Indirect self-reference
Encyclopedia
Indirect self-reference describes an object referring to itself
indirectly.
For example, define the function f such that f(x) = "x(x)". Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself.
This example is similar to the Scheme expression "((lambda(x)(x x)) (lambda(x)(x x)))", which is expanded to itself by beta reduction, and so its evaluation loops indefinitely despite the lack of explicit looping constructs. An equivalent example can be formulated in lambda-calculus.
Indirect self-reference is special in that its self-referential quality is not explicit, as it is in the sentence "this sentence is false." The phrase "this sentence" refers directly to the sentence as a whole. An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this."
An example will help to explain this. Suppose we define the quine of a phrase to be the quotation of the phrase followed by the phrase itself. So, the quine of:
is a sentence fragment
would be:
"is a sentence fragment" is a sentence fragment
which, incidentally, is a true statement.
Now consider the sentence:
"when quined, makes quite a statement" when quined, makes quite a statement
The quotation here, plus the phrase "when quined," indirectly refers to the entire sentence. The importance of this fact is that the remainder of the sentence, the phrase "makes quite a statement," can now make a statement about the sentence as a whole. If we had used a pronoun for this, we could have written something like "this sentence makes quite a statement."
It seems silly to go through this trouble when pronouns will suffice (and when they make more sense to the casual reader), but in systems of mathematical logic
, there is generally no analog of the pronoun. It is somewhat surprising, in fact, that self-reference can be achieved at all in these systems.
Upon closer inspection, it can be seen that in fact, the Scheme example above uses a quine, and f(x) is actually the quine function itself.
Indirect self-reference was studied in great depth by W. V. Quine (after whom the operation above is named), and occupies a central place in the proof of Gödel's incompleteness theorem. Among the paradoxical statements developed by Quine is the following:
"yields a false statement when preceded by its quotation" yields a false statement when preceded by its quotation
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
indirectly.
For example, define the function f such that f(x) = "x(x)". Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself.
This example is similar to the Scheme expression "((lambda(x)(x x)) (lambda(x)(x x)))", which is expanded to itself by beta reduction, and so its evaluation loops indefinitely despite the lack of explicit looping constructs. An equivalent example can be formulated in lambda-calculus.
Indirect self-reference is special in that its self-referential quality is not explicit, as it is in the sentence "this sentence is false." The phrase "this sentence" refers directly to the sentence as a whole. An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this."
An example will help to explain this. Suppose we define the quine of a phrase to be the quotation of the phrase followed by the phrase itself. So, the quine of:
is a sentence fragment
would be:
"is a sentence fragment" is a sentence fragment
which, incidentally, is a true statement.
Now consider the sentence:
"when quined, makes quite a statement" when quined, makes quite a statement
The quotation here, plus the phrase "when quined," indirectly refers to the entire sentence. The importance of this fact is that the remainder of the sentence, the phrase "makes quite a statement," can now make a statement about the sentence as a whole. If we had used a pronoun for this, we could have written something like "this sentence makes quite a statement."
It seems silly to go through this trouble when pronouns will suffice (and when they make more sense to the casual reader), but in systems of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, there is generally no analog of the pronoun. It is somewhat surprising, in fact, that self-reference can be achieved at all in these systems.
Upon closer inspection, it can be seen that in fact, the Scheme example above uses a quine, and f(x) is actually the quine function itself.
Indirect self-reference was studied in great depth by W. V. Quine (after whom the operation above is named), and occupies a central place in the proof of Gödel's incompleteness theorem. Among the paradoxical statements developed by Quine is the following:
"yields a false statement when preceded by its quotation" yields a false statement when preceded by its quotation
See also
- Fixed pointFixed point (mathematics)In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
- Fixed point combinatorFixed point combinatorIn computer science, a fixed-point combinator is a higher-order function that computes a fixed point of other functions. A fixed point of a function f is a value x such that x = f. For example, 0 and 1 are fixed points of the function f = x2, because 0 = 02 and 1 = 12...
- Willard Van Orman QuineWillard Van Orman QuineWillard Van Orman Quine was an American philosopher and logician in the analytic tradition...
- Quine (computing)
- IndirectionIndirectionIn computer programming, indirection is the ability to reference something using a name, reference, or container instead of the value itself. The most common form of indirection is the act of manipulating a value through its memory address. For example, accessing a variable through the use of a...
- Self-referenceSelf-referenceSelf-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
- Diagonal lemmaDiagonal lemmaIn mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions...
- Self-interpreterSelf-interpreterA self-interpreter, or metainterpreter, is a programming language interpreter written in the language it interprets. An example would be a BASIC interpreter written in BASIC...
- Gödel, Escher, BachGödel, Escher, BachGödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....
- Actor modelActor modelIn computer science, the Actor model is a mathematical model of concurrent computation that treats "actors" as the universal primitives of concurrent digital computation: in response to a message that it receives, an actor can make local decisions, create more actors, send more messages, and...