In the context of language,self-reference is used to denotea statement that refers to itself or its own referent. The most famousexample of a self-referential sentence is theliar sentence:“This sentence is not true.” Self-reference is often usedin a broader context as well. For instance, a picture could beconsidered self-referential if it contains a copy of itself (see theanimated image above); and a piece of literature could be consideredself-referential if it includes a reference to the work itself. Inphilosophy, self-reference is of specific interest in relation to theanalysis of language, but also to the analysis of the mind.Self-reference is also a field of special interest in mathematics andcomputer science, in particular in relation to the foundations ofthese sciences.

We also talk about self-reference when using the first-person pronoun“I” like in “I feel pain” or “I knowthis song”. Self-reference in the sense of an agent referring toherself as well as how self-attribution and self-identificationdiffers from attributing to or identifying others is covered inseveral other entries of this encyclopedia, includingself-knowledge,knowledge of the self,self-consciousness,Descartes’ epistemology, andintrospection.

A major part of the philosophical interest in self-reference iscentered around the paradoxes. Aparadox is a seemingly soundpiece of reasoning, based on apparently true assumptions, that stillleads to a contradiction (Quine, 1976). Consider again the liarsentence from above. It is a sentence \(L\) expressing “Thesentence \(L\) is not true.” We are led to a contradiction whentrying to determine whether \(L\) is true or not. If we first assume\(L\) to be true, then it must be expressing a true statement aboutthe world. As \(L\) expresses “The sentence \(L\) is nottrue,” we now have that \(L\) is not true, which is acontradiction. Suppose, conversely, that \(L\) is not true. Then theexpression “The sentence \(L\) is not true” is true. Butthe expression in the quotes is exactly the statement expressed by\(L\), so \(L\) must be true, again a contradiction. Hence,independently of whether we assume \(L\) to be true or not, we are ledto a contradiction. Thus we now have a contradiction obtained by aseemingly sound piece of reasoning based on apparently trueassumptions. It hence qualifies as a paradox. This paradox is known astheliar paradox. The liar sentence leads to a paradoxbecause it is self-referential, but self-reference is not asufficient condition for paradoxicality. Thetruth-tellersentence “This sentence is true” is not paradoxical,and neither is the sentence “This sentence contains fourwords” (it is false, though).

Most paradoxes of self-reference may be categorised as eithersemantic,set-theoretic orepistemic. Thesemantic paradoxes, like the liar paradox, are primarily relevant totheories of truth. The set-theoretic paradoxes are relevant to thefoundations of mathematics, and the epistemic paradoxes are relevantto epistemology. Even though these paradoxes are different in thesubject matter they relate to, they share the same underlyingstructure, and may often be tackled using the same mathematical means.

In the present entry, we will first introduce a number of the mostwell-known paradoxes of self-reference, and discuss their commonunderlying structure. Subsequently, we will discuss the profoundconsequences that these paradoxes have on a number of different areas:theories of truth, set theory, epistemology, foundations ofmathematics, computability. Finally, we will present the mostprominent approaches to solving the paradoxes.

• 1. Paradoxes of Self-Reference
  • 1.1 Semantic Paradoxes
  • 1.2 Set-Theoretic Paradoxes
  • 1.3 Epistemic Paradoxes
  • 1.4 Common Structures in the Paradoxes
  • 1.5 Paradoxes without Negation
  • 1.6 Paradoxes without Self-Reference
• 2. Why the Paradoxes Matter
  • 2.1 Consequences of the Semantic Paradoxes
  • 2.2 Consequences of the Set-Theoretic Paradoxes
  • 2.3 Consequences of the Epistemic Paradoxes
  • 2.4 Consequences Concerning Provability and Computability
• 3. Solving the Paradoxes
  • 3.1 Building Explicit Hierarchies
  • 3.2 Building Implicit Hierarchies
    • 3.2.1 Kripke’s Theory of Truth
    • 3.2.2 Extensions and Alternatives to Kripke’s Theory of Truth
    • 3.2.3 Implicit Hierarchies in Set Theories
  • 3.3 General Fixed Point Approaches
• 4. Recent Developments
• Bibliography
• Other Internet Resources
• Academic Tools
• Related Entries

1. Paradoxes of Self-Reference

1.1 Semantic Paradoxes

Paradoxes of self-reference have been known since antiquity. Thediscovery of the liar paradox is often credited to Eubulides theMegarian who lived in the 4th century BC. The liar paradox belongs tothe category ofsemantic paradoxes, since it is based on thesemantic notion of truth. Other well-known semantic paradoxes includeGrelling’s paradox, Berry’s paradox, and Richard’sparadox.

Grelling’s paradox involves a predicate defined asfollows. Say a predicate isheterological if it is not trueof itself, that is, if it does not itself have the property itexpresses. 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 tothe paradox is now:

Is “heterological” heterological?

It is easy to see that we obtain a contradiction independently ofwhether we answer “yes” or “no” to thisquestion (the argument runs more or less like in the liar paradox).Grelling’s paradox is self-referential, since the definition ofthe predicate heterological refers toall predicates,including the predicate heterological itself. Definitions such as thiswhich depends on a set of entities, at least one of which is theentity being defined, are calledimpredicative.

Berry’s paradox is another paradox based on animpredicative definition, or rather, an impredicative description.Some phrases of the English language are descriptions of naturalnumbers, for example, “the sum of five and seven” is adescription of the number 12. Berry’s paradox arises when tryingto determine the denotation of the following description:

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

The contradiction is that this description containing 93 symbolsdenotes a number which, by definition, cannot be denoted by anydescription containing less than 100 symbols. The description is ofcourse impredicative, since it implicitly refers toalldescriptions, including itself.

Richard’s paradox considers phrases of the Englishlanguage defining real numbers rather than natural numbers. Forexample, “the ratio between the circumference and diameter of acircle” is a phrase defining the number \(\pi\). Assume anenumeration of all such phrases is given (e.g. by putting them intolexicographical order). Now consider the phrase:

the real number whose \(n\)th decimal place is 1 whenever the \(n\)thdecimal 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 enumeratedphrases, say number \(k\) in this enumeration. But, at the same time,by definition, it differs from the number denoted by the \(k\)thphrase in the \(k\)th decimal place. Thus we have a contradiction. Thedefining phrase is obviously impredicative. The particularconstruction employed in this paradox is calleddiagonalisation. Diagonalisation is a general constructionand proof method originally invented by Georg Cantor (1891) to provethe uncountability of the power set of the natural numbers. It wasalso used as a basis for Cantor’s paradox, one of theset-theoretic paradoxes to be considered next.

1.2 Set-Theoretic Paradoxes

The best-known set-theoretic paradoxes are Russell’s paradox andCantor’s paradox.Russell’s paradox arises fromconsidering theRussell set \(R\) of all sets that are notmembers of themselves, that is, the set defined defined by \(R = \{ x\mid x \not\in x \}\). The contradiction is then derived by askingwhether \(R\) 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 \(R \inR\), again by definition of \(R\).

Cantor’s paradox is based on an application ofCantor’s theorem.Cantor’s theorem states thatgiven any finite or infinite set \(S\), the power set of \(S\) hasstrictly larger cardinality (greater size) than \(S\). The theorem isproved by a form of diagonalisation, the same idea underlyingRichard’s paradox. Cantor’s paradox considers the set ofall sets. Let us call this set theuniversal set and denoteit by \(U\). The power set of \(U\) is denoted\(\wp(U)\).Since \(U\) containsall setsit 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.

TheHypergame paradox is a more recent addition to the listof set-theoretic paradoxes, invented by Zwicker (1987). Let us call atwo-player gamewell-founded if it is bound to terminate in afinite number of moves. Tournament chess is an example of awell-founded game. We now definehypergame to be the game inwhich player 1 in the first move chooses a well-founded game to beplayed, and player 2 subsequently makes the first move in the chosengame. All remaining moves are then moves of the chosen game. Hypergamemust be a well-founded game, since any play will last exactly one movemore than some given well-founded game. However, if hypergame iswell-founded then it must be one of the games that can be chosen inthe first move of hypergame, that is, player 1 can choose hypergame inthe first move. This allows player 2 to choose hypergame in thesubsequent move, and the two players can continue choosing hypergamead infinitum. Thus hypergame cannot be well-founded,contradicting our previous conclusion.

1.3 Epistemic Paradoxes

The most well-know epistemic paradox is theparadox of theknower (Kaplan & Montague, 1960; Montague, 1963). Thisparadox has many equivalent formulations, one of them based on thesentence “This sentence is not known by anyone.” Let uscall this sentence theknower sentence, abbreviated \(KS.KS\) is obviously quite similar to the liar sentence, except thecentral concept involved is knowledge rather than truth. The reasoningleading to a contradiction from \(KS\) is a bit more complex than inthe liar paradox. First \(KS\) is shown to be true by the followingpiece 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 bysomeone. Since everything known is true (this is part of thedefinition of the concept of knowledge), \(KS\) is true, contradictingour 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 reasoningcapabilities. That is, an agent should be able to prove the truth of\(KS\), and thus come to know that \(KS\) holds. However, if \(KS\) isknown by someone, then what it expresses is not the case, and thus itcannot 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 basedon a sentence, \(KS\), referring directly to itself.

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

1.4 Common Structures in the Paradoxes

The paradoxes above are all quite similar in structure. In the case ofthe paradoxes of Grelling and Russell, this can be seen as follows.Define theextension of a predicate to be the set of objectsit is true of. For a predicate \(P\) we denote its extension byext\((P)\). Grelling’s paradox involves the predicateheterological, which is true of all those predicates that are not trueof themselves. Thus the extension of the predicate heterological isthe set \(\{ P \mid P \not\in\) ext\((P) \}\). Compare this to theRussell set \(R\) given by \(\{ x \mid x \not\in x \}\). The onlysignificant difference between these two sets is that the first isdefined on predicates whereas the second is defined on sets. Theproofs of contradictions based on these two sets also share the samestructure, as seen below (where “het” abbreviates“heterological”):

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

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

Note the similarity between this sequence of equivalences and thecorresponding sequences of equivalences derived for Russell’sand Grelling’s paradoxes above. Now consider the special case ofCantor’s theorem where \(S\) is the universal set. Then we cansimply choose \(f\) to be the identity function on \(\wp(S)\), since\(S\) is the universal set and hence \( \wp(S) \subseteq S\) (any setmust be a subset of the universal set). So \(f\) is the partialfunction \(f: \wp(S) \to \wp(S)\) defined by \(f(x) = x\). But then\(C\) above becomes the Russell set, and the sequence of equivalencesbecome the proof of contradiction in Russell’s paradox! ThusCantor’s paradox is nothing more than a slight variant ofRussell’s paradox; the core argument leading to thecontradiction is the same in both.

Priest (1994) gives even firmer evidence to the similarity between theparadoxes of self-reference by showing that they all fit into what heoriginally called theQualified Russell’s Schema, nowtermed theInclosure Schema. The idea behind it goes back toRussell himself (1905) who also considered the paradoxes ofself-reference to have a common underlying structure. Given twopredicates predicates \(P\) and \(Q\), and a possibly partial function\(\delta\), the Inclosure Schema consists of the following twoconditions:

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 bycondition 1, we have both \(\delta(w) \not\in w\) and \(\delta(w) \inw\), by 2a and 2b, respectively. Thus any triple \((P,Q,\delta)\)satisfying the Inclosure Schema will produce a paradox. Priest showshow most of the well-known paradoxes of self-reference fit into theschema. Below we will consider only a few of these paradoxes, startingwith 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 thecontradiction obtained from the schema becomes Russell’sparadox.

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

• \(P(x)\) is the predicate “\(x\) is a real definable by aphrase in English.”
• \(Q(y)\) is the predicate “\(y\) is a denumerable set ofreals definable by a phrase in English.”
• \(\delta\) is the function that maps any denumerable set \(y\) ofreals 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. (Anyenumeration of the elements in \(y\) will do.)

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

The liar paradox also fits Russell’s schema, albeit in aslightly 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 belongto 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: “thissentence 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 underlyingstructure—independent of whether they are semantic,set-theoretic or epistemic. Priest (1994) argues that they should thenalso share a common solution. Priest calls this theprinciple ofuniform solution: “same kind of paradox, same kind ofsolution.” Whether the Inclosure Schema can in full generalitycount as a necessary and sufficient condition for self-referentialparadoxicality is however disputable (Slater, 2002; Abad, 2008;Badici, 2008; Zhong, 2012, and others), hence not all authors agree onthe principle of uniform solution either.

Thesorites paradox is a paradox that on the surface does not involve self-reference atall. However, Priest (2010b, 2013) argues that it still fits theinclosure 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 asthe paradoxes of self-reference. This has led Colyvan (2009), Priest(2010) and Weber (2010b) to all advance a dialetheic approach tosolving the Sorites paradox. This approach to the Sorites paradox hasbeen attacked by Beall (2014a, 2014b) and defended by Weber et al.(2014). Cobreros et al. (2015) investigate the notion ofpermissive consequence with an aim to give a unifiedtreatment of the paradoxes of vagueness (like the sorites paradox) andthe paradoxes of self-reference. The permissive consequence relationis a weakened version of the classical consequence relation in thesetting of multi-valued logic: it only requires that when the premisestake value 1 (are true-only), then the conclusion must not take value0 (is not false-only). An even more recent unified diagnosis ofsemantic and soritical paradoxes is by Bruni & Rossi (2023),identifying their source in a general form of indiscernibility.

1.5 Paradoxes without Negation

Most paradoxes considered so far involve negation in an essential way,e.g. sentences saying of themselves that they arenot true orknowable. The central role of negation will become even clearer whenwe formalise the paradoxes of self-reference in Section 2 below.Curry’s paradox is a similar paradox of self-referencethat however does not directly involve negation. A semantic variant ofCurry’s paradox comes from the followingCurry sentence\(C\): “If this sentence is true then \(F\)”, where \(F\)can be any statement, for instance an obviously false one. Suppose theCurry sentence \(C\) is true. Then it expresses a true fact, that is,if \(C\) is true then \(F\). However, we already assumed \(C\) to betrue, so we can infer \(F\), using Modus Ponens. We have now provedthat if we assume \(C\) to be true, then \(F\) follows. This isexactly what the Curry sentence itself expresses. In other words, wehave proved that the Curry sentence itself is true! But then we alsohave that \(F\) is true, and this is a paradox, since \(F\) can be anystatement, including things that are obviously false. We can forinstance easily prove that Santa Claus exists by simply letting \(F\)be the sentence “Santa Claus exists” (Boolos, 1993;Smullyan, 2006). In a classical logical setting where the implication\(C \rightarrow F\) is equivalent to \(\neg C \vee F\), Curry’sparadox still implicitly involves negation, but Curry’s paradoxis still independently interesting since it goes through with fewerassumptions about the underlying logic than the liar paradox. See theentry onCurry’s paradox for more details.

1.6 Paradoxes without Self-Reference

Most classical paradoxes of self-reference involvedirectself-reference as in the liar paradox where a sentence refers directlyto itself. However, it is easy to construct paradoxes that only employindirect self-reference, i.e., sentences that refer to othersentences that refer to yet other sentences in such a way as to form aloop back to the original sentence. One example is thepostcardparadox, often attributed to Philip Jourdain (1879–1919),though according to Roy Sorensen (2003, p. 332), the true inventor isG.G. Berry (1867–1928), the Oxford librarian to whom also theearlier mentioned Berry’s paradox is accredited. In the postcardparadox, the front side of a postcard reads “the sentence on theback side is true”, whereas the back side reads “thesentence on the front side is false”. For the sentence on thefront to be true, the sentence on the back needs to be true, but forthe sentence on the back to be true, the sentence on the front needsto be false. This is a contradiction, achieved similarly as in theliar paradox. There are even much earlier examples of indirectself-reference in the literature: Sophism 9 in John Buridan’s14th century Sophismata (Buridan [SD], Hughes 1982) is structurallyequivalent to the postcard paradox.

In 1985, Yablo succeeded in constructing a semantic paradox that doesnot involve self-reference at all, not even indirect self-reference.Instead, it consists of an infinite chain of sentences, each sentenceexpressing the untruth of all the subsequent ones. More precisely, foreach natural number \(i\) we define \(S_i\) to be the sentence“for all \(j\gt i, S_j\) is not true”. We can then derivea contradiction as follows:

Yablo calls this paradox the \(\omega\)-liar, but othersusually refer to it asYablo’s paradox. Note that noneof the sentences \(S_i\) refer to themselves (not even indirectly),but only to the ones that occur later in the sequence. Yablo’sparadox is semantic, but as shown by Yablo (2006), similarset-theoretic paradoxes involving no self-reference can be formulatedin certain set theories.

Yablo’s paradox demonstrates that we can have logical paradoxeswithout self-reference—only a certain kind ofnon-wellfoundedness is needed to obtain a contradiction. There areobviously structural differences between the ordinary paradoxes ofself-reference and Yablo’s paradox: The ordinary paradoxes ofself-reference involve a cyclic structure of reference, whereasYablo’s paradox involve an acyclic, but non-wellfounded,structure of reference. More precisely, we can think the referentialstructure underlying a paradox as a directed graph. The vertices ofthis graph are sentences, and there is a directed edge from sentence\(S\) to sentence \(T\) if \(S\) refers directly to \(T\). Thereferential structure of the liar is then a graph with a singlereflexive loop. The referential structure of the postcard paradox is acyclic graph with 2 vertices each having an edge to the other vertex.All paradoxes of direct or indirect self-reference have cyclicstructures of reference (their underlying graphs are cyclic). It isdifferent with Yablo’s paradox. The referential structure inYablo’s paradox is isomorphic to the usual less-than ordering onthe natural numbers, which is a strict total order (contains nocycles). Even though there is this difference, Yablo’s paradoxstill share most properties with the ordinary paradoxes ofself-reference. When solving paradoxes we might thus choose toconsider them all under one, and refer to them asparadoxes ofnon-wellfoundedness. In the following we will however stick tothe termparadoxes of self-reference, even though most ofwhat we say will apply to Yablo’s paradox and related paradoxesof non-wellfoundedness as well.

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

Significant amounts of newer work on self-reference has gone intotrying to make a complete graph-theoretical characterisation of whichstructures of reference admit paradoxes, including Rabern and Macauley(2013), Cook (2014) and Dyrkolbotn and Walicki (2014). A completecharacterisation is still an open problem (Rabern, Rabern andMacauley, 2013), but it seems to be a relatively widespread conjecturethat all paradoxical graphs of reference are either cyclic or containa Yablo-like structure. The conjecture has been confirmed for certainsub-classes of infinite graphs (Walicki, 2019), but it is still openwhether it holds for arbitrary graphs. If the conjecture is indeedtrue, it means that in terms of structures of reference, all paradoxesof reference are either liar-like or Yablo-like. What it exactly meansfor a structure (graph) to be Yablo-like can be defined in severaldifferent, but equivalent, ways, including: 1) the graph contains thereference graph of the Yablo paradox, \( (\omega,\lt) \), as afinitary minor (Beringer & Schindler, 2017); 2) the graph containsa ray (an infinite path) on which there are infinitely many verticeseach having infinitely many disjoint paths to infinitely many othervertices on the ray (Walicki, 2019).

Even though the structure of reference involved in Yablo’sparadox does not contain any cycles (each sentence only refers tolater sentences in the sequence), it is still being discussed whetherthe 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 otherYablo-like paradoxes that are not self-referential in the sense ofPriest. In the analysis of Yablo’s paradox, it is essential tonote that it involves aninfinite sequence of sentences,where each sentence refers toinfinitely many othersentences. To formalise it in a setting of propositional logic, it ishence necessary to use infinitary propositional logic (see the entryoninfinitary logic). Any finitary variant of Yablo’s sequence—where everysentence only refers to finitely many later sentences in thesequence—must necessarily be consistent (non-paradoxical) due tothe compactness theorem in propositional logic (every finite subset ofsentences in the sequence induces a well-founded reference relationand the sentences can hence consistently be assigned truth-valuesbottom-up). In finitary first- and second-order arithmetic, one caninstead attempt to formalise Yablo’s paradox by a unarypredicate \(S(x)\) where, for every natural numbers \(i\),\(S({\inumeral})\) represents the formalisation of the \(i\)thsentence \(S_i\) of the Yablo sequence (where \(\inumeral{ }\) is thenumeral representing \(i)\). How and whether the Yablo paradox cantruthfully be represented this way, and how it relates to compactnessof the underlying logic, has been investigated by Picollo (2013).

Yablo’s paradox has also inspired the creation of similarparadoxes involving non-wellfounded, acyclic structures of referencein other areas than truth, e.g. the “Yabloesque” variantof the Brandenburger-Keisler paradox of epistemic game theory byBaşkent (2016), a variant concerning provability byCieśliński and Urbaniak (2013), and a variant in the contextof Gödel’s incompleteness theorems by Leach-Krouse(2014).

2. Why the Paradoxes Matter

After having presented a number of paradoxes of self-reference anddiscussed some of their underlying similarities, we will now turn to adiscussion of theirsignificance. The significance of aparadox is its indication of a flaw or deficiency in our understandingof the central concepts involved in it. In case of the semanticparadoxes, it seems that it is our understanding of fundamentalsemantic concepts such astruth (in the liar paradox andGrelling’s paradox) anddefinability (in Berry’sand Richard’s paradoxes) that are deficient. In case of theset-theoretic paradoxes, it is our understanding of the concept of aset. If we fully understood these concepts, we should be ableto deal with them without being led to contradictions.

To illustrate this, consider the case of Zeno’s classicalparadox onAchilles and the Tortoise (see the entryZeno’s paradoxes for details). In this paradox we seem able to prove that the tortoisecan win a race against the 10 times faster Achilles if given anarbitrarily small head start. Zeno used this paradox as an argumentagainst the possibility of motion. It has later turned out that theparadox rests on an inadequate understanding of infinity. Moreprecisely, it rests on an implicit assumption that any infinite seriesof positive reals must have an infinite sum. The later developments ofthe mathematics of infinite series has shown that this assumption isinvalid, and thus the paradox dissolves. The original acceptance ofZeno’s argument as a paradox was a symptom that the concept ofinfinity was not sufficiently well understood at the time. In analogy,it seems reasonable to expect that the existence of semantic andset-theoretic paradoxes is a symptom that the involved semantic andset-theoretic concepts are not yet sufficiently well understood. Or,at least, that we need to revise our view on what counts as“natural assumptions”. Russell’s paradox rests onthe assumption that any predicate on mathematical objects determines aset consisting of exactly the objects that satisfy the predicate, andthe liar paradox rests on the assumption that it is possible for alanguage to contain its own truth predicate. As a response to theparadoxes, these otherwise seemingly sensible assumptions have beenrevised, see Section 3 below.

Another possible answer could be that it is our understanding of thevery concept of “contradiction” that is flawed. Thereasoning involved in the paradoxes of self-reference all end up withsome contradiction, a sentence concluded to be both true and false. Weconsider this an impossibility, hence the paradox, but maybe wedon’t have to?Dialetheism is the view that there canbe “true contradictions”, meaning that it is notimpossible for a sentence to be both true and false. If adopting theview of dialetheism, all the paradoxes of self-reference dissolve andinstead become existence proofs of certain “dialetheia”:sentences being both true and false. Priest (1987) is a strongadvocate of dialetheism, and uses his principle of uniform solution(see Section 1.4 above) to defend the dialetheist solution. See theentries ondialetheism andparaconsistent logic for more information.

Currently, no commonly agreed upon solution to the paradoxes ofself-reference exists. They continue to pose foundational problems insemantics and set theory. No claim can be made to a solid foundationfor these subjects until a satisfactory solution to the paradoxes hasbeen 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, inconsistentsystems linger as the paradoxes will be formalisable in thesesystems.

2.1 Consequences of the Semantic Paradoxes

The liar paradox is a significant barrier to the construction offormal theories of truth as it produces inconsistencies in thesepotential theories. A substantial amount of research in self-referenceconcentrates on formal theories of truth and ways to circumvent theliar paradox. There are two articles that have influenced the work onformal theories of truth and the liar paradox more than any other:Alfred Tarski’s “The Concept of Truth in FormalisedLanguages” (1935) and Saul Kripke’s “Outline of aTheory of Truth” (1975). Below we first introduce some of theideas and results of Tarski’s article. Kripke’s article isdiscussed in Section 3.2.

Tarski gives a number of conditions that, as he puts it, anyadequate definition of truth must satisfy. The central ofthese conditions is what is now most often referred to asSchemaT (or theT-schema orConvention T or theTarski biconditionals):

Here \(T\) is the predicate intended to express truth and \(\langle\phi \rangle\) is aname for the sentence \(\phi\). Applyingthe predicate \(T\) to the name \(\langle \phi \rangle\) gives theexpression \(T\langle \phi \rangle\) intended to represent the phrase“\(\phi\) is true”. Schema \(T\) thus expresses that forevery sentence \(\phi , \phi\) holds if and only if the sentence“\(\phi\) is true” holds. The \(T\)-schema is usuallyregarded as a set of sentences within a formal theory. It is customaryto use first-order arithmetic, that is, first-order predicate logicextended with a set of standard axioms for arithmetic like PA (PeanoArithmetic) or Robinson’s Q. What is being said in the followingwill apply to any such first-order formalisation of arithmetic. Inthis setting, \(\langle \phi \rangle\) above denotes theGödel code of \(\phi\), and \(T\langle \phi \rangle\) isshort for \(T(\langle \phi \rangle)\). The reader not familiar withGödel codings (also known as Gödel numberings) can justthink of the mapping \(\langle \cdot \rangle\) as a naming device orquotation mechanism for formulae—just like quotation marks innatural 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 formaltheory containing his schema T, and thus any such theory must beinconsistent. This result is often referred to asTarski’stheorem on the undefinability of truth. The result is basically aformalisation of the liar paradox within first-order arithmeticextended with the \(T\)-schema. In order to construct such aformalisation it is necessary to be able to formulate self-referentialsentences (like the liar sentence) within first-order arithmetic. Thisability is provided by the diagonal lemma.

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

Here the notation \(S \vdash \alpha\) means that \(\alpha\) isprovable in the theory S, and \(\phi \langle \psi \rangle\) is shortfor \(\phi(\langle \psi \rangle)\). Assume a formula \(\phi(x)\) isgiven that is intended to express some property of sentences –truth, for instance. Then the diagonal lemma gives the existence of asentence \(\psi\) 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 thusexpresses that \(\psi\) is equivalent to the sentence expressing that\(\psi\) has property \(\phi\). One can therefore think of \(\psi\) asa sentenceexpressing of itself that it has property\(\phi\). In the case of truth, it would be a sentence expressing ofitself that it is true. The sentence \(\psi\) is of course notself-referential in a strict sense, but mathematically it behaves likeone. It is therefore possible to use sentences generated by thediagonal lemma to formalise paradoxes based on self-referentialsentences, like the liar. The diagonal lemma is sometimes called thefixed-point lemma, since the equivalence \(\psi\leftrightarrow \phi \langle \psi \rangle\) can be seen as expressingthat \(\psi\) is a fixed-point of \(\phi(x)\).

A theory in first-order predicate logic is calledinconsistent if a logical contradiction is provable in it.Tarski’s theorem (on the undefinability of truth) may now bestated and proved.

Note that the contradiction \(T\langle \lambda \rangle \leftrightarrow\neg T\langle \lambda \rangle\) above expresses: The liar sentence istrue if and only if it is not. Compare this to the informal liarpresented in the beginning of the article. Tarski’s theoremshows that, in the setting of first-order arithmetic, it is notpossible to give what Tarski considers to be an “adequate theoryof truth”. The central question then becomes: How may the formalsetting or the requirements for an adequate theory of truth bemodified to regain consistency—that is, to prevent the liarparadox from trivialising the system? There are many different answersto this question, as there are many different ways to regainconsistency. In Section 3 we will review the most influentialapproaches.

2.2 Consequences of the Set-Theoretic Paradoxes

The set-theoretic paradoxes constitute a significant challenge to thefoundations of mathematics. They show that it is impossible to have aconcept of set satisfying theunrestricted comprehensionprinciple (also calledfull comprehension orunrestricted abstraction):

Unrestricted comprehension:
\(\forall u (u \in \{ x \mid \phi(x)\} \leftrightarrow \phi(u))\), forall formulae \(\phi(x)\).

In an informal setting, the formulae \(\phi(x)\) could be allowed tobe arbitrary predicates. In a more formal setting they would beformulae of e.g. a suitable first-order language. The unrestrictedcomprehension principle says that for any property (expressed by\(\phi)\) there is theset of those entities that satisfy theproperty. This sounds as a very reasonable principle, and it more orless captures the intuitive concept of a set. Indeed, it is theconcept of set originally brought forward by the father of set theory,Georg Cantor (1895), himself. Unfortunately, the principle is notsound, as it gives rise to Russell’s paradox. Consider theproperty 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 comprehensionprinciple:

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

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

Theorem (Inconsistency of Naive Set Theory).
Any theory containing the unrestricted comprehension principle isinconsistent.

Compare this theorem with Tarski’s theorem. Tarski’stheorem expresses that if we formalise the intuitively most obviousprinciple concerning truth we end up with an inconsistent theory. Thetheorem above expresses that the same thing happens when formalisingthe intuitively most obvious principle concerning set existence andmembership.

Given the inconsistency of unrestricted comprehension, the objectivebecomes to find a way to restrict either the comprehension principleitself or the underlying logical principles to regain a consistenttheory, that is, a set theory that will not be trivialised byRussell’s paradox. Many alternative set theories excluding theunrestricted comprehension principle have been developed during thelast century, among them the type theory of Russell and Whitehead,Simple Type theory (ST), Gödel-Bernays set theory (GB),Zermelo-Fraenkel set theory (ZF), and Quine’s New Foundations(NF). These are all believed to be consistent, although no simpleproofs of their consistency are known. At least they all escape theknown paradoxes of self-reference. We will return to a discussion ofthis in Section 3.

2.3 Consequences of the Epistemic Paradoxes

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

A1.

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

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

A2.

\(K\langle K\langle \phi \rangle \rightarrow \phi \rangle\), forall sentences \(\phi\).

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

A3.

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

Furthermore, knowability must be closed under logicalconsequences:

A4.

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

Now, the principles A1–A4 is all it takes to formalise theparadox of the knower. More precisely, we have the following theorem,from Montague (1963), whose proof may be given in the following form(see Bolander 2004).

The above proof directly mimicks the reasoning underlying the paradoxof the knower. Montague’s theorem shows that in the setting offirst-order arithmetic we cannot have a theory of knowledge orknowability satisfying even the basic principles A1–A4.Montague’s theorem is a generalisation of Tarski’stheorem. If a predicate symbol \(K\) satisfies Tarski’s schema\(T\) then it is easy to see that it will also satisfy axiom schemasA1–A4. Thus axiom schemas A1–A4 constitute a weakening ofthe \(T\)-schema, and Montague’s theorem shows that even thismuch weaker version of the \(T\)-schema is sufficient to produceinconsistency. One possible reply might be that even axiom schemasA1–A4 are too strong, and should be weakened further. However,as in the previous formalisations of paradoxes, it’s not clearhow to weaken the assumptions further, as all assumptions seemsensible and natural for the concept we’re formalising(knowability in this case). Another possible way out of theinconsistency result might be to preserve the principles in theircurrent form, but only apply them to a subset of the availablesentences (meaning that A1–A4 are only required to hold forsentences \(\phi \in S\) for some subset \(S\)). The principles shouldhold for all “normal” sentences, but we might not want toinsist on them holding for certain pathological sentences expressingself-referential statements. It is however not at all clear that wecan sensibly point out exactly which sentences are normal and whichare pathological. But we might still be able to findsomesensible subsets to which we can instantiate our principles and stillhave a consistent theory. Revières and Levesque (1988) showedthat the principles stay consistent when only instantiated over theso-calledregular sentences, and this result was latergeneralised to the more inclusive class of so-calledRPQsentences (Morreau and Kraus, 1998). Most solutions to theparadox-driven inconsistency results are actually of this kind:stratify or limited the applicability of the principles that lead toinconsistency, whether it’s the principles of truth, setexistence, knowability or something fourth. We will look much moreinto this in Section 3.

Formalising knowledge as a predicate in a first-order logic isreferred to as thesyntactic treatment of knowledge.Alternatively, one can choose to formalise knowledge as a modaloperator in a suitable modal logic. This is referred to as thesemantic treatment of knowledge (see the entry onepistemic logic). In the semantic treatment of knowledge, one generally avoids problemsof self-reference, and thus inconsistency, but it is at the expense ofthe expressive power of the formalism (the problems of self-referenceare avoided by propositional modal logic not admitting anythingequivalent to the diagonal lemma for constructing self-referentialformulas). In fact, the regular sentences mentioned above are achievedexactly by a translation of first-order modal logic into predicatelogic, so only instantiating the principles over the regular sentencesis a way of following the syntactic treatment of knowledge, but stillensure consistency by limiting the knowledge operator to theexpressivity it has in the semantic treatment.

2.4 Consequences Concerning Provability and Computability

The central argument given in the proof of Tarski’s theorem isclosely related to the central argument in Gödel’s firstincompleteness theorem (Gödel, 1931). Gödel’s theoremcan be given the following formulation.

Gödel’s first incompleteness theorem.
If first-order arithmetic is \(\omega\)-consistent, then it isincomplete.

A theory is called \(\omega\)-consistent if the followingholds for every formula \(\phi(x)\) containing \(x\) as its only freevariable: If \(\vdash \neg \phi(n)\) (meaning: \(\neg \phi(n)\) isprovable) for every natural number \(n\) then \(\not\vdash \existsx\phi(x)\) (meaning: \(\exists x\phi(x)\) is not provable).\(\omega\)-consistency is a stronger condition than ordinaryconsistency, so any \(\omega\)-consistent theory will also beconsistent. A theory isincomplete if it contains a formulawhich can neither be proved nor disproved.

In the proof above we reduced Gödel’s incompletenesstheorem to an application of Tarski’s theorem in order to showthe close link between the two (this version of the proof is due toBolander, 2002). Gödel was well aware of this link, and indeed itseems that Gödel even proved Tarski’s theorem before Tarskihimself did (Feferman, 1984). Gödel’s theorem can beinterpreted as demonstrating a limitation in what can be achieved bypurely formal procedures. It says that if first-order arithmetic is\(\omega\)-consistent (which it is believed to be), then there must bearithmetical sentences that can neither be proved nor disproved by theformal procedures of first-order arithmetic. One might at first expectthis limitation to be resolvable by the inclusion of additionalaxioms, but Gödel showed that the incompleteness result stillholds when first-order arithmetic is extended with an arbitrary finiteset of axiom schemas (or, more generally, an arbitrary recursive setof axioms). Thus we obtain a general limitation result saying thatthere cannot exist a formal proof procedure by which any givenarithmetical sentence can be proven to hold or not to hold. For moredetails on Gödel’s incompleteness theorem, see the entry onKurt Gödel.

The limitation result of Gödel’s theorem is closely relatedto another limitation result known as theundecidability of thehalting problem. This is a result stating that there arelimitations to what can be computed. We will present this result inthe following. The result is based on the notion of aTuringmachine, which is a generic model of a computer program runningon a computer having unbounded memory. Thus any program running on anycomputer can be thought of as a Turing machine (see the entry onTuring machines for more details). When running a Turing machine, it will eitherterminate after a finite number of computation steps, or it willcontinue running forever. In case it terminates after a finite numberof computation steps, we say that ithalts. Thehaltingproblem is the problem of finding a Turing machine that candecide whether other Turing machines halt or not. We say that a Turingmachine \(H\)decides the halting problem if the followingholds:

\(H\) takes as input a pair \((\langle A\rangle ,x)\) consisting ofthe Gödel code \(\langle A\rangle\) of a Turing machine \(A\) andan arbitrary string \(x. H\) returns the answer “yes” ifthe Turing machine \(A\) halts when given input \(x\), and“no” otherwise.

Thus if a Turing machine \(H\) decides the halting problem, we can useit to determine for an arbitrary Turing machine \(A\) and arbitraryinput \(x\) whether \(A\) will halt on input \(x\) or not. Theundecidability of the halting problem is the followingresult, due to Turing (1937), stating that no such machine can exist:

From the two theorems above we see that in the areas of provabilityand computability, the paradoxes of self-reference turn intolimitation results: there are limits to what can be proven and whatcan be computed. This is actually quite similar to what happened inthe areas of semantics, set theory and epistemology: The paradoxes ofself-reference turned into theorems showing that there are limits towhich properties we can consistently assume a truth predicate to have(Tarski’s theorem), a set theory to have (inconsistency of naiveset theory), and a knowledge predicate to have (Montague’stheorem). It is hard to accept these limitation results because mostof them conflict with our intuitions and expectations. The centralrole played by self-reference in all of them potentially makes themeven harder to accept, at least it definitely makes them morepuzzling. However, we are forced to accept them, and forced to acceptthe 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 onehas to weaken some of the assumptions leading to the contradiction. Itis very difficult to choose which assumptions to weaken, since each ofthe explicitly stated assumptions underlying a paradox appears to be“obviously true”—otherwise it would not qualify as aparadox. Below we will take a look at the most influential approachesto solving the paradoxes.

So far the presentation has been structured according to type ofparadox, that is, the semantic, set-theoretic and epistemic paradoxeshave been dealt with separately. However, it has also beendemonstrated that these three types of paradoxes are similar inunderlying structure, and it has been argued that a solution to oneshould be a solutions to all (the principle of uniform solution).Therefore, in the following, the presentation will be structured notaccording to type of paradox but according to type of solution. Eachtype of solution considered in the following can be applied to any ofthe paradoxes of self-reference, although in most cases theconstructions involved were originally developed with only one type ofparadox 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 tohis paradox—as well as Tarski’s original solution to hisundefinability of truth problem—was to buildhierarchies. In Russell’s case, this led totypetheory. In Tarski’s case, it led to what is now known asTarski’s hierarchy of languages. In both cases, theidea is to stratify the universe of discourse (sets, sentences) intolevels. In type theory, these levels are calledtypes. Thefundamental idea of type theory is to introduce the constraint thatany set of a given type may only contain elements of lower types (thatis, may only contain sets which are located lower in thestratification). 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 thefollowing way. Assume one wants to equip a language \(L_0\) with atruth predicate. From Tarski’s theorem (Section 2.1) it is knownthat this truth predicate cannot be part of the language \(L_0\)itself—at least not as long as we want the truth predicate tosatisfy schema \(T\). Instead one builds a hierarchy of languages,\(L_0, L_1, L_2,\ldots\), where each language \(L_{i+1}\) has a truthpredicate \(T_{i+1}\) that only applies to the sentences of \(L_j,j\le i\). In this hierarchy, \(L_0\) is called theobjectlanguage and, for all \(i, L_{i+1}\) is called themeta-language of \(L_i\). This hierarchy effectively blocksthe liar paradox, since now a sentence can only express the truth oruntruth of sentences at lower levels, and thus a sentence such as theliar that expresses itsown untruth cannot be formed.

Russell’s type theory can be regarded as a solution toRussell’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 tothe liar paradox. It is the same stratification idea that underliesboth of Russell’s and Tarski’s solutions. The point tonote is that Russell’s paradox and the liar paradox dependcrucially on circular notions (self-membership andself-reference). By making a stratification in which anobject may only contain or refer to objects at lower levels,circularity disappears. In the case of the epistemic paradoxes, asimilar stratification could be obtained by making an explicitdistinction between first-order knowledge (knowledge about theexternal world), second-order knowledge (knowledge about first-orderknowledge), third-order knowledge (knowledge about second-orderknowledge), and so on. This stratification actually comes for free inthe semantic treatment of knowledge, where knowledge is formalised asa modal operator.

Building explicit hierarchies is sufficient to avoid circularity, andthus sufficient to block the standard paradoxes of self-reference.However, there also exist paradoxes such as Yablo’s that do notrely on circularity and self-reference. Such paradoxes can also beblocked by a hierarchy approach, but it is necessary to furtherrequire the hierarchy to be well-founded, that is, to have a lowestlevel. Otherwise, the paradoxes of non-wellfoundedness can still beformulated. For instance, Yablo’s paradox may be formalised in adescending hierarchy of languages. A descending hierarchy oflanguages consists of languages \(L_0, L_{-1}, L_{-2},\ldots\) whereeach language \(L_{-i}\) has a truth predicate that only applies tothe sentences of the languages \(L_{-j}, j\gt i\). Similarly, aset-theoretic paradox of non-wellfoundedness may be formulated in atype theory allowing negative types. The conclusion drawn is that astratification of the universe is not itself sufficient to avoid allparadoxes—the stratification also has to be well-founded.

Building an explicit (well-founded) hierarchy to solve the paradoxesis today by most considered an overly drastic and heavy-handedapproach. The hierarchies introduce a number of complicatingtechnicalities not present in a “flat universe”, and eventhough the paradoxes do indeed disappear, so do all non-paradoxicaloccurrences of self-reference. Kripke (1975) gives the followingillustrative example taken from ordinary discourse. Let \(N\) be thefollowing 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 beon a higher level than all of Jones’ utterances, and,conversely, the sentence \(J\) would have to be on a higher level thanall 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 higherlevel than\(J\), and \(J\) on a higher leverthan \(N\). This is obviously not possible, so in a hierarchy like theTarskian, 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 \(J\) makesreference to a totality including \(N\). Nevertheless, in most cases\(N\) and \(J\) are harmless, and do not produce a paradox. A paradoxis only produced in the special case where all of Jones’utterances except possibly \(J\) are true, and exactly half ofNixon’s utterances are false, disregarding \(N\). Kripke usesthe fact that \(N\) and \(J\) are only problematic in a certainspecial case as an argument against an approach that altogetherexcludes the possibility of formulating \(N\) and \(J\).

Another argument against the hierarchy approach is that explicitstratification is not part of ordinary discourse, and thus it might beconsidered somewhatad hoc to introduce it into formalsettings with the sole purpose of circumventing the paradoxes. Nothaving an explicit stratification in ordinary discourse obviouslydoesn’t imply the non-existence of an underlying, implicit,stratification, but at least it’s not explicitly represented inour syntax.

The arguments given above are among the reasons the work of Russelland Tarski has not been considered to furnish the final solutions tothe paradoxes. Many alternative solutions have been proposed. Onemight for instance try to look forimplicit hierarchiesrather thanexplicit hierarchies. An implicit hierarchy is ahierarchy not explicitly reflected in the syntax of the language. Inthe following section we will consider some of the solutions to theparadoxes obtained by such implicit stratifications.

3.2 Building Implicit Hierarchies

Tarski’s hierarchy approach to the semantic paradoxes dominatedthe field until 1975, when Kripke published his famous and highlyinfluential paper, “Outline of a Theory of Truth”. Thispaper has greatly shaped most later approaches to theories of truthand the semantic paradoxes. It should be noted, however, that ideasquite similar to Kripke’s were developed simultaneously andindependently by Martin and Woodruff (1975), and that a parallelapproach in a set-theoretic setting was developed independently byGilmore (1974).

3.2.1 Kripke’s Theory of Truth

Kripke’s ideas are based on an analysis of the problems involvedin Tarski’s hierarchy approach. Kripke lists a number ofarguments against having a language hierarchy in which each sentencelives at a fixed level, determined by its syntactic form. He proposesan 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 atruth predicate. To explain Kripke’s construction, someadditional technical machinery is required.

At each stage in Kripke’s construction, the truth predicate isonlypartially defined, that is, it only applies to some ofthe sentences of the language. To deal with such partially definedpredicates, athree-valued logic is employed, that is, alogic which operates with a third value,undefined, inaddition to the truth valuestrue andfalse. Oftenthe third value is denoted “\(u\)” or“\(\bot\)” (bottom). A partially defined predicate onlyreceives one of the classical truth values,true orfalse, when it is applied to one of the terms for which thepredicate has been defined, and otherwise it receives the valueundefined. There are several different three-valued logicsavailable, differing in how they treat the third value. Here only oneof them is briefly described, calledKleene’s strongthree-valued logic. More detailed information on this and relatedlogics can be found in the entry onmany-valued logic.

In Kleene’s strong three-valued logic, the value \(\bot\)(undefined) can be interpreted as “not yetdefined”. So one could think of formulae with the value \(\bot\)as formulae that actually have a classical truth value (trueorfalse), but which has just not been determined yet. Thisinterpretation ofundefined is reflected in the truth tablesfor the logic, given below. The upper truth table is for disjunction,the lower for negation:

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

To handle partially defined truth predicates, it is necessary tointroduce the notion of partial models. In apartial modelfor a first-order language, each \(n\)-place predicate symbol \(P\) isinterpreted by a pair \((U,V)\) of disjoint \(n\)-place relations onthe domain of the model. \(U\) is called theextension of\(P\) and \(V\) itsanti-extension. In the model, \(P\) istrue of the objects in \(U\), false of the objects in \(V\), andundefined otherwise. In this way, any atomic sentence receives one ofthe truth valuestrue,false orundefinedin the model. Non-atomic formulae are given truth values in the modelby using Kleene’s strong three-valued logic for evaluating theconnectives.

Given the definition of a partial model, apartially interpretedlanguage is a pair \((L,M)\) where \(L\) is a first-orderlanguage and \(M\) is a partial model of \(L\). Kripke recursivelydefines a sequence of partially interpreted languages \(L_0, L_1,L_2,\ldots\), only differing in their interpretation of the truthpredicate \(T\). The first language, \(L_0\), is taken to be anarbitrary language in which both the extension and anti-extension of\(T\) are the empty set. Thus in \(L_0\), the truth predicate iscompletely undefined. For any \(\alpha\), the language \(L_{\alpha+1}\) is like \(L_{\alpha}\) except that \(T\) is interpreted by theextension/anti-extension pair \((U,V)\) given by:

• \(U\) is the set of Gödel codes \(\langle \phi \rangle\) ofsentences \(\phi\) true in \(L_{\alpha}\).
• \(V\) is the set of Gödel codes \(\langle \phi \rangle\) ofsentences \(\phi\) 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 \(L_0, L_1, L_2,\ldots\) ofpartially interpreted languages such that \(T\) is interpreted in\(L_{\alpha +1}\) as the truth predicate for \(L_{\alpha}\). This isjust like Tarski’s hierarchy of languages, except that herethere is no syntactic distinction between the different languages andtheir truth predicates.

The sequence \(L_0, L_1, L_2,\ldots\) has an important property: Foreach \(\alpha\), the interpretation of \(T\) in \(L_{\alpha +1}\)extends the interpretation of \(T\) in \(L_{\alpha}\), that is, boththe extension and anti-extension of \(T\) increase (or stay the same)when moving from \(L_{\alpha}\) to \(L_{\alpha +1}\). This means thatone can define a new partially interpreted language \(L_{\omega}\) byletting the extension of \(T\) be the union of all the extensions of\(T\) in \(L_0, L_1, L_2,\ldots\); and similarly for theanti-extension. Thus in \(L_{\omega}\), the interpretation of \(T\)extends the interpretation that \(T\) receives in all previouslanguages. This furnishes a strategy for continuing the iterativeconstruction of a truth predicate into the transfinite: For eachsuccessor ordinal \(\alpha +1\), define \(L_{\alpha +1}\) from\(L_{\alpha}\) exactly as in the finite case above; and for each limitordinal \(\sigma\), 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 inthis context, see the entry onthe revision theory of truth). A simple cardinality consideration now shows that this transfinitesequence of languages will eventuallystabilise: There is anordinal \(\gamma\) such that \(L_{\gamma} = L_{\gamma +1}\). Hence thefollowing 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 itsown truth predicate: Any sentence \(\phi\) is true (false) if and onlyif the sentence expressing its truth, \(T\langle \phi \rangle\), istrue (false). The equivalence (5) is nothing more than a semanticanalogue of Tarski’s schema \(T\) in a three-valued setting. Theconstruction of the language \(L_{\gamma}\) was one of the majorcontributions of Kripke (1975). It shows that in a three-valuedlogical setting it is actually possible for a language to contain itsown 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, alanguage with no undefined sentences), then \(L_{\gamma}\) wouldsatisfy schema T, by (5) above. However, it immediately contradictsTarski’s theorem that such a totally interpreted language canexist.

Among the sentences that receive the valueundefined in\(L_{\gamma}\) is the liar sentence. The solution to the liar paradoximplicit in Kripke’s theory is this: Since both assuming thatthe liar sentence is true and that it is false leads to acontradiction it must be neither, it isundefined. The liarsentence is said to suffer from atruth-value gap. The ideaof avoiding the liar paradox by allowing truth-value gaps did in factappear several times in the literature before Kripke’s paper,but Kripke was among the first to make it an integral part of agenuine theory.

As with the hierarchy solution to the liar paradox, the truth-valuegap solution is by many considered to be problematic. The maincriticism is that by using a three-valued semantics, one gets aninterpreted language which is expressively weak. One cannot, forinstance, in any of Kripke’s languages have a predicateexpressing the property of beingundefined. This is in factnoted by Kripke himself. If a partially interpreted language containedsuch a predicate, the followingstrengthened liar sentencewithin the language could be formulated: “This sentence iseither false or undefined”. The strengthened liar sentence istrue if and only if false or undefined, so we have a new paradox,called thestrengthened liar paradox. The problem with thestrengthened liar paradox is known as arevenge problem:Given any solution to the liar, it seems we can come up with a newstrengthened paradox, analogous to the liar, that remains unsolved.The idea is that whatever semantic status the purported solutionclaims the liar sentence to have, if we are allowed freely to refer tothis semantic status in the object language, we can generate a newparadox.

The inability of the Kripkean language to express its ownundefined predicate also means that we cannot in the Kripkeanobject-language express a statement such as: “The liar sentenceis undefined”. In fact in Kripke’s language\(L_{\gamma}\), the liar sentence \(is\) undefined, so the previoussentence expresses a truth about \(L_{\gamma}\) that cannot beexpressed within \(L_{\gamma}\) itself (hence the language isexpressively incomplete). To express the true statement “Theliar sentence is undefined”, we are forced to ascend into ameta-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 toconstruct languages containing their own truth predicate and notsuffering from the revenge problem of strengthened liars. Many ofthese attempts have focused on modifying or extending the underlyingstrong three-valued logic, e.g. modifying the semantics of theconditional (Field, 2003, 2008) or allowing an unbounded number oftruth-values (Cook, 2007; Schlenker, 2010; Tourville and Cook, 2016).

Kripke’s theory circumvents the liar paradox by assigning it thevalueundefined. An alternative way to circumvent the liarparadox would be to assign it the valueboth true and falsein a suitable paraconsistent logic. This would be the correct solutionaccording to the dialetheist view, cf. Section 2. One of the simplestparaconsistent logics is LP, which is a three-valued logic with thesame truth tables as Kleene’s strong three-valued logicpresented above—the only difference is that the third truthvalue is interpreted asboth true and false rather thanundefined. A reason for preferring a paraconsistent logicover a partial logic is that paradoxical sentences such as the liarcan then be modelled astrue contradictions (dialetheia)rather than truth-value gaps. We refer again to the entries ondialetheism andparaconsistent logic for more information.

The choice is betweentruth-value gaps andtruth-valuegluts: A truth-value gap is a statement with no truth-value,neither true or false (likeundefined in Kleene’sstrong three-valued logic), and a truth-value glut is a statement withseveral truth-values, e.g. both true and false (like in theparaconsistent logic LP). There are also arguments in favour ofallowing both gaps and gluts, e.g. by letting the set of truth-valuesform of a bilattice (Fitting, 2006; Odintsov and Wansing, 2015). Thesimplest non-trivial bilattice has exactly four values, which in thecontext of truth-values are interpreted as:true,false, \(\bot\) (neither true nor false), and \(\top\) (bothtrue and false).

For a more extensive discussion of Kripke’s theory, itssuccessors and rivals, see the entry onthe liar paradox.

3.2.3 Implicit Hierarchies in Set Theories

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

NF comprehension:
\(\forall u(u \in \{ x \mid \phi(x)\} \leftrightarrow \phi(u))\), forallstratified formulae \(\phi(x)\).

A formula \(\phi\) isstratified if there exists a mapping\(\sigma\) (astratification) 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 asubformula of \(\phi\) then \(\sigma(v) = \sigma(u)\). Obviously theformula \(x \not\in x\) is not stratified, and thus the NFcomprehension principle cannot be used to formulate Russell’sparadox in the theory. Quine’s New Foundations is essentiallyobtained from type theory by hiding the types from the syntax. Thus,the theory still makes use of a hierarchy approach to avoid theparadoxes, but the hierarchy is made implicit by not representing itin the syntax of formulae. Cantini (2015) has investigated thepossibility of mimicking this implicit hierarchy approach in thecontext of theories of truth (achieving an implicitly representedTarskian truth hierarchy).

Zermelo-Fraenkel set theory (ZF) is another theory that builds on theidea of an implicit hierarchy to circumvent the paradoxes. However, itdoes so in a much less direct way than NF. In ZF, sets are builtbottom-up, starting with the empty set and iterating a construction ofbigger 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 toKripke’s iterative construction, the procedure is continued intothe transfinite using the union operator at the limit ordinal levels.The hierarchy obtained is called thecumulative hierarchy.One of the axioms of ZF, the axiom of foundation, states that everyset of ZF lives on a certain level in this cumulative hierarchy. Inother words, the axiom of foundation states that there are no sets inZF besides the ones that can be constructed bottom-up by the iterativeprocedure just described. Since in a cumulative hierarchy, there canbe no sets containing themselves, no universal set, and nonon-wellfounded sets, none of the known paradoxes can immediately beformulated in the theory. This does obviously not in itself ensure theconsistency of ZF, but at least it illustrates how the idea of a sethierarchy plays a significant role in ZF as well. ZF has a privilegedstatus among set theories, as it is today the most widely acknowledgedcandidate for a formal foundation of mathematics.

3.3 General Fixed Point Approaches

Kripke’s iterative construction of a truth predicate presentedabove may be viewed as an instance of a more generalfixed pointapproach towards building formal theories of truth. Fixed pointapproaches have become central to contemporary formal theories oftruth. The main idea is to have atruth revision operator andthen look for fixed points of this operator. At heart of such fixedpoint approaches is some suitablefixed point theoremguaranteeing the existence of fixed points for certain kinds ofoperators. 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 \(\tau(f) = f\).

Accpo is a partial order \((D,\lt)\) in which every totallyordered subset of \(D\) has a least upper bound. The totally orderedsubsets of \(D\) are calledchains in \(D\). Amonotoneoperator on \((D,\lt)\) is a mapping \(\tau : D \rightarrow D\)satisfying:

Kripke’s construction fits into the fixed point theorem above inthe following way. First note that the set of partially interpretedlanguages that only differ on the interpretation of \(T\) forms accpo: Simply define the ordering on these languages by \(L_1 \le L_2\)iff the interpretation of \(T\) in \(L_2\) extends the interpretationof \(T\) in \(L_1\) (that is, the extension and anti-extension of\(T\) in \(L_1\) are included in those of \(L_2)\). Then define atruth 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’sconstruction, then \(L_{\alpha +1} = \tau(L_{\alpha})\). The idea ofthis truth revision operator \(\tau\) is that if \(\tau(L)=L'\) then\(L'\) will be a language in which \(T\) is interpreted as the truthpredicate for \(L\). If therefore \(\tau(L)=L\) for some \(L\), thatis, if \(L\) is a fixed point of \(\tau\), then \(L\) will be alanguage containing its own truth predicate. This motivates the searchfor fixed points of \(\tau\). Since \(\tau\) is easily seen to bemonotone, by the fixed point theorem it has a least fixed point. It isnot 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 fixedpoints for monotone operators.

The point of introducing the additional machinery was not just torediscover the language \(L_{\gamma}\). The point is rather to haveprovided a much more general and abstract framework which may lead tonew theories of truth and give further insights into the semanticparadoxes. 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 byconsidering alternative ways of making the set of interpretedlanguages into a ccpo. One may for instance add an additional truthvalue and consider the situation in a four-valued logic, as consideredby Fitting (1997); or one may remove the third truth valueundefined and construct a ccpo in a completely classicalsetting. In the classical setting, attention is restricted to thetotally interpreted languages (languages in which every sentence iseither 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 \(L_1\) isincluded 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 ona suitably defined revision operator, it is fairly easy to prove theexistence of a totally interpreted language containing apositivedefinition of truth. By this is meant that the interpretedlanguage has a predicate \(T\) satisfying the following restrictedversion of the \(T\)-schema:

(7)

\(\phi \leftrightarrow T(\langle \phi \rangle)\), for allpositive sentences \(\phi\),

where the positive sentences are those built without using negation\((\neg)\). Since (7) is satisfiable in a totally interpretedlanguage, the first-order theory containing the sentences of (7) asaxioms must be consistent. This should be contrasted withTarski’s theorem stating that theunrestricted\(T\)-schema is inconsistent. If the unrestricted comprehensionprinciple is similarly restricted to the positive formulae, we alsoget a consistent theory. This was originally shown by Gilmore(1974).

The fixed point approach is also the point of departure of therevision theory of truth developed by Belnap and Gupta(1993). The revision theory of truth is the most influential theory oftruth and the semantic paradoxes that has been developed since thetheory of Kripke. The revision theory considers the standard truthrevision operator \(\tau\) defined by (6) as an operator on thetotally 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 fullschema \(T\), directly contradicting Tarski’s theorem. Since\(\tau\) doesn’t have a fixed point on the totally interpretedlanguages, the revision theory instead considers transfinite sequences\(L_1, L_2\), … , \(L_{\omega}, L_{\omega +1}\), … oftotally 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 eventuallystabilise on a classical truth value (true or false), or it will neverstabilise. An example of a sentence that will never stabilise is theliar 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 correctlymodels the behaviour of the liar sentence as one that never stabiliseson a truth value. In the revision theory it is argued that this givesa more correct account of truth and self-reference than Kripke’stheory in which the liar sentence is simply assigned the valueundefined. Both the revision theory of truth and Kripke-stylefixed-point theories are still being actively researched (Gupta andStandefer, 2017; Hsiung, 2017; Schindler, 2017). A full account of therevision theory can be found in the entry onthe revision theory of truth.

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

4. Recent Developments

Murzi and Massimiliano (2015) give an overview of recent developmentsin approaches to solving the paradoxes: paracompleteness (allowingtruth-value gaps), paraconsistency (allowing truth-value gluts),substructural logics (weakening the logical principles of classicallogic), and the revenge problems that these approaches will or couldlead to. Recent developments in substructural logics as a cure to theparadoxes include French (2016) (dropping reflexivity), Caret, Colinand Weber (2015), Shapiro and Lionel (2015), Mares and Paoli (2014)(dropping contraction), and Cobreros, Égré, Ripley andvan Rooij (2014) (dropping transitivity). More recently, a specialvolume ofSynthese has been devoted to substructuralapproaches to paradox (December 2021 issue, with an introduction byZardini, 2021). The volume by Achourioti et al. (eds., 2015) hasseveral papers on self-reference and how to avoid paradoxes in thecontext of theories of truth.

Volker Halbach and Albert Visser (2014a, 2014b) have made a verydetailed study of self-reference in arithmetic, studying what it meansfor a sentence of arithmetic to ascribe itself a property, and howthis depends on the chosen encoding, the details of fixed-pointconstruction etc. Albert Visser (2019) is also among the authors whohave studied how little self-reference we can get away with whenproving classical theorems like Gödel’s secondincompleteness theorem.

There is also still work on trying to characterise what exactly itmeans for something to be a paradox (Hsiung, 2022). This is somewhatin the spirit of Priest’s Inclosure Schemes and thegraph-theoretical classifications of paradoxicality mentioned above.Finding the exact necessary and sufficient ingredients required forparadox is still an open problem. There has also been further work indefining exactly it means for something to be self-referential(Picollo, 2018, 2020). Grabmayr, Halbach and Ye (2023) distinguishesbetween genuine and accidental self-reference, and hence tries toprovide a more fine-grained analysis of self-referentiality. A morefine-grained analysis of self-referentiality can also be made bytaking a closer look at the kind of Gödel numbering we use(Grabmayr and Visser, 2023; Kripke, 2023).

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Other Internet Resources

• Logical Paradoxes, by B. Hartley Slater, in theInternet Encyclopedia ofPhilosophy.
• Truth, by Bradley Dowden and Norman Swartz, in theInternet Encyclopediaof Philosophy.

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