Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 S - Self-reference


Self-reference: In logic: A statement refers to itself if it contains a term that denotes the very statement.

The occurence and possibility of self-reference creates many problems, and also is at the foundation of some deep theorems.

The problems may be indicated by the statement "This statement is false". Intuitively, it seems as if, in case the statement is true, it is false, whereas if it is false it seems as if it must be true. Hence, if any statement is true or false, the statement just mentioned seems to show that cannot be so.

The given intuitive reasoning can be blocked in various ways, but many of these lead to problems. In any case, the root problem is how to represent in logic such terms as "this statement", since standard logic does not contain indexical terms like "this" (or indeed "my" or "now" or "here").

Some of the deep theorems related to self-reference are Gödel's Incompleteness Theorems, that turn around the statement "This statement is unprovable".

Gödel showed that one can arithmetize logic, and that one can define arithmetical functions that do what "this" does in natural language. He then shows that in the resulting system, which requires only standard logic plus basic arithmetics, the statement "This statement is unprovable" is neither provably true (for then it would not be provable) nor provably false (for then it should be provably so), and that consequently that one can in standard basic mathematics, supposing that is consistent, construct sentences that are not provable (as argued) but that are true (since they say what is in fact so). And by a similar argument one can show that within such a system one cannot prove a statement like "This system is provably consistent", if indeed the system is consistent. (It may be provably consistent by another system, though.)

There are many problems related to self-reference, in part because it involves a use of language that is not only about the domain the language is taken to represent, but also about the (formal) language itself, and in part because this leads to unclarities, (apparent) circularities, subtleties, and the limits of what one can represent with a language.

At least some of these problems are also relevant to psychology, for much of human self-consciousness is strongly tied up with language, and by statements that relate language to a domain or reality. Thus, one example of such a statement is "I am a theory of my brain, that my brain constructs to account for its input (experiences)". This may be true - but how does it do it, and what is logically involved in such subtly and multiply self-referring statements?

From a logical point of view, many books of the mathematician, logician and philosopher Raymond Smullyan are highly relevant, since he constructed many very subtle yet clear formal systems to account for Gödelian arguments of many kinds, and also wrote a number of delightful book with logic puzzles concerning the same subject, that are far less technical, yet also very revealing. The most technical, latest, and most comprehensive of his books relating to these problems is "Diagonalization and Self-Reference".

See also: Gödel's Theorems, Paradox, Theory of Types


Gödel, Smullyan

 Original: Dec 21, 2007                                                Last edited: 12 December 2011.   Top