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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.
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
(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 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-
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
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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27
... The sentence and its reference relations can be visualized as in Fig. 1.2. for "this sentence is truth" (Bolander, 2002) S 1 S 2 ran(R) truth "is"-relation dom(R) (Bolander, 2002) The loop at T hints at a direct self-reference, because it means that T itself refers to the sentence T , so that there is a self-referential relation (T, T ) ∈ R. ...
... The sentence and its reference relations can be visualized as in Fig. 1.2. for "this sentence is truth" (Bolander, 2002) S 1 S 2 ran(R) truth "is"-relation dom(R) (Bolander, 2002) The loop at T hints at a direct self-reference, because it means that T itself refers to the sentence T , so that there is a self-referential relation (T, T ) ∈ R. ...
Chapter
Prediction is involved in many recurrent tasks individuals are confronted with and can be seen as the result of the interaction between “judgement, intuition, and educated guesswork.” Even when forecasts rely on mathematical methods the central role of intuition cannot be denied as it supervises e.g. the choice of variables that belong to the model, their initial value and their functional specification. This chapter deepens a central aspect for human cognition and problem-solving, namely that individuals make use of bounded rational heuristics for taking decisions under uncertainty. Heuristics are simplified procedures for assessing probabilities. They are based on rules of thumb. They rely on mental clues which selectively orient the search process and enable the individual to reach her goals when time, informational and computational capabilities are constrained. Although in some cases bounded rational heuristics can be made responsible for the sub-optimality of outcomes and for behavioural biases. In some other cases it represents an essential support for carrying on inference when complexity overloads the individual cognitive and computational capabilities. It enables the individual to reach better solutions than otherwise. There are mainly two different approaches to subjective judgement and bounded rational heuristics, namely the “heuristics and biases” approach, pioneered by Kahneman and Tversky, and the “ecological rationality” approach, with Gigerenzer as one of its most influential proponents.
... Gödel's original example of an undecidable statement arose out of selfreference. Both Gödel Gödel and Turing rely on self-reference to prove their theorems (Bolander 2002(Bolander , 2014. (Even if self-reference isn't essential to prove incompleteness (Chaitin et al. 2011), it remains an important tool in the construction of Gödel-like examples.) ...
... Self-reference arises from meta-statements: statements about themselves. What Gödel ingeniously discovered was that formal models can be reflected inside themselves (Bolander, 2002). A formal model of consumer's choice, therefore, will have a meta-statement about its own statements. ...
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The work of Gödel and Turing, among others, shows that there are fundamental limits to the possibility of formal quantification of natural and social phenomena . Both our knowledge and our ignorance are, to a large extent, not amenable to quantification. Disregard of these limits in the economic sphere might lead to underestimation of risk and, consequently, to excessive risk-taking. If so, this would expose markets to undue instability and turbulence. One major lesson of the Global Financial Crisis, therefore, is to reform economic methodology to expand beyond formal reasoning.
... The sentence and its reference relations can be visualized as in Fig. 1.2. for "this sentence is truth" (Bolander, 2002) S 1 S 2 ran(R) truth "is"-relation dom(R) (Bolander, 2002) The loop at T hints at a direct self-reference, because it means that T itself refers to the sentence T , so that there is a self-referential relation (T, T ) ∈ R. ...
... The sentence and its reference relations can be visualized as in Fig. 1.2. for "this sentence is truth" (Bolander, 2002) S 1 S 2 ran(R) truth "is"-relation dom(R) (Bolander, 2002) The loop at T hints at a direct self-reference, because it means that T itself refers to the sentence T , so that there is a self-referential relation (T, T ) ∈ R. ...
Chapter
The phenomenon of referring is pervasive and regards all fields of human thought and activity, so much that it appears to be an inescapable basis of all that can be thought, conceptualized and expressed. The human capability of referring creates the basis for ordering the subjective perception of the world, for interpreting events, for interacting with others, etc., thus creating the basis for all activities which regard human cognition and which are essential for individual survival. Being able to establish self-references is even a necessary prerequisite for self-change and behavioural adjustment. Furthermore, the reflexive capacity underlies basic problemsolving abilities and makes mental adaptiveness possible. Consciousness (in the form of self-consciousness) can be identified as the main source of reflexivity for human thought and action. Individuals think and are simultaneously conscious of their thought, so that all discourses are both directed to outward reality (the external world) and to the inner reality of the individual who formulates them, since she is conscious of expressing them. Therefore it can be said that each human discourse, being a human way of thought formulation, has a self-referring nature. This chapter is dedicated to the analysis of the polyvalent concept of “self-reference.” After its definition which will be accompanied by an overview of the different kinds of reference relations some common varieties and possible taxonomies for self-reference will be presented. The polymorphism of self-reference will be illustrated by its implications for formal and natural language. Logical consistency of self-reference in its different forms and contexts of appearance will then be discussed, in that the relation between self-reference and paradoxes will be deepened and some guidelines for testing the legitimacy of self-references will be extrapolated. The chapter concludes discussing the role of self-reference for human understanding as well as for social and individual decision making.
... Gödel ingeniously discovered was that formal models can be reflected inside themselves (Bolander, 2002). ...
... On the other hand, the concept of selfreference -an object that refers to it -has been one of the central themes used in the study of logic (Bolander, 2002). The simplest well-known example of self-reference is the so-called the Liar's Paradox which may be written as follows: ...
... The WLIMES approach combines the advantages of a multi-scale multi-agents multi-temporality methodology (MES) based on a 'dynamic' category theory and a situation and context aware computational logic for active self-organizing networks (WLI) to systemically integrate theoretical and applied research. The results of this effort can be combined with other related research (Goranson and Cardier, 2013;Goranson et al., 2015;Cardier et al., ;Gunji et al, 2007Gunji et al, , 2008Marchal, 1991Marchal, , 1994Marchal, , 2000Kauffman, 1987;Smoryński, 2002;Bolander, 2002;Perlis, D. 2006;Tozzi et al., 2017; for use within the current methodology and practices of theoretical biology and personalized medicine to deepen and to enhance the understanding of life. ...
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The goal of this paper is to advance an extensible theory of living systems using an approach to biomathematics and biocomputation that suitably addresses self-organized, self-referential and anticipatory systems with multi-temporal multi-agents. Our first step is to provide foundations for modelling of emergent and evolving dynamic multi-level organic complexes and their sustentative processes in artificial and natural life systems. Main applications are in life sciences, medicine, ecology and astrobiology, as well as robotics, industrial automation, man-machine interface and creative design. Since 2011 over 100 scientists from a number of disciplines have been exploring a substantial set of theoretical frameworks for a comprehensive theory of life known as Integral Biomathics. That effort identified the need for a robust core model of organisms as dynamic wholes, using advanced and adequately computable mathematics. The work described here for that core combines the advantages of a situation and context aware multivalent computational logic for active self-organizing networks, Wandering Logic Intelligence (WLI), and a multi-scale dynamic category theory, Memory Evolutive Systems (MES), hence WLIMES. This is presented to the modeller via a formal augmented reality language as a first step towards practical modelling and simulation of multi-level living systems. Initial work focuses on the design and implementation of this visual language and calculus (VLC) and its graphical user interface. The results will be integrated within the current methodology and practices of theoretical biology and (personalized) medicine to deepen and to enhance the holistic understanding of life.
... One of the central ideas in the study of logic has been the notion of self-reference (i.e., something that references itself) (Bolander, 2002). For instance, this particular approach has been used to establish some of the most important results in mathematical logic, including Gödel's incompleteness theorem, which showed that sufficiently complex formal axiomatic systems cannot be both consistent and complete (Gödel, 1931). ...
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The self-reference method has yielded some of the most important results in the study of mathematical logic and computation. The physical realization of the self-reference method has been said to be the self-observation of consciousness, which suggests a non-computable element (i.e., there exists a natural phenomenon that cannot be computed). However, the contradiction or paradox in self-reference may be solved by adopting cyclical time. Contrary to the familiar notion of time as a linear progression, the cyclical concept of time suggests that time circulates. In this cyclical time model (i.e., t0 → t1 →...tN →t0), the non-computable element in self-observation can be considered to be computable.
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The rich body of physical theories defines the foundation of our understanding of the world. Its mathematical formulation is based on classical Aristotelian (binary) logic. In the philosophy of science the ambiguities, paradoxes, and the possibility of subjective interpretations of facts have challenged binary logic, leading, among other developments, to Gotthard Günther’s theory of polycontexturality (often also termed ’transclassical logic’). Günther’s theory explains how observers with subjective perception can become aware of their own subjectivity and provides means to describe contradicting or even paradox observations in a logically sound formalism. Here we summarize the formalism behind Günther’s theory and apply it to two well-known examples from physics where different observers operate in distinct and only locally valid logical systems. Using polycontextural logic we show how the emerging awareness of these limitations of logical systems entails the design of mathematical transformations, which then become an integral part of the theory. In our view, this approach offers a novel perspective on the structure of physical theories and, at the same time, emphasizes the relevance of the theory of polycontexturality in modern sciences.
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In 1993, the American logic S. Yablo was proposed an original infinitive formulation of the classical ≪Liar≫ paradox. It questioned the traditional notion of self-reference as the basic structure for semantic paradoxes. The article considers the arguments underlying two different approaches to analysis of proposals of the ≪Infinite Liar≫ and understanding of the genuine sources for semantic paradoxes. The first approach (V. Valpola, G.-H. von Wright, T. Bolander, etc.) imposes responsibility for the emergence of semantic paradoxes on the negation of the truth predicate. It deprives the ≪Infinite Liar≫ sentences of consistent truth values. The second approach is based on a modified version of anaphoric prosententialism (D. Grover, R. Brandom, etc.). The concepts of truth and falsehood are treated as special anaphoric operators. Logical constructs similar to the ≪Infinite Liar≫ do not attribute any definite truth values to sentences from which they are composed, but only state certain types of relations between the semantic content of such sentences.
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