Quine's Paradox
Encyclopedia
Quine's paradox is a paradox
concerning truth values, attributed to 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 demonstrative
s or indexicals
(i.e. it does not explicitly refer to itself). The paradox can be expressed as follows:
If the paradox is not clear, consider each part of the above description of the paradox incrementally:
With these tools, we may now reconsider the description of the paradox. It can be seen to assert the following:
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.
("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, concluded that the problem was with the use of demonstrative
word "this" or its replacements. Once we properly analyze this sort of self-reference
, they claimed, 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.
Any system, such as English, that contains entities such as words or sentences that can be used to apply to themselves, must contain this type of paradox. There is no way to eliminate the paradoxes, short of a severe crippling of the language.
, 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 use a higher hierarchy each of their parts'. The form Clause about falsehood0' yields falsehood1" will be grammatically correct, and Denoting0 phrase' denotes0 itself" – wrong.
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.
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 the proofs of Gödel's incompleteness theorems
.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
concerning truth values, attributed to Willard Van Orman Quine
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
. It is related to the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
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 demonstrative
Demonstrative
In linguistics, demonstratives are deictic words that indicate which entities a speaker refers to and distinguishes those entities from others...
s or indexicals
Indexicality
In linguistics and in philosophy of language, an indexical behavior or utterance points to some state of affairs. For example, I refers to whoever is speaking; now refers to the time at which that word is uttered; and here refers to the place of utterance...
(i.e. it does not explicitly refer to itself). The paradox can be expressed as follows:
- "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, we may now reconsider the description of the paradox. 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 paradoxLiar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
("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, concluded that the problem was with the use of demonstrative
Demonstrative
In linguistics, demonstratives are deictic words that indicate which entities a speaker refers to and distinguishes those entities from others...
word "this" or its replacements. Once we properly analyze this sort of self-reference
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...
, they claimed, the paradox no longer arises.
Quine's construction demonstrates that paradox of this kind arises independently of such direct self-reference, for, no lexeme
Lexeme
A lexeme is an abstract unit of morphological analysis in linguistics, that roughly corresponds to a set of forms taken by a single word. For example, in the English language, run, runs, ran and running are forms of the same lexeme, conventionally written as RUN...
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
Possessive pronoun
A possessive pronoun is a part of speech that substitutes for a noun phrase that begins with a possessive determiner . For example, in the sentence These glasses are mine, not yours, the words mine and yours are possessive pronouns and stand for my glasses and your glasses, respectively...
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.
Any system, such as English, that contains entities such as words or sentences that can be used to apply to themselves, must contain this type of paradox. There is no way to eliminate the paradoxes, short of a severe crippling of the language.
Application
Quine suggested an unnatural linguistic resolution to such logical antinomiesAntinomy
Antinomy literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology....
, inspired by Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
's Type theory
Type theory
In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
and Tarski's work. His system would attach levels to a line of problematic expressions such as falsehood and denote. Entire sentences would use a higher hierarchy each of their parts'. The form Clause about falsehood0' yields falsehood1" will be grammatically correct, and Denoting0 phrase' denotes0 itself" – wrong.
George Boolos
George Boolos
George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
, 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.
In Gödel, Escher, Bach: An Eternal Golden Braid
Gödel, Escher, Bach
Gödel, 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"....
, author Douglas Hofstadter
Douglas Hofstadter
Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...
suggests that the Quine sentence in fact uses an indirect type of self-reference. He then shows that indirect self-reference is crucial in the proofs of Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
.
See also
- Quine, a computer program that produces its source codeSource codeIn computer science, source code is text written using the format and syntax of the programming language that it is being written in. Such a language is specially designed to facilitate the work of computer programmers, who specify the actions to be performed by a computer mostly by writing source...
as output - Grelling paradox
- Russell paradox
- List of paradoxes