Self-reference: In logic: A
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