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 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
The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters".
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".
A quine is a computer program that takes no input and produces a copy of its own source code as its only output. The standard terms for these programs in the computability theory and computer science literature are "self-replicating programs", "self-reproducing programs", and "self-copying programs".
In analytic philosophy and computer science, referential transparency and referential opacity are properties of linguistic constructions, and by extension of languages. A linguistic construction is called referentially transparent when for any expression built from it, replacing a subexpression with another one that denotes the same value does not change the value of the expression. Otherwise, it is called referentially opaque. Each expression built from a referentially opaque linguistic construction states something about a subexpression, whereas each expression built from a referentially transparent linguistic construction states something not about a subexpression, meaning that the subexpressions are ‘transparent’ to the expression, acting merely as ‘references’ to something else. For example, the linguistic construction ‘_ was wise’ is referentially transparent but ‘_ said _’ is referentially opaque.
Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields.
A strange loop is a cyclic structure that goes through several levels in a hierarchical system. It arises when, by moving only upwards or downwards through the system, one finds oneself back where one started. Strange loops may involve self-reference and paradox. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter in Gödel, Escher, Bach, and is further elaborated in Hofstadter's book I Am a Strange Loop, published in 2007.
Gödel, Escher, Bach: an Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter.
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics.
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder, or a wildcard character that stands for an unspecified symbol.
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F". The paradox requires only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.
In analytic philosophy, a fundamental distinction is made between the use of a term and the mere mention of it. Many philosophical works have been "vitiated by a failure to distinguish use and mention". The distinction can sometimes be pedantic, especially in simple cases where it is obvious.
In mathematical logic, the diagonal lemma 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. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem.
In philosophy and logic, a deflationary theory of truth is one of a family of theories that all have in common the claim that assertions of predicate truth of a statement do not attribute a property called "truth" to such a statement.
Quine's paradox is a paradox concerning truth values, stated by Willard Van Orman Quine. It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals. The paradox can be expressed as follows:
Self-referential humor, also known as self-reflexive humor, self-aware humor, or meta humor, is a type of comedic expression that—either directed toward some other subject, or openly directed toward itself—is self-referential in some way, intentionally alluding to the very person who is expressing the humor in a comedic fashion, or to some specific aspect of that same comedic expression. Here, meta is used to describe that the joke explicitly talks about other jokes, a usage similar to the words metadata, metatheatrics and metafiction. Self-referential humor expressed discreetly and surrealistically is a form of bathos. In general, self-referential humor often uses hypocrisy, oxymoron, or paradox to create a contradictory or otherwise absurd situation that is humorous to the audience.
A truth-bearer is an entity that is said to be either true or false and nothing else. The thesis that some things are true while others are false has led to different theories about the nature of these entities. Since there is divergence of opinion on the matter, the term truth-bearer is used to be neutral among the various theories. Truth-bearer candidates include propositions, sentences, sentence-tokens, statements, beliefs, thoughts, intuitions, utterances, and judgements but different authors exclude one or more of these, deny their existence, argue that they are true only in a derivative sense, assert or assume that the terms are synonymous, or seek to avoid addressing their distinction or do not clarify it.
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs.
Logophoricity is a phenomenon of binding relation that may employ a morphologically different set of anaphoric forms, in the context where the referent is an entity whose speech, thoughts, or feelings are being reported. This entity may or may not be distant from the discourse, but the referent must reside in a clause external to the one in which the logophor resides. The specially-formed anaphors that are morphologically distinct from the typical pronouns of a language are known as logophoric pronouns, originally coined by the linguist Claude Hagège. The linguistic importance of logophoricity is its capability to do away with ambiguity as to who is being referred to. A crucial element of logophoricity is the logophoric context, defined as the environment where use of logophoric pronouns is possible. Several syntactic and semantic accounts have been suggested. While some languages may not be purely logophoric, logophoric context may still be found in those languages; in those cases, it is common to find that in the place where logophoric pronouns would typically occur, non-clause-bounded reflexive pronouns appear instead.
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