Indirect self-reference: Difference between revisions

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{{Afd-merge to|Self-reference|Indirect self-reference|12 December 2025}}
'''Indirect self-reference''' describes an object [[self-reference|referring to itself]] indirectly. For example, the "this sentence is false." contains a direct self-reference, in which 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."
{{Short description|Object referring to itself indirectly}}
'''Indirect self-reference''' describes an object [[self-reference|referring to itself]] indirectly. For example, the "this sentence is false." contains a direct self-reference, in which 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 indirect reference; and expression that effectively still referred to the sentence, but did not use the pronoun "this."<ref name=":0">{{Citation |last=Bolander |first=Thomas |title=Self-Reference and Paradox |date=2024 |work=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/fall2024/entries/self-reference/ |access-date=2025-12-05 |edition=Fall 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>


If the [[quine (computing)|quine]] of a phrase is defined to be the quotation of the phrase followed by the phrase itself, then the quine of:
Indirect self-reference can be defined rigorously in terms of cycles in a graph of reference relationships.<ref>{{Cite web |last=Bolander |first=Thomas |date=2005 |title=Self-reference and logic |url=https://www.imm.dtu.dk/~tobo/essay.pdf}}</ref>
is a sentence fragment
would be:
"is a sentence fragment" is a sentence fragment
which, incidentally, is a true statement.


Now consider the sentence:
An example of this is the [[postcard paradox]], in which a sentence refers to another sentence which in turn references the original one.<ref name=":0" />
"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 a pronoun were used for this, the sentence would be the directly self-referencing "this sentence makes quite a statement." In natural language, pronouns are straightforwardly used and indirect self-references are uncommon, but in systems of [[mathematical logic]], there is generally no analog of the pronoun.
Indirect self-reference was studied in great depth by [[Willard Van Orman Quine|W. V. Quine]] and occupies a central place in the proof of [[Gödel's incompleteness theorem]].<ref>{{Cite web |last=Silva |first=Matheus |date= December 1, 2025 |title=On the circularity of Gödel's incompleteness proofs |url=https://philarchive.org/rec/SILOTC-8 |access-date=2025-12-05 |website=philarchive.org |language=en}}</ref>
 
Indirect self-reference was studied in great depth by [[Willard Van Orman Quine|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==
==See also==
* [[Diagonal lemma]]
{{portal |Mathematics |Philosophy}}
* [[Fixed point (mathematics)]]
* {{annotated link|Diagonal lemma}}
* [[Fixed-point combinator]]
* {{annotated link|Fixed point (mathematics)}}
* [[Gödel, Escher, Bach]]
* {{annotated link|Fixed-point combinator}}
* [[Indirection]]
* {{annotated link|Gödel, Escher, Bach}}
* [[Quine's paradox]]
* {{annotated link|Indirection}}
* [[Self-hosting (compilers)]]
* {{annotated link|Quine's paradox}}
* [[Self-interpreter]]
* {{annotated link|Self-hosting (compilers)}}
* {{annotated link|Self-interpreter}}


==References==
==References==

Latest revision as of 14:52, 12 December 2025

Script error: No such module "Unsubst". Template:Short description Indirect self-reference describes an object referring to itself indirectly. For example, the "this sentence is false." contains a direct self-reference, in which 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 indirect reference; and expression that effectively still referred to the sentence, but did not use the pronoun "this."[1]

Indirect self-reference can be defined rigorously in terms of cycles in a graph of reference relationships.[2]

An example of this is the postcard paradox, in which a sentence refers to another sentence which in turn references the original one.[1]

Indirect self-reference was studied in great depth by W. V. Quine and occupies a central place in the proof of Gödel's incompleteness theorem.[3]

See also

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References

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  1. ↑ a b Script error: No such module "citation/CS1".
  2. ↑ Script error: No such module "citation/CS1".
  3. ↑ Script error: No such module "citation/CS1".

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