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# Self-reference and Logic

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Tarski's schema T plays a central role in each of these formalizations. a In particular, we show that each of the classical paradoxes of self-reference can be reduced to lIf 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. aTarski's schema T is the set of all first-order logical equivalences T(r-g TM) - g where g is any sentence and rg is a term denoting g. 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-r
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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.
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.
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.
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.
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
(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
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
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
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.
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.
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.
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 odel’s Incompleteness Theorem
We now show how schema T is related to 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, 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 ` (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-
` 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
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 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.
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 odel’s Incompleteness Theorem is
closely related to reﬂection. What 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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17
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18
Answer: A self-referential question which is its own answer.
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