that \(T\) is interpreted in
\(L\_{\alpha +1}\) as the truth predicate for
\(L\_{\alpha}\). This is just like Tarski's
hierarchy of languages, except that here there is no syntactic
distinction between the different languages and their truth
predicates.



The sequence \(L\_0, L\_1,
L\_2,\ldots\) has an important property: For each
\(\alpha\), the interpretation of \(T\) in
\(L\_{\alpha +1}\) extends the interpretation of
\(T\) in \(L\_{\alpha}\), that is, both the
extension and anti-extension of \(T\) increase (or stay the same)
when moving from \(L\_{\alpha}\) to
\(L\_{\alpha +1}\). This means that one can define a new
partially interpreted language \(L\_{\omega}\) by letting
the extension of \(T\) be the union of all the extensions of
\(T\) in \(L\_0, L\_1,
L\_2,\ldots\); and similarly for the anti-extension.
Thus in \(L\_{\omega}\), the interpretation of \(T\)
extends the interpretation that \(T\) receives in all previous
languages. This furnishes a strategy for continuing the iterative
construction of a truth predicate into the transfinite: For each
successor ordinal \(\alpha +1\), define
\(L\_{\alpha +1}\) from \(L\_{\alpha}\)
exactly as in the finite case above; and for each limit ordinal
\(\sigma\), define \(L\_{\sigma}\) from the preceding
languages \((L\_i)\_{i\lt \sigma}\) in
the same way as \(L\_{\omega}\) was defined (for a
detailed explanation of the ordinal numbers and their use in this
context, see the entry on
 the revision theory of truth).
 A simple cardinality consideration now shows that this transfinite
sequence of languages will eventually *stabilise*: There is an
ordinal \(\gamma\) such that \(L\_{\gamma} = L\_{\gamma +1}\). Hence the following instance of (4) is
obtained:


* (5)

\(\phi\) is true (false) in \(L\_{\gamma} \Leftrightarrow T\langle \phi
\rangle\) is true (false) in \(L\_{\gamma}\).



This shows that \(L\_{\gamma}\) is actually a language
containing its own truth predicate: Any sentence \(\phi\) is true (false)
if and only if the sentence expressing its truth,
\(T\langle \phi \rangle\), is true (false). The equivalence (5) is
nothing more than a semantic analogue of Tarski's schema
\(T\) in a three-valued setting. The construction of the language
\(L\_{\gamma}\) was one of the major contributions of
Kripke (1975). It shows that in a three-valued logical setting it is
actually possible for a language to contain its own truth predicate.
It is easy to see that the third value, *undefined,* is
essential to making things work: If \(L\_{\gamma}\) had
been a totally interpreted language (that is, a language with no
undefined sentences), then \(L\_{\gamma}\) would satisfy
schema T, by (5) above. However, it immediately contradicts
Tarski's theorem that such a totally interpreted language can
exist.



Among the sentences that receive the value *undefined* in
\(L\_{\gamma}\) is the liar sentence. The solution to the
liar paradox implicit in Kripke's theory is this: Since both
assuming that the liar sentence is true and that it is false leads to
a contradiction it must be neither, it is *undefined*. The liar
sentence is said to suffer from a *truth-value gap*. The idea
of avoiding the liar paradox by allowing truth-value gaps did in fact
appear several times in the literature before Kripke's paper,
but Kripke was among the first to make it an integral part of a
genuine theory.



As with the hierarchy solution to the liar paradox, the truth-value
gap solution is by many considered to be problematic. The main
criticism is that by using a three-valued semantics, one gets an
interpreted language which is expressively weak. One cannot, for
instance, in any of Kripke's languages have a predicate
expressing the property of being *undefined*. This is in fact
noted by Kripke himself. If a partially interpreted language contained
such a predicate, the following *strengthened liar sentence*
within the language could be formulated: "This sentence is
either false or undefined". The strengthened liar sentence is
true if and only if false or undefined, so we have a new paradox,
called the *strengthened liar paradox*. The problem with the
strengthened liar paradox is known as a *revenge problem*:
Given any solution to the liar, it seems we can come up with a new
strengthened paradox, analogous to the liar, that remains unsolved.
The idea is that whatever semantic status the purported solution
claims the liar sentence to have, if we are allowed freely to refer to
this semantic status in the object language, we can generate a new
paradox. 



The inability of the Kripkean language to express its own
*undefined* predicate also means that we cannot in the Kripkean
object-language express a statement such as: "The liar sentence
is undefined". In fact in Kripke's language
\(L\_{\gamma}\), the liar sentence \(is\) undefined,
so the previous sentence expresses a truth about
\(L\_{\gamma}\) that cannot be expressed within
\(L\_{\gamma}\) itself (hence the language is
expressively incomplete). To express the true statement "The
liar sentence is undefined", we are forced to ascend into a
meta-language of \(L\_{\gamma}\). As Kripke (1975)
himself puts it: "The ghost of the Tarski hierarchy is still
with us." 


#### 3.2.2 Extensions and Alternatives to Kripke's Theory of Truth



Succeeding the work of Kripke, many attempts have been made to
construct languages containing their own truth predicate and not
suffering from the revenge problem of strengthened liars. Many of
these attempts have focused on modifying or extending the underlying
strong three-valued logic, e.g. modifying the semantics of the
conditional (Field, 2003, 2008) or allowing an unbounded number of
truth-values (Cook, 2007; Schlenker, 2010; Tourville and Cook, 2016).




Kripke's theory circumvents the liar paradox by assigning it the
value *undefined*. An alternative way to circumvent the liar
paradox would be to assign it the value *both true and false*
in a suitable paraconsistent logic. This would be the correct solution
according to the dialetheist view, cf. Section 2. One of the simplest
paraconsistent logics is LP, which is a three-valued logic with the
same truth tables as Kleene's strong three-valued logic
presented above--the only difference is that the third truth
value is interpreted as *both true and false* rather than
*undefined*. A reason for preferring a paraconsistent logic
over a partial logic is that paradoxical sentences such as the liar
can then be modelled as *true contradictions* (dialetheia)
rather than truth-value gaps. We refer again to the entries on
 dialetheism
 and
 paraconsistent logic
 for more information.



The choice is between *truth-value gaps* and *truth-value
gluts*: A truth-value gap is a statement with no truth-value,
neither true or false (like *undefined* in Kleene's
strong three-valued logic), and a truth-value glut is a statement with
several truth-values, e.g. both true and false (like in the
paraconsistent logic LP). There are also arguments in favour of
allowing both gaps and gluts, e.g. by letting the set of truth-values
form of a bilattice (Fitting, 2006; Odintsov and Wansing, 2015). The
simplest non-trivial bilattice has exactly four values, which in the
context of truth-values are interpreted as: *true*,
*false*, \(\bot\) (neither true nor false), and \(\top\) (both
true and false). 



For a more extensive discussion of Kripke's theory, its
successors and rivals, see the entry on
 the liar paradox.


#### 3.2.3 Implicit Hierarchies in Set Theories



Building implicit rather than explicit hierarchies is also an idea
that has been employed in set theory. New Foundations (NF) by Quine
(1937) is a modification of simple type theory where the
stratification into syntactic types has been replaced by a
stratification on the comprehension principle:



**NF comprehension:**
  

\(\forall u(u \in \{ x \mid \phi(x)\} \leftrightarrow \phi(u))\), for
all *stratified* formulae \(\phi(x)\).




A formula \(\phi\) is *stratified* if there exists a mapping
\(\sigma\) (a *stratification*) from the variables of \(\phi\) to the
natural numbers such that if \(u \in v\) is a
subformula of \(\phi\) then \(\sigma(v) = \sigma(u)+1\)
and if \(u = v\) is a subformula of \(\phi\) then
\(\sigma(v) = \sigma(u)\). Obviously the formula
\(x \not\in x\) is not stratified, and thus the NF
comprehension principle cannot be used to formulate Russell's
paradox in the theory. Quine's New Foundations is essentially
obtained from type theory by hiding the types from the syntax. Thus,
the theory still makes use of a hierarchy approach to avoid the
paradoxes, but the hierarchy is made implicit by not representing it
in the syntax of formulae. Cantini (2015) has investigated the
possibility of mimicking this implicit hierarchy approach in the
context of theories of truth (achieving an implicitly represented
Tarskian truth hierarchy). 



Zermelo-Fraenkel set theory (ZF) is another theory that builds on the
idea of an implicit hierarchy to circumvent the paradoxes. However, it
does so in a much less direct way than NF. In ZF, sets are built
bottom-up, starting with the empty set and iterating a construction of
bigger and bigger sets using the operations of union and power set.
This produces a hierarchy with the empty set on the lowest level,
level 0, and with the power set operation producing a set of level
\(\alpha +1\) from a set of level \(\alpha\). Analogous to Kripke's
iterative construction, the procedure is continued into the
transfinite using the union operator at the limit ordinal levels. The
hierarchy obtained is called the *cumulative hierarchy*. One of
the axioms of ZF, the axiom of foundation, states that every set of ZF
lives on a certain level in this cumulative hierarchy. In other words,
the axiom of foundation states that there are no sets in ZF besides
the ones that can be constructed bottom-up by the iterative procedure
just described. Since in a cumulative hierarchy, there can be no sets
containing themselves, no universal set, and no non-wellfounded sets,
none of the known paradoxes can immediately be formulated in the
theory. This does obviously not in itself ensure the consistency of
ZF, but at least it illustrates how the idea of a set hierarchy plays
a significant role in ZF as well. ZF has a privileged status among set
theories, as it is today the most widely acknowledged candidate for a
formal foundation of mathematics.


### 3.3 General Fixed Point Approaches



Kripke's iterative construction of a truth predicate presented
above may be viewed as an instance of a more general *fixed point
approach* towards building formal theories of truth. Fixed point
approaches have become central to contemporary formal theories of
truth. The main idea is to have a *truth revision operator* and
then look for fixed points of this operator. At heart of such fixed
point approaches is some suitable *fixed point theorem*
guaranteeing the existence of fixed points for certain kinds of
operators. There are several different fixed point theorems available.
Consider now one of the simpler ones.



**Fixed point theorem**.
  

Let \(\tau\) be a monotone operator on a chain complete partial order
(henceforth ccpo). Then \(\tau\) has a least fixed point, that is, there
is a least f such that \(\tau(f) = f\).




A *ccpo* is a partial order \((D,\lt)\) in which every
totally ordered subset of \(D\) has a least upper bound. The
totally ordered subsets of \(D\) are called *chains* in
\(D\). A *monotone operator* on \((D,\lt)\) is a
mapping \(\tau : D \rightarrow D\) satisfying:



\[ 
d\_1 \le d\_2 \Rightarrow \tau(d\_1) \le \tau(d\_2),
 \text{ for all } d\_1, d\_2 \in D.
\]


Kripke's construction fits into the fixed point theorem above in
the following way. First note that the set of partially interpreted
languages that only differ on the interpretation of \(T\) forms a
ccpo: Simply define the ordering on these languages by
\(L\_1 \le L\_2\) iff the
interpretation of \(T\) in \(L\_2\) extends the
interpretation of \(T\) in \(L\_1\) (that is, the
extension and anti-extension of \(T\) in \(L\_1\)
are included in those of \(L\_2)\). Then define a truth
revision operator \(\tau\) on these languages by:


* (6)

\(\tau(L) = L'\), where \(T\langle \phi \rangle\) is true (false) in
\(L'\) iff \(\phi\) is true (false) in \(L\).



Note that if \(L\_{\alpha}\) is one of the languages in
Kripke's construction, then \(L\_{\alpha +1} = \tau(L\_{\alpha})\). The idea of this truth revision
operator \(\tau\) is that if \(\tau(L)=L'\) then
\(L'\) will be a language in which \(T\) is
interpreted as the truth predicate for \(L\). If therefore
\(\tau(L)=L\) for some \(L\), that is, if
\(L\) is a fixed point of \(\tau\), then \(L\) will be a
language containing its own truth predicate. This motivates the search
for fixed points of \(\tau\). Since \(\tau\) is easily seen to be monotone,
by the fixed point theorem it has a least fixed point. It is not hard
to see that this fixed point is exactly the language
\(L\_{\gamma}\) constructed in Kripke's theory of
truth. Kripke's construction is thus recaptured in the setting
of fixed points for monotone operators.



The point of introducing the additional machinery was not just to
rediscover the language \(L\_{\gamma}\). The point is
rather to have provided a much more general and abstract framework
which may lead to new theories of truth and give further insights into
the semantic paradoxes. It turns out that the truth revision operator
\(\tau\) defined above has many interesting fixed-points in addition to
\(L\_{\gamma}\). It is also possible to obtain new
theories of truth by considering alternative ways of making the set of
interpreted languages into a ccpo. One may for instance add an
additional truth value and consider the situation in a four-valued
logic, as considered by Fitting (1997); or one may remove the third
truth value *undefined* and construct a ccpo in a completely
classical setting. In the classical setting, attention is restricted
to the totally interpreted languages (languages in which every
sentence is either true or false), and an ordering on them is defined
by: \(L\_1 \le L\_2\) holds iff the
extension of the truth predicate in \(L\_1\) is included
in the extension of the truth predicate in \(L\_2\),
that is, iff \(L\_2\) points out at least as many
sentences as true as \(L\_1\). This gives a ccpo. By
using the fixed point theorem in this setting on a suitably defined
revision operator, it is fairly easy to prove the existence of a
totally interpreted language containing a *positive definition of
truth*. By this is meant that the interpreted language has a
predicate \(T\) satisfying the following restricted version of
the \(T\)-schema:


* (7)

\(\phi \leftrightarrow T(\langle \phi \rangle)\), for all
*positive* sentences \(\phi\),



where the positive sentences are those built without using negation
\((\neg)\). Since (7) is satisfiable in a totally interpreted language,
the first-order theory containing the sentences of (7) as axioms must
be consistent. This should be contrasted with Tarski's theorem
stating that the *unrestricted* \(T\)-schema is
inconsistent. If the unrestricted comprehension principle is similarly
restricted to the positive formulae, we also get a consistent theory.
This was originally shown by Gilmore (1974).



The fixed point approach is also the point of departure of the
*revision theory of truth* developed by Belnap and Gupta
(1993). The revision theory of truth is the most influential theory of
truth and the semantic paradoxes that has been developed since the
theory of Kripke. The revision theory considers the standard truth
revision operator \(\tau\) defined by (6) as an operator on the
*totally* interpreted languages. On these languages \(\tau\)
doesn't have a fixed point: If it had such a fixed point
\(L\) then \(L\) would be a totally interpreted language
satisfying the full schema \(T\), directly contradicting
Tarski's theorem. Since \(\tau\) doesn't have a fixed point
on the totally interpreted languages, the revision theory instead
considers transfinite sequences \(L\_1,
L\_2\), ... ,
\(L\_{\omega},
L\_{\omega +1}\), ... of totally interpreted
languages satisfying:


* For any successor ordinal \(\alpha +1,
L\_{\alpha +1} = \tau(L\_{\alpha})\).
* For any limit ordinal \(\sigma\) and any sentence \(\phi\), if \(\phi\)
stabilises on the value true (false) in the sequence
\((L\_{\alpha})\_{\alpha \lt \sigma}\) then
\(\phi\) is true (false) in \(L\_{\sigma}\).



In such a sequence, each sentence \(\phi\) will either eventually
stabilise on a classical truth value (true or false), or it will never
stabilise. An example of a sentence that will never stabilise is the
liar sentence: If the liar sentence is true in one of the languages
\(L\_{\alpha}\) it will be false in
\(L\_{\alpha +1}\), and vice versa. The revision theory
thus gives an account of truth that correctly *models* the
behaviour of the liar sentence as one that never stabilises on a truth
value. In the revision theory it is argued that this gives a more
correct account of truth and self-reference than Kripke's theory
in which the liar sentence is simply assigned the value undefined.
Both the revision theory of truth and Kripke-style fixed-point
theories are still being actively researched (Gupta and Standefer,
2017; Hsiung, 2017; Schindler, 2017). A full account of the revision
theory can be found in the entry on
 the revision theory of truth.
 



Studying self-referential phenomena as fixed-points is not limited to
theories of truth. For instance, in the context of epistemic
paradoxes, the Brandenburger-Keisler paradox has been cast as a
fixed-point result by Abramsky and Zvesper (2015).


## 4. Recent Developments



Murzi and Massimiliano (2015) gives an overview of recent developments
in approaches to solving the paradoxes: paracompleteness (allowing
truth-value gaps), paraconsistency (allowing truth-value gluts),
substructural logics (weakening the logical principles of classical
logic), and the revenge problems that these approaches will or could
lead to. Recent developments in substructural logics as a cure to the
paradoxes include French (2016) (dropping reflexivity), Caret, Colin
and Weber (2015), Shapiro and Lionel (2015), Mares and Paoli (2014)
(dropping contraction), and Cobreros, Egre, Ripley and van Rooij
(2014) (dropping transitivity). The volume by Achourioti et al. (eds.,
2015) has several papers on self-reference and how to avoid paradoxes
in the context of theories of truth. 



Volker Halbach and Albert Visser (2014a, 2014b) has made a very
detailed study of self-reference in arithmetic, studying what it means
for a sentence of arithmetic to ascribe itself a property, and how
this depends on the chosen encoding, the details of fixed-point
construction etc.