Self-reference

Self-reference is used to denote any situation in which some one 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. That self-reference can lead to paradoxes is the main reason why so much effort has been put into understanding, modeling, 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.

I 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.

PARADOXES: Richard’s Paradox, Berry’s Paradox, Liar’s Paradox.

REFLECTION:
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.


UNIVERSALITY:
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.

If R is the reference relation of our natural language then the sentence
“All sentences are false”

will be universal. The problem about universality is that reflection and universality together necessarily lead to self-reference, and thereby is likely to give rise to paradoxes.


UNGROUNDEDNESS AND SELF-REFERENCE:
Self-reference often occurs in situations that have an ungrounded nature.
If we take the dictionary example, we can give a simple example of ungroundedness. Let R be the reference relation of Webster’s 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 → . . .

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 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.


IMPORTANCE OF SELF-REFERENCE
Although it is very difficult to tame self-reference (it can be done is some cases. Ex. Russell’s Paradox), but it has become very important as it is showing up in almost all sciences.

Mathematics – Russell’s Paradox, Gödel’s incompleteness theorem.
Physics - Heisenberg’s Uncertainty principle, Space-time and rotation.
Computer Sciences – Recursion, self-modifying codes, AI – agent acting on itself.

Studying self-reference can also help in the study of mind i.e. subjectivity, because that is also a self-reference as in that case “mind will be studying itself”.