Quine's paradox

Last updated

Quine's paradox is a paradox concerning truth values, stated by Willard Van Orman Quine. [1] 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 (i.e. it does not explicitly refer to itself). The paradox can be expressed as follows:

Contents

"yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

If the paradox is not clear, consider each part of the above description of the paradox incrementally:

it = yields falsehood when preceded by its quotation
its quotation = "yields falsehood when preceded by its quotation"
it preceded by its quotation = "yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

With these tools, the description of the paradox may now be reconsidered; it can be seen to assert the following:

The statement "'yields falsehood when preceded by its quotation' yields falsehood when preceded by its quotation" is false.

In other words, the sentence implies that it is false, which is paradoxical—for if it is false, what it states is in fact true.

Motivation

The liar paradox ("This sentence is false", or "The next sentence is true. The previous sentence is false") demonstrates essential difficulties in assigning a truth value even to simple sentences. Many philosophers attempting to explain the liar paradox – for examples see that article – concluded that the problem was with the use of demonstrative word "this" or its replacements. Once we properly analyze this sort of self-reference, according to those philosophers, the paradox no longer arises.

Quine's construction demonstrates that paradox of this kind arises independently of such direct self-reference, for, no lexeme of the sentence refers to the sentence, though Quine's sentence does contain a lexeme which refers to one of its parts. Namely, "its" near the end of the sentence is a possessive pronoun whose antecedent is the very predicate in which it occurs. Thus, although Quine's sentence per se is not self-referring, it does contain a self-referring predicate. [2]

Application

Quine suggested an unnatural linguistic resolution to such logical antinomies, inspired by Bertrand Russell's type theory and Tarski's work. His system would attach levels to a line of problematic expressions such as falsehood and denote. Entire sentences would stand higher in the hierarchy than their parts. The form "'Clause about falsehood0' yields falsehood1" will be grammatically correct, and "'Denoting0 phrase' denotes0 itself" – wrong. [1]

George Boolos, inspired by his student Michael Ernst, has written that the sentence might be syntactically ambiguous, in using multiple quotation marks whose exact mate marks cannot be determined. He revised traditional quotation into a system where the length of outer pairs of so-called q-marks of an expression is determined by the q-marks that appear inside the expression. This accounts not only for ordered quotes-within-quotes but also to, say, strings with an odd number of quotation marks. [3]

In Gödel, Escher, Bach: An Eternal Golden Braid, author Douglas Hofstadter suggests that the Quine sentence in fact uses an indirect type of self-reference. He then shows that indirect self-reference is crucial in many of the proofs of Gödel's incompleteness theorems. [4]

See also

Related Research Articles

The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters".

<span class="mw-page-title-main">Epimenides paradox</span> Paradox revealing a problem with self-reference in logic

The Epimenides paradox reveals a problem with self-reference in logic. It is named after the Cretan philosopher Epimenides of Knossos who is credited with the original statement. A typical description of the problem is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:

Epimenides was a Cretan who made the immortal statement: "All Cretans are liars."

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 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".

<span class="mw-page-title-main">Quine (computing)</span> Self-replicating program

A quine is a computer program which 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".

<span class="mw-page-title-main">Self-reference</span> Sentence, idea or formula that refers to itself

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.

<span class="mw-page-title-main">Willard Van Orman Quine</span> American philosopher and logician (1908–2000)

Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He served as the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978.

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.

<i>Gödel, Escher, Bach</i> 1979 book by Douglas Hofstadter

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.

Indirect self-reference describes an object referring to itself indirectly.

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.

<span class="mw-page-title-main">George Boolos</span> American philosopher and mathematical logician

George Stephen Boolos was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language.

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".

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.

Meta is an adjective meaning "more comprehensive" or "transcending".

This article gives a sketch of a proof of Gödel's first incompleteness theorem. This theorem applies to any formal theory that satisfies certain technical hypotheses, which are discussed as needed during the sketch. We will assume for the remainder of the article that a fixed theory satisfying these hypotheses has been selected.

<i>I Am a Strange Loop</i> 2007 book by Douglas Hofstadter

I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop to explain the sense of "I". The concept of a strange loop was originally developed in his 1979 book Gödel, Escher, Bach.

In the end, we are self-perceiving, self-inventing, locked-in mirages that are little miracles of self-reference.

References

  1. 1 2 Quine, W.V.O (1962). "Paradox". Scientific American. 206 (4): 84. Bibcode:1962SciAm.206d..84Q. doi:10.1038/scientificamerican0462-84. reprinted as "The Ways of Paradox". The Ways of Paradox and Other Essays. Cambridge: Harvard University Press. 1966. pp. 1–21.
  2. ↑ Quine, W. V. O. (1987). "Paradoxes". Quiddities: An Intermittently Philosophical Dictionary. Harvard University Press. pp. 145–149. ISBN   0-674-74352-0.
  3. ↑ Boolos, George (1995). Leonardi, P; Santambrogio, M (eds.). On Quine: New Essays. Cambridge University Press. pp. 283–2296. ISBN   978-0-521-47091-9. Reprinted in Boolos, George (1998). "Quotational Ambiguity". Logic, Logic and Logic. Harvard University Press. pp. 392–405. ISBN   0-674-53766-1.
  4. ↑ Hofstadter, Douglas (1979). Gödel, Escher, Bach: An Eternal Golden Braid . New York: Basic Books.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.