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Self-reference and Logic

Thomas Bolander

22nd August 2005

Self-reference is used to denote any situation in which someone or something

refers to itself. Object that refer to themselves are called self-referential. Any

object that we can think of as referring to something—or that has the ability

to refer to something—is potentially self-referential. This covers objects such as

sentences, thoughts, computer programs, models, pictures, novels, etc.

The perhaps most famous case of self-reference is the one found in the Liar

sentence:

“This sentence is not true”.

The Liar sentence is self-referential because of the occurrence of the indexical

“this sentence” in the sentence. It is also paradoxical.

1

That self-reference

can lead to paradoxes is the main reason why so much eﬀort has been put

into understanding, modelling, and “taming” self-reference. If a theory allows

for self-reference in one way or another it is likely to be inconsistent because

self-reference allows us to construct paradoxes, i.e. contradictions, within the

theory. This applies, as we will see, to theories of sets in mathematics, theories

of truth in the philosophy of language, and theories of introspection in artiﬁcial

intelligence, amongst others.

This essay consists of two parts. The ﬁrst is called “Self-reference” and the

second is called “Logic”. In the ﬁrst part we will try to give an account of the

situations in which self-reference is likely to occur. These can be divided into

situations involving reﬂection, situations involving universality, and situations

involving ungroundedness.

2

In the second part we will turn to a more formal

treatment of self-reference, by formalizing a number of the situations involving

self-reference as theories of ﬁrst-order predicate logic. It is shown that Tarski’s

schema T plays a central role in each of these formalizations.

3

In particular,

we show that each of the classical paradoxes of self-reference can be reduced to

1

If the sentence is true, what it states must be the case. But it states that it itself is not

true. Thus, if it is true, it is not true. On the contrary assumption, if the sentence is not true,

then what it states must not be the case and, thus, it is true. Therefore, the sentence is true

iﬀ it is not true.

2

Often cases of self-reference will ﬁt into more than one of these categories.

3

Tarski’s schema T is the set of all ﬁrst-order logical equivalences

T ( ϕ ) ↔ ϕ

where ϕ is any sentence and ϕ is a term denoting ϕ.

1

schema T. This leads us to a discussion of schema T, the problems it gives rise

to, and how to circumvent these problems.

The ﬁrst part of the essay does not require any training in mathematical

logic.

Part I: Self-Reference

We start out by taking a closer look at paradoxes related to self-reference.

1 Paradoxes

A paradox is a “seemingly sound piece of reasoning based on seemingly true as-

sumptions, that leads to a contradiction (or other obviously false conclusion)”

(Audi, 1995). A classical example is Zeno’s Paradox of Achilles and the

Tortoise in which we seem to be able to prove that the tortoise can win any

race against the much faster Achilles, if only the tortoise is given an arbitrarily

small head-start (cf. (Erickson and Fossa, 1998) for a detailed description of this

paradox). Another classical paradox is the Liar Paradox, which is the contra-

diction derived from the Liar sentence. Among the paradoxes we can distinguish

those which are related to self-reference. These are called the paradoxes of

self-reference. The Liar Paradox is one of these, and below we consider a few

of the others.

1.1 Grelling’s Paradox

A predicate is called heterological if it is not true of itself, that is, if it does

not itself have the property that it expresses. Thus the predicate “long” is

heterological, since it is not itself long (it consists only of four letters), but the

predicate “short” is not heterological. The question that leads to the paradox

is now:

Is “heterological” heterological?

It is easy to see that we run into a contradiction independently of whether we

answer ‘yes’ or ‘no’ to this question.

Grelling’s paradox is self-referential, since the deﬁnition of the predicate

“heterological” refers to all predicates, including the predicate “heterological”

itself.

1.2 Richard’s Paradox

Some phrases of the English language denote real numbers. For example, “the

ratio between the circumference and diameter of a circle” denotes the number π.

Assume that we have given an enumeration of all such phrases (e.g. by putting

them into lexicographical order). Now consider the phrase

2

“the real number whose nth decimal place is 1 if the nth decimal

place of the nth phrase is 2, otherwise 1”.

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 kth phrase in the kth decimal place.

Richard’s paradox is self-referential, since the deﬁned phrase refers to all

phrases that deﬁne real numbers, including itself.

1.3 Berry’s Paradox

Berry’s Paradox is obtained by considering the phrase

“the least natural number not speciﬁable by a phrase containing

fewer than 100 symbols”.

The contradiction is that that natural number has just been speciﬁed using only

87 symbols!

The paradoxes may seem simply like amusing quibbles. We may think of

them as nothing more than this when they are part of our imprecise natural

language and not part of theories. When the reasoning and assumptions involved

in the paradoxes are not attempted to be made completely explicit and precise,

we might expect contradictions to be derivable because of this lack of precision.

But having a theory—mathematical, philosophical or otherwise—containing a

contradiction is of course devastating for the theory. It shows the entire theory

to be inconsistent (unsound). The problem is that it turns out that in many

of the intuitively correct theories in which some kind of self-reference is taking

place, we can actually reconstruct the above paradoxes, and thereby show these

theories to be inconsistent. This applies to the naive theories of truth, sets, and

introspection as we will later see.

Before we turn to a more thorough study of the situations in which self-

reference is to be expected to occur, we put a bit more structure on our notion

of self-reference by introducing reference relations.

2 Reference Relations

Reference can be thought of as a relation R between a class of referring objects

and a class of objects being referred to. R is called a reference relation, and

it is characterized by the property that

(a, b) ∈ R iﬀ b is referred to by a.

The domain of R, that is, the set of a’s for which there is a b with (a, b) ∈ R,

is denoted dom(R). The range of R, that is, the set of b’s for which there is an

a with (a, b) ∈ R, is denoted ran(R). The relation R can be depicted as a graph

3

small car

cars

big car

dom(R)

ran(R)

Figure 1: A reference relation.

T

truth ”is”-relation

Figure 2: Reference relation for T .

on dom(R) ∪ ran(R), in which there is an edge from a ∈ dom(R) to b ∈ ran(R)

iﬀ (a, b) ∈ R. If e.g.

A = {“small car”, “big car”, “cars”}

and

B =

,

we could have the reference relation depicted on Figure 1. If dom(R)∩ran(R) =

∅, as above, a referring object will always be isolated from the object it refers

to, since these two objects will be members of two distinct and disjoint classes.

Self-reference is thus only possible when dom(R) ∩ ran(R) 6= ∅.

Let T be the self-referential sentence

“This sentence is true”

(T for truth teller). The sentence refers to

(i) the sentence itself

(ii) the “is”-relation

(iii) the concept of truth.

Graphically, this could be represented by the reference relation in Figure 2.

4

S

2

S

1

”is”-relation

truth

dom(R)

ran(R)

Figure 3: Reference relation for S

1

and S

2

.

Notice the loop at T . The loop means that

(T, T ) ∈ R,

that is, T is referred to by T , which is exactly the condition for T being self-

referential. This leads us to the following deﬁnition:

An object a ∈ dom(R) is called directly self-referential if

there is a loop at a in (the graph of) the reference relation.

Now consider the following two sentences, S

1

and S

2

,

S

1

: The sentence S

2

is true.

S

2

: The sentence S

1

is true.

The reference relation for these two sentences become as depicted in Figure 3.

Here the set of referring objects is dom(R) = {S

1

, S

2

} and the set of objects

referred to is

ran(R) = {S

1

, S

2

, “is”-relation, truth} .

Notice that dom(R) ∩ ran(R) 6= ∅. None of these sentences are directly self-

referential, but S

1

refers to S

2

which in turn refers back to S

1

, and vice versa.

This gives a cycle in the graph consisting of the nodes S

1

and S

2

, and the two

edges connecting them. We consider both of S

1

and S

2

to be indirectly self-

referential since each of them refers to itself through the other sentence. Thus

we deﬁne:

An object a ∈ dom(R) is called indirectly self-referential if

a is contained in a cycle in (the graph of) the reference relation.

5

Kripke gives a very nice example of indirect self-reference in (Kripke, 1975). S

1

is the following statement, made by Jones,

S

1

: Most of Nixon’s assertions about Watergate are false.

and S

2

is the following statement, made by Nixon,

S

2

: Everything Jones says about Watergate is true.

The reference relation for this pair of sentences will contain that of Figure 3,

i.e. we have again a cycle between S

1

and S

2

.

Let us consider a few additional examples of indirect self-reference. In the

following, when an object is either directly or indirectly self-referential, we often

simply call it self-referential.

2.1 Naive Set Theory

In naive set theory (as conceived in the early works of Georg Cantor. See e.g.

(Cantor, 1932)) the concept of a set can be deﬁned in the following way:

By a set we understand any collection of mathematical objects (in-

cluding sets).

We see that the concept of a set is deﬁned in terms of mathematical objects

which can themselves be sets. This means that what we have is a self-referential

deﬁnition of the concept of a set. This self-reference makes the deﬁned concept

inconsistent, as we will see from Cantor’s Paradox, introduced in Section 4.

2.2 Dictionary Reference

In a dictionary, the referring objects are the deﬁnienda, that is, the expressions

or words being deﬁned, and the objects referred to are the deﬁnientia, that is,

the expressions or words that deﬁne the deﬁnienda. In Webster’s 1828 dictionary

the word “regain” is deﬁned as:

regain : to recover, as what has escaped or been lost.

At the same time, the word “recover” is deﬁned as:

recover : to regain; to get or obtain that which was lost.

Using only the words in italic, the reference relation for the above two dictionary

deﬁnitions become as depicted in Figure 4. Since the deﬁnition of “regain” refers

to the word “recover” and the deﬁnition of “recover” refers to the word “regain”,

there is a cycle between these two words in the graph. Each is deﬁned through

the other in an indirectly self-referential way. This means that unless we know

the meaning of one of these words in advance, the dictionary deﬁnition will not

be able to give us the full meaning of the other word.

6

regain

recover

get

escaped

obtain

lost

Figure 4: A dictionary reference relation.

This becomes even worse if we consider an English dictionary of the entire

English language. Since every word is simply deﬁned in terms of other words,

we will not from the dictionary be able to learn the meaning of any of the

words, unless we know the meaning of some of them in advance. This makes

a dictionary insuﬃcient as a deﬁnition of meaning for a language, as noted by

Wittgenstein in the so-called Blue Book ((Wittgenstein, 1958)). Wittgenstein’s

way out was to think of a dictionary as supplied with a set of ostensive deﬁni-

tions. An ostensive deﬁnition of a word is a deﬁnition “by pointing out” the

referent of the word—e.g. to say the word “banjo” while pointing to a banjo.

Wittgenstein’s ideas are related to ideas of groundedness of ungroundedness

of reference relations, as we will see in Section 5. But before that we will relate

self-reference to reﬂection and universality.

3 Reﬂection and Self-Reference

Self-reference is often an epiphenomenon of reﬂection of some kind. The word

reﬂection actually means “bending back”. We use reﬂection to denote situations

such as: viewing yourself in a mirror; exercising introspection (that is, reﬂecting

on yourself and your own thoughts and feelings); having a theory which is

contained in its own subject matter; having a picture which contains a picture

of itself (Figure 6). Reﬂection can also be considered as a name for all the

situations in which someone or something views itself “from the outside”. In

the framework of reference relations, we can choose to deﬁne:

A reference relation R is said to have reﬂection if dom(R) ⊆

ran(R).

By this deﬁnition, a reference relation has reﬂection iﬀ every referring object is

also an object that is referred to. That is, if R is the reference relation of some

7

Figure 5: Reﬂection means “bending back”.

Figure 6: A picture containing itself.

8

Figure 7: An agent in Blocks World.

theory, then that theory can refer (represent, describe) not only objects of the

“external world” but also all the objects of the theory itself.

Reﬂection does not in itself necessarily lead to self-reference, though self-

reference often comes together with reﬂection. We do only have self-reference if

we among the elements of dom(R) can point out an element r which refers to

r. Reﬂection means that every element r of dom(R) is referred to by another

element q of dom(R), but for all such pairs (q, r) we might have q 6= r. In

Section 4 we will show, though, that if reﬂection is combined with universality,

then self-reference cannot be avoided.

Below we will consider some important examples of reﬂection.

3.1 Artiﬁcial Intelligence

A very explicit form of reﬂection is involved in the construction of artiﬁcial

intelligence systems such as for instance robots. Such systems are called agents.

Reﬂection enters the picture when we want to allow agents to reﬂect upon

themselves and their own thoughts, beliefs, and plans. Agents that have this

ability we call introspective agents.

An artiﬁcial intelligence agent is most often equipped with some formal

language which it uses for representing its experiences and beliefs, and which it

uses for reasoning about its environment. That is, such an agent has a model

of the world it inhabits which is represented by a set of formal sentences.

Consider an agent situated in a blocks world

4

as depicted in Figure 7. The

agent’s task in this world is to move blocks to obtain some goal conﬁguration

(e.g. building a tower consisting of all blocks placed in a speciﬁc order). The

agent’s beliefs about this world could be represented in the agent by formal

4

“Blocks worlds” are the classical example domains used in artiﬁcial intelligence.

9

sentences such as

on(black box, ﬂoor)

on(dotted box, black box)

on(white box, ﬂoor)

on(agent, ﬂoor).

For the agent to be introspective, though, it should also contain sentences con-

cerning the agent’s own beliefs. If the agent believes the sentence

on(black box, ﬂoor)

to be part of its own model of the world, that could e.g. be represented by the

sentence

agent(pon(black box, ﬂoor)q).

Now, the referring objects in this situation are obviously the sentences that

make up the agent’s model of the world. So if R denotes the reference relation of

the agent, then dom(R) consists of all these sentences. The object referred to in

the case of a sentence like on(black box, ﬂoor) is the black box on the ﬂoor, while

the object referred to in the case of a sentence like agent(pon(black box, ﬂoor)q)

is the sentence on(black box, ﬂoor). If ϕ is any sentence then agent(pϕq) is a

sentence referring to ϕ. This means that the set of objects referred to, ran(R),

contains every sentence, i.e. we have dom(R) ⊆ ran(R). By our deﬁnition, this

means that R has reﬂection. This reﬂection—that the agent can refer to any of

its own referring objects—turns out to provide a major theoretical obstacle to

the construction of introspective agents, as we will see in Section 11.

3.2 Philosophy of Language

One of the major problems in the philosophy of language is to give a deﬁnition

of truth for natural languages. Tarski suggests that every adequate theory of

truth should give a predicate “true” satisfying

ϕ is true iﬀ ϕ

where ϕ is any sentence. In such a theory of truth we would also have reﬂection,

since the referring objects are sentences, and any sentence ϕ can be referred to

by the sentence

“ϕ is true”.

Reﬂection is in itself not enough to give self-reference. In both examples

above we had reﬂection but no self-reference, since there were no cycles in the

reference relations. The problem is, though, that reﬂection often comes together

with universality, and when we have both reﬂection and universality then self-

reference cannot be avoided. Universality is the subject of the following section.

10

4 Universality and Self-Reference

When we make a statement about all entities in the world, this will necessarily

also cover the statement itself. Thus such statements will necessarily be self-

referential. We call such statements universal (as we call formulas of the form

∀xϕ(x) in predicate logic). Actually, we will use the term “universal” to de-

note any statement concerning all entities in the relevant domain of discourse.

Correspondingly, in the framework of a reference relation R, we can deﬁne:

An object a ∈ dom(R) is called universal if (a, b) ∈ R for all

b ∈ ran(R).

If R is the reference relation of our natural language then the sentence

“All sentences are false” (1)

will be universal. The problem about universality is that reﬂection and uni-

versality together necessarily lead to self-reference, and thereby is likely to give

rise to paradoxes. To see that reﬂection and universality together lead to self-

reference, assume R has reﬂection and that a ∈ dom(R) is a universal object.

Then we have (a, b) ∈ R for all b ∈ ran(R), and since dom(R) ⊆ ran(R) we

especially get (a, a) ∈ R. That is, we have the following result:

Assume R has reﬂection and that a ∈ dom(R) is universal. Then

a is self-referential.

Universality enters the picture in the two examples of reﬂection previously given

if we want the agent to be able to express universal statements about its envi-

ronment or if we want to be able to apply the truth predicate to sentences that

concern all sentences of the language (like e.g. the sentence (1)). In such cases

self-reference cannot be avoided, and as we will see in the second part of the

essay this will allow the paradoxes to surface and produce contradictions in the

involved theories.

The problem sketched is not in any way only related to theories of agent

introspection and truth. Any theory that is part of its own subject matter

has reﬂection. Thus, if these theories make use of universal statements as well,

then these theories contain self-referential statements, and then the paradoxes of

self-reference will not be far away. Thus, self-reference is a problem to be taken

seriously by any theory that is part of its own subject matter. This applies

to theories of cognitive science, psychology, semiotics, mathematics, sociology,

system science, cybernetics, computer science.

Note, that each of the paradoxes of self-reference considered in Section 1

involves both reﬂection and universality, since they all refer to the totality of

objects of their own type: the predicate “heterological” refers to all predicates;

the phrase deﬁning a real number in Richard’s paradox refers to all phrases

deﬁning real numbers; the phrase specifying a natural number in Berry’s para-

dox refers to all phrases specifying natural numbers.

11

Let us conclude this section by considering another example of a universal

object in a reﬂective setting. In the naive theory of sets (cf. Section 2.1) we can

consider the set U of all sets. U is certainly a universal object, since it refers

to all other sets.

5

At the same time, the theory of sets is reﬂective since for

the reference relation R of sets, dom(R) and ran(R) are both the class of all

sets. Thus U is a self-referential object, and this leads to trouble. Cantor have

proved that the cardinality

6

of any set is smaller than the cardinality of the set

of subsets of this set. This result is called Cantor’s Theorem.

7

Let us see

what happens if we apply Cantor’s Theorem to the set U. First of all, we note

that the set of all subsets of U is U itself, since U contains all sets. But then,

by Cantor’s Theorem, the cardinality of U is smaller than the cardinality of U,

which is a contradiction. This contradiction is known as Cantor’s Paradox.

Cantor’s Paradox proves that the naive theory of sets is inconsistent.

5 Ungroundedness and Self-Reference

Self-reference often occurs in situations that have an ungrounded nature. Given

a reference relation R, we can deﬁne ungroundedness in the following way:

An object a ∈ dom(R) is called ungrounded if there is an

inﬁnite path starting at a in the graph of the reference relation

R. Otherwise a is called grounded.

Note, that if dom(R)∩ran(R) = ∅, that is, if referring objects and objects being

referred to are completely separated, then all elements are grounded.

If we take the dictionary example of Section 2.2, we can give a simple example

of ungroundedness. Let R be the reference relation of Webster’s 1828 dictionary,

that is, let R contain all pairs (a, b) for which b is a word occurring in the

deﬁnition of a. Since every word of the dictionary refers to at least one other

word, every word will be the starting word of an inﬁnite path of R. Here is a

ﬁnite segment of one of these paths, taken from the 1828 dictionary:

regain → recover → lost → mislaid → laid →

position → placed → ﬁxed → . . .

Now the problem that Wittgenstein considered can be stated in the following

simple manner: in a dictionary all words are ungrounded. Since there are only

ﬁnitely many words in the English language, any inﬁnite path of words will

contain repetitions. If a word occurs at least twice on the same path, it will

5

It is natural to think of the objects being referred to by a set as the elements of the set.

6

The cardinality of a set is a measure of its size.

7

It is interesting at this point to note that the argument leading to Cantor’s Theorem—a

so-called diagonal argument (which he was the ﬁrst to use)—has basically the same structure

as Richard’s Paradox.

12

be contained in a cycle. Thus, in any dictionary of the entire English language

there will necessarily be words deﬁned indirectly in terms of themselves. That

is, any such dictionary will contain (indirect) self-reference.

Ungroundedness does not always lead to self-reference, but self-reference is

very often a byproduct of ungroundedness. So whenever one encounters un-

groundedness, one should be very careful to ensure that this ungroundedness

does not lead to self-reference and paradoxes.

Actually, as showed by Steven Yablo in (Yablo, 1993), ungroundedness can

lead to paradoxes even in cases where we do not have self-reference. Yablo’s

Paradox is obtained by considering an inﬁnite sequence of sentences S

1

, S

2

, . . .

deﬁned by:

S

1

: All sentences S

i

with i > 1 are false.

S

2

: All sentences S

i

with i > 2 are false.

S

3

: All sentences S

i

with i > 3 are false.

.

.

.

The reference relation for these sentences looks like this:

S

1

//

))

))

))

S

2

//

55

77

S

3

//

55

S

4

//

S

5

//

· · ·

As one sees, there is no self-reference involved, but we still get a paradox:

Assume S

i

is true for some i. Then all S

j

for j > i must be false. In particular,

S

i+1

must be false. But since S

j

is false for all j > i + 1, S

i+1

must also be true.

This is a contradiction. Therefore all S

i

must be false. But then S

1

should be

true, which is again a contradiction.

It should be noted that even though ungroundedness does not always lead to

self-reference, self-reference always leads to ungroundedness: any self-referential

object a is contained in a cycle, and we get an inﬁnite path from a by passing

through this cycle repeatedly.

6 Vicious and Innocuous Self-Reference

Not all self-reference leads to paradoxes. There is no paradox involved in a

self-referential sentence like

“This sentence is true”. (2)

We can assume either that the sentence is true or that it is false, and neither of

the cases will lead into contradiction. But as soon as we introduce a “not” in

the sentence, that is, consider the following sentence instead

“This sentence is not true” (3)

13

we get a paradox (the Liar Paradox). Self-reference that leads to paradoxes we

call vicious self-reference and self-reference that does not we call innocuous

self-reference. It can be shown that self-reference can only be vicious if it

involves negation or something equivalent (as the “not” in (3)). This means, for

instance, that none of the paradoxes of self-reference considered above could

be carried through if the occurrence of negation in their central deﬁnitions

where removed (e.g. if we removed the “not” in the deﬁnition of heterological

of Grelling’s Paradox).

Part II: Logic

We now turn to a more formal treatment of self-reference, by formalizing some

of the situations considered in the ﬁrst part of the essay as theories of ﬁrst-order

predicate logic (henceforth simply called ﬁrst-order theories). We will assume

that all considered ﬁrst-order theories contain the standard numerals:

8

¯

0,

¯

1,

¯

2,

¯

3, . . .

We use p·q to range over coding schemes. By a coding scheme we understand

any injective mapping from sentences into numerals. That is, if ϕ is a sentence

then pϕq is the numeral ¯n for some natural number n. pϕq is a name for ϕ;

we call it the code number of ϕ. If ψ(x) is a formula containing x as its

only free variable then ψ(pϕq) is a sentence expressing that “ϕ has the property

expressed by ψ”. In this sense, ψ(pϕq) refers to ϕ.

Schema T is, as before, deﬁned as the theory containing each of the equiv-

alences

T (pϕq) ↔ ϕ

where T is a ﬁxed one-place predicate symbol and ϕ is any sentence.

The aim of this part of the essay is to show that schema T is taking a central

position in almost all situations in which we have self-reference. Indeed, schema

T can be thought of as a unifying principle of all the diﬀerent occurrences of

self-reference.

7 Formalizing Paradoxes

We now try to formalize some of the most famous paradoxes of self-reference

to show how these involve schema T. As mentioned, a paradox is a “seemingly

sound piece of reasoning based on seemingly true assumptions that lead to a

contradiction”. Formalizing a paradox means to reconstruct it inside a formal

theory (in our case, a ﬁrst-order theory). This involves ﬁnding formal coun-

terparts to each of the elements involved in the informal paradox. The formal

counterpart of a “piece of reasoning” is a formal proof and the formal counter-

part of an “assumption” is an axiom. Thus the formal counterpart of a piece of

reasoning leading to a contradiction will be a formal proof of the inconsistency

of the theory in question. Thus:

8

Actually, any inﬁnite collection of closed terms would do.

14

A formalization of a paradox is a formal proof of the incon-

sistency of the theory in which the axioms are the formal coun-

terparts of the assumptions of the paradox.

7.1 The Liar Paradox

As already mentioned, the Liar Paradox is the contradiction that emerges from

trying to determine whether the Liar sentence

“This sentence is false”

is true or false. We will now try to formalize this paradox.

In general, sentences that are directly self-referential can be put in the fol-

lowing form:

“This sentence has property P ”. (4)

The assumption that characterizes such a sentence is that the term “this sen-

tence” refers to the sentence itself. Another way of stating this assumption is

to say that (4) should satisfy the following equivalence

This sentence has property P

⇔ “This sentence has property P ” has property P ,

(5)

that is, replacing the term “this sentence” by the sentence itself will not change

the meaning of the sentence. Formally, this assumption can be expressed as the

axiom

P (t) ↔ P (pP (t)q) (6)

where t is a term having the intended interpretation: “this sentence”. This

equivalence corresponds to the equivalence (5), in that “this sentence” have

been replaced by t and the quotes “·” have been replaced by p·q.

The Liar Paradox also rests on the assumption that our language has a truth

predicate. The formal counterpart of this assumption is that our theory includes

schema T. In the Liar sentence, P is the property “not true”. Let therefore P

in (6) denote the formula ¬T (x). Then, in the theory consisting of schema T

and (6), we get the following proof:

1. ¬T (t) ↔ ¬T (p¬T (t)q) (6) with P being ¬T

2. T (p¬T (t)q) ↔ ¬T (t) instance of schema T

3. T (p¬T (t)q) ↔ ¬T (p¬T (t)q) by 1. and 2.

This proves the theory consisting of (6) and schema T to be inconsistent, which

is our formalization of the Liar Paradox.

7.2 Grelling’s Paradox

We will now formalize Grelling’s paradox. Recall that Grelling’s Paradox is

the paradox that emerges when trying to answer whether “heterological” is

15

heterological. The formal counterpart of a predicate is a formula. A formula

ϕ(x) is then heterological if it is “not true of itself”, that is, if

¬T (pϕ(pϕq)q)

holds, where T is a truth predicate. So to formalize Grelling’s paradox we again

need to have schema T among our axioms. We also need axioms that allow

us to apply a formula to itself (that is, the code of itself). To obtain this, we

introduce a function symbol app and axioms

app (pϕ(x

1

)q, τ) = pϕ(τ)q (7)

for all formulas ϕ and all terms τ. These axioms ensure us that app(pϕ(x

1

)q, τ)

denotes the result of “applying” ϕ(x

1

) to τ (that is, instantiating ϕ(x

1

) with

τ). Now we can formalize the predicate “heterological” as the formula het(x

1

)

given by

het(x

1

) =

df

¬T (app(x

1

, x

1

)) .

To obtain the contradiction we should ask whether het(phet(x

1

)q) holds or not.

We get the following proof:

1. het(phet(x

1

)q) ↔ ¬T (app (phet(x

1

)q, phet(x

1

)q)) by def. of het(x

1

)

2. het(phet(x

1

)q) ↔ ¬T (phet(phet(x

1

)q)q) by 1. and (7)

3. het(phet(x

1

)q) ↔ T (phet(phet(x

1

)q)q) instance of schema T

4. ¬T (phet(phet(x

1

)q)q) ↔ T (phet(phet(x

1

)q)q) by 2. and 3.

This proves the theory consisting of (7) and schema T to be inconsistent, which

is our formalization of Grelling’s paradox.

Richard’s Paradox is formalized in much the same way as Grelling’s Paradox,

though the formalization becomes slightly more technical. For these reasons we

choose to leave out a formalization of Richard’s Paradox in this essay.

7.3 Berry’s Paradox

Obviously, to formalize Berry’s Paradox, we need axioms formalizing a reason-

able part of arithmetic. Apart from this we only need a formal counterpart of

the notion of speciﬁability (the formal counterpart of a “phrase” naturally being

a formula). We can use the same trick as we did in the previous examples. A

formula ϕ(x) speciﬁes the number n iﬀ ϕ(m) holds exactly when m = n. If

we want to deﬁne a formula spec(x, y) such that spec(pϕ(x)q, n) holds precisely

when ϕ(x) speciﬁes n, then it should look like

spec(x, y) =

df

∀z (z = y ↔ T (app(x, z)))

where T and app are deﬁned as before. We will not go further into the details of

formalizing this paradox, but refer to (Boolos, 1989) in which this is carried out.

We just note that again schema T is central to the formalization. The notion

16

of speciﬁability could not have been formalized without schema T or something

equivalent.

We have now shown how to formalize several of the most famous paradoxes

of self-reference, and, as we have seen, these formalized paradoxes all turn out

to be reducible to schema T. That is, all these paradoxes have a common core

which is schema T. What we can conclude is that:

(i) That schema T can be extracted from all these paradoxes helps us see the

close formal relationship between the paradoxes of self-reference.

(ii) That all these paradoxes can be extracted from schema T helps us to

see the importance of schema T in understanding the paradoxes of self-

reference, and in understanding self-reference in general.

Below we consider some examples of occurrences of schema T in the philos-

ophy of language, mathematics, and artiﬁcial intelligence.

8 The Naive Theory of Truth

As mentioned, Tarski thought of his schema T as describing the principle that

any theory of truth should satisfy. The ﬁrst-order theory consisting only of

schema T is consistent. But for schema T to be a sensible principle of truth we

must expect it to be consistent also when added to any consistent, “realistic”

ﬁrst-order theory. It should be a principle of truth working no matter which

domain of discourse we would like to apply truth to. But, unfortunately, because

of self-reference it is not so. In the formalizations of the paradoxes above we have

seen several examples showing that schema T becomes inconsistent when added

to even quite weak and harmless axioms (at least harmless when these axioms

are taken by themselves or together with standard theories for arithmetic, set

theory, or the like). In fact, it can easily be shown that all of the axioms assumed

above in addition to schema T are interpretable in Peano Arithmetic, that is,

they can all be translated into equivalent axioms of Peano Arithmetic.

9

This

gives us the famous Tarski’s Theorem:

Peano Arithmetic extended with schema T is inconsistent.

Note the interesting fact that any of the above paradoxes can be used to prove

Tarski’s Theorem—one just needs to show that the axioms of the formalized

paradox are interpretable in PA (Peano Arithmetic). This shows that the

contradiction derivable from the formalized paradox can be carried through

in PA + schema T.

9

For a precise deﬁnition of “interpretable in” we refer to (Mendelson, 1997) or a similar

introduction to mathematical logic. At this point it is enough to note that when an axiom

A is interpretable in a theory K it means that any proof in K + A can be translated into a

corresponding proof in K. It should be noted that to prove the interpretability we need to

choose our coding scheme

· with care.

17

That schema T becomes inconsistent when standard arithmetic is added is

a very serious drawback for the theory of truth expressed through schema T. It

gives rise to an important problem of how we can restrict schema T to regain

the essential consistency. This is the question that we take up in Section 12.

But let us ﬁrst consider some more examples of situations in which schema

T turns up, which makes the reasons to ﬁnd consistent ways to restrict schema

T even more urgent.

9 G¨odel’s Incompleteness Theorem

We now show how schema T is related to G¨odel’s famous First Incompleteness

Theorem.

A version of the Incompleteness Theorem states that

If PA is ω-consistent

10

then it is incomplete.

11

To prove this, we can show that the assumption that PA is both ω-consistent

and complete leads to a contradiction. On the basis of the formalizations of

paradoxes that we have been considering, we see that this could be proved by

showing that if PA were both ω-consistent and complete then some paradox

would be formalizable in PA. This was, roughly, G¨odel’s idea.

12

He constructed

a formula Bew (for “Beweis”) in his theory satisfying, for all ϕ and all n,

` Bew(¯n, pϕq) ⇔ n denotes a proof of ϕ. (8)

Assuming the theory to be ω-consistent and complete we can prove that

` ∃xBew(x, pϕq) ⇔ ` ϕ

for every sentence ϕ. The proof runs like this: First we prove the implica-

tion from left to right. If ` ∃xBew(x, pϕq) then there is some n such that

6` ¬Bew(¯n, pϕq), by ω-consistency. By completeness we get ` Bew(¯n, pϕq) for

this n. By (8) above we get that n denotes a proof of ϕ. That is, ϕ is provable,

so we have ` ϕ. To prove the implication from right to left, note that if ` ϕ

then there must be an n such that ` Bew(¯n, pϕq), by (8). From this we get

` ∃xBew(x, pϕq), as required. This concludes the proof.

Now, when we have

` ∃xBew(x, pϕq) ⇔ ` ϕ

in a complete theory, we must also have

` ∃xBew(x, pϕq) ↔ ϕ.

10

A theory is called ω-consistent if, for every formula ϕ(x) containing x as its only free

variable, if ` ¬ϕ(¯n) for every natural number number n, then it is not the case that ` ∃xϕ(x).

11

A theory is incomplete if it contains a formula which can neither be proved nor disproved.

12

Though he considered a diﬀerent formal theory, P.

18

If we let the formula ∃xBew(x, pϕq) be abbreviated by T (pϕq) then these equiv-

alences read

` T (pϕq) ↔ ϕ

which is schema T!

That is, if we assume PA (or a related theory) to be ω-consistent and com-

plete then schema T turns out to be interpretable in it. Now, Tarski’s Theorem

shows that there exists no such consistent theory. This gives us a proof of

G¨odel’s Incompleteness Theorem. Furthermore, in the same way that one could

use any of the paradoxes of self-reference to prove Tarski’s Theorem, one can

use ones favorite paradox of self-reference to prove G¨odel’s Theorem.

To summarize the process: ﬁrst you assume your theory to be both ω-

consistent and complete. Then you show that this makes schema T interpretable

in the theory. Having schema T means that you can choose any paradox of self-

reference and formalize it in the theory. The formalized paradox produces a

contradiction in the theory, and thus shows that the theory cannot be both

ω-consistent and complete.

G¨odel himself actually had a footnote in his 1931 article, in which he proved

the Incompleteness Theorem ((G¨odel, 1931)), saying that any paradox of self-

reference

13

could be used to prove the Incompleteness Theorem.

The reason that we have a result such as G¨odel’s Incompleteness Theorem is

closely related to reﬂection. What G¨odel ingeniously discovered was that formal

theories can be reﬂected inside themselves, since numerals can be used to refer

to formulas through the use of a coding scheme, p·q, and by means of these

codes provability can be restated inside the theories as arithmetical properties.

10 Axiomatic Set Theory

Schema T also plays a central role in axiomatic set theory. By the full ab-

straction principle we understand the set of formulas on the form

∀x (x ∈ {y | ϕ(y)} ↔ ϕ(x))

14

where ϕ is any formula. When Gottlob Frege tried to give a foundation for

mathematics (set theory) through his works “Die Grundlagen der Arithmetik

(1884)” and “Grundgesetze der Arithmetik (1893,1903)”, the full abstraction

principle were among his axioms. But in 1902 his system was shown to be

inconsistent by Bertrand Russell. Russell constructed a paradox of self-reference

which was formalizable within Frege’s system. Russell’s Paradox runs like

this:

Let M be the set of all sets that are not members of themselves. Is

M a member of itself or not?

13

He used the term “epistemic” about these paradoxes.

14

The formula can be read: “for all sets x, x is in the set of y’s for which ϕ(y) holds if and

and only if ϕ(x) holds”.

19

From each answer to this question the opposite follows. Notice the similarity

between this paradox and Grelling’s paradox considered in Section 1.1. Rus-

sell’s Paradox can be formalized in any system containing the full abstraction

principle. We let M = {y | y 6∈ y}, that is, M = {y | ϕ(y)} where ϕ(y) = y 6∈ y.

The abstraction principle instantiated by the formula ϕ now becomes

∀x (x ∈ {y | y 6∈ y} ↔ x 6∈ x) .

Letting x = {y | y 6∈ y}, we get

{y | y 6∈ y} ∈ {y | y 6∈ y} ↔ {y | y 6∈ y} 6∈ {y | y 6∈ y}

which is a contradiction. Thus Frege’s system, or indeed any system containing

the full abstraction principle, is inconsistent.

The discovery of this inconsistency lead to extensive research in how the full

abstraction principle could be restricted to regain consistency.

Actually, as we will now show, every instance of schema T can be interpreted

in the corresponding instance of the abstraction principle. This means that if we

can prove that a set of instances of schema T is inconsistent, then we have also

proven that the corresponding set of instances of the abstraction principle is in-

consistent. In other words, proving consistency results about restricted versions

of schema T will also give corresponding consistency results about restricted

versions of the abstraction principle.

The result is the following:

Every instance of schema T:

T (pϕq) ↔ ϕ

can be interpreted in the corresponding instance of the abstrac-

tion principle:

∀x (x ∈ {y | ϕ} ↔ ϕ) .

The proof is quite simple. If we have got a theory containing

∀x (x ∈ {y | ϕ} ↔ ϕ)

then T (pϕq) can be interpreted in it by the following extension by deﬁnitions:

15

pϕq = {y | ϕ}

T (x) ↔

¯

0 ∈ x.

Since we have

∀x (x ∈ {y | ϕ} ↔ ϕ)

15

We refer again to (Mendelson, 1997) for a deﬁnition of the concept of “extension by

deﬁnitions”.

20

we get in particular

(

¯

0 ∈ {y | ϕ} ↔ ϕ)

which is the same as

T (pϕq) ↔ ϕ,

using the deﬁnitions of p·q and T . This proves T (pϕq) ↔ ϕ to be interpretable

in ∀x (x ∈ {y | ϕ} ↔ ϕ).

11 Agent Introspection

We now turn to our last example of an occurrence of schema T in a situation

dealing with self-reference. We consider again the problem of constructing in-

trospective agents, as introduced in Section 3.1. Since the agent’s model of the

world is supposed to consist of a set of sentences, we can think of this model as

being a formal theory K. This could be a theory in any kind of formal language,

but at this point we will assume that it is a theory in a ﬁrst-order language.

Then, for the agent to believe that e.g. the black box is on the ﬂoor would

correspond to having

K ` on(black box, ﬂoor). (9)

If the agent has introspection, it also has beliefs about its own model of the

world. If it believes the sentence in (9) to be contained in its own model of the

world we would have

K ` agent (pon(black box, ﬂoor)q) .

Now, if we assume that all of the agent’s beliefs about itself to be correct, we

should have

K ` agent(pϕq) ⇔ K ` ϕ

for all sentences ϕ. Of course, not all of an agent’s beliefs about itself will

necessarily always be correct. But even so, the agent might believe this to be

the case; and that would correspond to having

K ` agent(pϕq) ↔ ϕ

for all sentences ϕ. Using “T ” instead of “agent” this gives us, once again,

schema T!

That is, if an agent has introspection and believes this introspection to be

correct, then it will necessarily contain schema T in its model of the world.

As we know from Tarski’s Theorem and our formalized paradoxes this is very

diﬃcult to obtain without running into contradictions. At least, it is extremely

sensitive to what other axioms we have in K. This is a major drawback in the

design of introspective agents.

We have to expect that any kind of axioms could be in K, depending on the

environment of the agent and its beliefs about it. The set of axioms of K could

even change over time due to changes in the environment. If K includes schema

21

T it means that the agent could suddenly become inconsistent as a consequence

of changes in the external world. This seems to prove that it is not possible

for an introspective agent consistently to obtain and retain the belief that its

introspection is correct.

This conclusion appears very counterintuitive, but again it has to do with

the paradoxes of self-reference. If the agent has introspection, and believes

this introspection to be correct, it can construct paradoxes of self-reference

concerning its own beliefs, and these paradoxes make the agent inconsistent.

The problem is now to ﬁnd ways to treat agent introspection such that

this introspection will not lead into inconsistency. It seems that we have two

possibilities: either to ensure that the agent will not be able to make self-

referential statements (which would be a restriction on its introspective abilities)

or to restrict its logical abilities such that self-referential statements could be

assumed consistently. Such restrictions are the subject of the following section.

12 Taming Self-Reference

We have now seen that schema T occurs as the natural principle in a large num-

ber of situations of very diﬀerent kinds. Schema T is the underlying principle

in the naive theories of truth, sets, and agent introspection. But unfortunately,

schema T is also the underlying principle in the paradoxes of self-reference,

which means that most of the theories we are interested in become inconsistent

when schema T is added. Since the inconsistency of schema T is a consequence

of the presence of self-referential sentences, there seems to be two possible ways

to get rid of the problem: ban self-referential sentences in our language or

weaken the underlying logic so that these sentences will do no harm. That is,

the diﬀerent ways to restrict schema T in order to ensure consistency seems to

divide into the following two major categories:

(i) Cutting away the problematic part (i.e. getting rid of the viciously self-

referential sentences).

(ii) Making the problematic part unproblematic (i.e. ensure that self-reference

does not lead to disaster).

12.1 Cutting Away the Problematic Part

Cutting away the problematic part means to restrict the set of instances of

schema T such that the viciously self-referential sentences are excluded from

entering the schema. By the T-scheme over M , where M is a set of sentences,

we understand the following set of equivalences:

T (pϕq) ↔ ϕ, for all ϕ ∈ M .

If M does not contain sentences that are viciously self-referential, it can be

proven that the T-scheme over M can consistently be added to any consistent

22

theory.

16

This is because banning the viciously self-referential sentences from

schema T makes it impossible to reconstruct the paradoxes of self-reference

within the theory.

One very coarse way of disallowing self-reference was proposed by Tarski

himself ((Tarski, 1956)): M should not be allowed to contain any sentence in

which the predicate symbol T occurs. Note that this will ensure that none of

the proofs of the formalized paradoxes considered in Section 7 can be carried

through. This restriction is suﬃcient to reestablish consistency, but it is at the

expense of a substantial loss of the expressive power of schema T. It means that

iterated truth like in

“It is true that it is not true that n is a prime number”

that formally looks like this

T (p¬T (prime(¯n))q)

will not be treated correctly by the restricted T-scheme.

Several less coarse solutions have been proposed in the literature since Tarski.

First of all, one notes that not all self-reference is vicious, so we can allow self-

referential sentences in M as long as they are not vicious. As mentioned, for

self-reference to be vicious, it needs to involve negation. A sentence in which

the predicate symbol T is not in the scope of negation (¬) is called a positive

sentence. Positive sentences can be self-referential, but only of the innocuous

kind. Donald Perlis and Solomon Fefermann showed independently ((Perlis,

1985), (Feferman, 1984)) that the T-schema over a set of positive sentences can

consistently be added to any consistent theory.

Another way to exclude viciously self-referential sentences is to make restric-

tions on universality. As we saw in Section 4, reﬂection only necessarily leads

to self-reference when it is combined with universality. Refraining from having

universal sentences about truth like e.g.

“All sentences are true”

in M we can again obtain a consistent, restricted T-scheme. More precisely, in

(Bolander, 2002) it is shown that if none of the sentences of M contain T (x) as

a sub-formula with x quantiﬁed, then the T-scheme over M can consistently be

added to any consistent theory.

Finally, the method of restricting negation and the method of restricting

universality can be combined to get an even stronger T-scheme. M can consis-

tently be allowed to contain any sentence in which T (x) does not occur in the

scope of negation (see (Bolander, 2002)).

16

That is, can consistently be added to any consistent theory that does not in advance

contain axioms for the T predicate.

23

12.2 Making the Problematic Part Unproblematic

Another way of ensuring consistency is to stick with self-reference (i.e. all in-

stances of schema T) but to make sure that self-reference does not get the chance

to become paradoxical. Such solutions seem again to divide into two categories:

(i) Restricting the form of schema T.

(ii) Restricting the underlying logic.

Below we consider each of these methods.

Restricting the Form of Schema T

Instead of having bi-implications

T (pϕq) ↔ ϕ (10)

in some cases it is suﬃcient to have e.g. the following implications

T (pϕq) → ϕ and T (pϕq) → T (pT (pϕq)q).

Some of these restrictions on the form of schema T will form consistent exten-

sions to any consistent theory, even if we do not restrict the set of sentences

that these schemas are instantiated with. Results of this type can be found in

e.g. (Montague, 1963), (Thomason, 1980), and (McGee, 1985). Another pos-

sibility is to use a weak equivalence operator in (10) instead of the classical

bi-implication operator ↔. A result concerning such a weak equivalence opera-

tor can be found in (Feferman, 1984).

Restricting the underlying logic

Theories containing schema T become inconsistent because in them we can

construct self-referential sentences that turn out to be true iﬀ they are false.

If we change the underlying logic such that sentences are allowed either to be

nor true nor false, or both true and false, the self-referential sentences will

no longer be able to prove the theories to be inconsistent. Kripke considers

in (Kripke, 1975) the possibility of allowing sentences to have no truth-value,

that is, to be neither true nor false. His trick is then to only assign truth-

values to the grounded sentences of the language (cf. Section 5). By this, he

ensures that no self-referential sentence will be given a truth-value (since every

self-referential sentence is ungrounded). This corresponds to the fact that in a

dictionary, as considered in Section 5, we can only, from the dictionary alone,

assign meaning to the grounded words. The ungrounded words, among these

the self-referentially deﬁned ones, will be “undecided” (not be assigned any

meaning). Kripke’s theory can be used to construct formal systems in which we

consistently have schema T, but in which the underlying logic is restricted (we

cannot have classical negation, for instance, since this requires every sentence to

be either true or false). Graham Priest (in (Priest, 1989) and others) proposes

that we should allow sentences to be both true and false, because this is, in a

sense, what paradoxical self-referential sentences are.

24

13 Conclusion

The paradoxes of self-reference still have no ﬁnal solution that is generally agreed

upon. This makes them, in a sense, genuine paradoxes. The presence of a

paradox is always a symptom that some part of our fundamental understanding

of a subject is crucially ﬂawed. In Zeno’s Paradox it was the understanding

of inﬁnity that was deﬁcient. In the paradoxes of self-reference it seems that

what we do not yet have a proper understanding of is the fundamental relation

between something that refers (or represents) and something that is referred

to (or represented) when these two can not be completely separated. As long

as this relationship is not entirely grasped we will probably not get to a full

understanding of the paradoxes of self-reference and their consequences for the

theories of truth, sets, agent introspection, etc.

In Zeno’s Paradox it was not an explicitly stated assumption that later

proved to be defective. In the paradox it was implicitly assumed that “inﬁnitely

many things can not happen in ﬁnite time”, but it was not until the development

of the mathematical calculus that this assumption could be made explicit and

rejected. In the case of Zeno’s Paradox it was thus not simply a question of

ﬁnding the failing assumption involved in the paradox. It was rather a question

of discovering a new dimension of the world that had hitherto been hidden to

the human eye. A similar thing might very well be the case for the paradoxes of

self-reference. The right solution (assuming there is one) to the paradoxes is not

to remove or restrict any of our explicit assumptions (that is, restrict schema

T, underlying logic, or similar), but to discover a new dimension of the problem

that will in the end give more, not fewer, axioms in some kind of extended

logic. This new dimension is then expected to make explicit some assumptions

about the general relations between referring objects and objects referred to;

assumptions that a now invisible to us.

17

Finally: What would be a suitable concluding remark in an essay like this?

18

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18

Answer: A self-referential question which is its own answer.

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