Pinocchio paradox
Encyclopedia
The Pinocchio paradox arises when Pinocchio
says: "My nose grows now", and is a version of the liar paradox
. In philosophy
and logic
, the liar paradox consists of the statement "This sentence is false." Any attempts to assign a classical binary truth value to this statement leads to a contradiction, or paradox
. This occurs because if the statement "This sentence is false" is true, then it is false; this would mean that it is technically true, but also that it is false, and so on ad infinitum
.
Although the Pinocchio paradox belongs to the Liar paradox tradition, it is a special case because it has no semantic predicates
, as for example "My sentence is false" does.
The Pinocchio paradox has nothing to do with Pinocchio
being a known liar. If Pinocchio is to say: "I am getting sick", then, it could be either true or false, but Pinocchio's sentence "My nose grows now" can be neither truth nor false hence this and only this sentence creates the Pinocchio's Liar paradox.
and specializes in Logic and the Philosophy of Logic. Peter Eldridge-Smith explained the liar paradox to Veronique and Veronique's older brother, and asked the children to come up with their own versions of the famous paradox. In a few minutes Veronique suggested: "Pinocchio
says ‘My nose will be growing’". Dr. Eldridge-Smith liked the formulation of the paradox suggested by his daughter and wrote an article on the subject. The article was published in the journal Analysis
, and the Pinocchio paradox became popularized on the web
.
is a hero of the children's novel The Adventures of Pinocchio by Italian
author Carlo Collodi
. Pinocchio, an animated puppet, is punished for each lie that he tells by undergoing further growth of his nose. There are no restrictions on the length of Pinocchio's nose. It grows as he tells lies and at one point grows so long that he can not even get his nose "through the door of the room".
The present tense of the same sentence "My nose is growing now" or "My nose grows", appears to provide a better opportunity to generate the Liar paradox.
The sentence "My nose grows" could be either true or false.
Assume the sentence: "My nose grows now" is true:
Assume the sentence: "My nose grows now" is false:
And just to make it easier, as Dr. Eldridge-Smith states, "Pinocchio’s nose is growing if and only if it is not growing", which makes Pinocchio's sentence to be "a version of the Liar."
Dr. Eldridge-Smith argues that because the phrases "is not true" and "is growing" are not synonyms, the Pinocchio paradox is not a semantic paradox:
Dr. Eldridge-Smith believes that Alfred Tarski's
theory, in which he states that Liar paradoxes should be diagnosed as arising only in languages that are "semantically closed". By this he means a language in which it is possible for one sentence to predicate the truth (or falsehood) of a sentence in the same language should not be applied to the Pinocchio paradox:
In his next article, "Pinocchio against the dialetheists", Dr. Eldridge-Smith states: "If it is a true contradiction that Pinocchio's nose grows and does not grow, then such a world is metaphysically impossible, not merely semantically impossible." He then reminds the readers that, when, (in Buridan's bridge
) Socrates
asked if he may cross a bridge, Plato
responded that he may cross the bridge only "if in the first proposition that you would utter you speak the truth. But surely, if you speak falsely I shall throw you into water." Socrates replied, "You are going to throw me into the water". Socrates's response is a sophism
that puts Plato in a difficult situation. He could not throw Socrates into water, because by doing this Plato would have violated his promise to let Socrates to cross the bridge, if he speaks the truth. On the other hand, if Plato would have allowed Socrates to cross the bridge, it would have meant that Socrates said untruth, when he replied "You are going to throw me into the water", and therefore should have been thrown into the water. In other words Socrates could be allowed to cross the bridge if and only if he could be not.
, while admitting that he has not read the articles published in Analysis, says that he does not see a paradox in the future tense of the sentence "My nose will grow now", or in the present tense of the sentence "My nose grows now".
Dr. Vallicella argues that the future tense sentence cannot generate the Liar paradox because this sentence cannot be ever treated as a falsity. He explains his point with this example: "Suppose I predict that tomorrow morning, at 6 AM, my blood pressure will be 125/75, but my prediction turns out false: my blood pressure the next morning is 135/85. No one who heard my prediction could claim that I lied when I made it even if I had the intention of deceiving my hearers. For although I made (what turned out to be) a false statement with the intention to deceive, I had no way of knowing exactly what my blood pressure would be the next day." The same explanation could be used to explain Pinocchio's sentence. Even if his prediction that his nose will grow turns out to be false, it is impossible to claim that he has lied.
Then Dr. Vallicella explains why he does not see the Liar paradox in Pinocchio's sentence if the present tense is used:
However, Dr. Vallicella's argument can be criticized in the following way: Unlike Pinocchio, Dr. Vallicella's blood pressure does not respond to the veracity of his own statements. However, Pinocchio, operating within the framework of having observed that his nose grows when and only when he lies would be making an inductively reasoned statement which he believes to be true based on his past experiences.
According to Fabio, what is known and taken premised on the story of Pinocchio is simply:
"If Pinocchio lies, then his nose grows."
Thus, as Pinocchio lies, it follows that his nose grows. We can formulate the following set of possibilities and, based on the premise, eliminate impossible cases:
- Pinocchio lies and his nose does not grow: this possibility is discarded because whenever he lies, his nose grows.
- Pinocchio lies and his nose grows: this possibility is discarded, because if he lied in the sentence "my nose will grow now," it means that his nose should not grow.
- Pinocchio tells the truth and his nose does not grow: this possibility is discarded, because if he told the truth in the sentence "my nose will grow now," it means that his nose should grow.
- Pinocchio tells the truth and his nose grows: nothing prevents such a possibility from occurring, based on the premise.
- Pinocchio's nose shrink.
So the only viable possibility is that the sentence is true and that his nose grows.
According to Fabio, the premise "if Pinocchio lies, then his nose grows," says nothing about what happens to Pinocchio's nose when he tells the truth, meaning that his nose might grow even as he told the truth.
Unfortunately, this stance becomes problematic, and the paradox still remains in tact, if we consider the following premise: "if, and only if, Pinnochio lies, then his nose grows".
Pinocchio
The Adventures of Pinocchio is a novel for children by Italian author Carlo Collodi, written in Florence. The first half was originally a serial between 1881 and 1883, and then later completed as a book for children in February 1883. It is about the mischievous adventures of Pinocchio , an...
says: "My nose grows now", and is a version of the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
. In philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
and logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the liar paradox consists of the statement "This sentence is false." Any attempts to assign a classical binary truth value to this statement leads to a contradiction, or paradox
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...
. This occurs because if the statement "This sentence is false" is true, then it is false; this would mean that it is technically true, but also that it is false, and so on ad infinitum
Ad infinitum
Ad infinitum is a Latin phrase meaning "to infinity."In context, it usually means "continue forever, without limit" and thus can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses...
.
Although the Pinocchio paradox belongs to the Liar paradox tradition, it is a special case because it has no semantic predicates
Predicate (grammar)
There are two competing notions of the predicate in theories of grammar. Traditional grammar tends to view a predicate as one of two main parts of a sentence, the other being the subject, which the predicate modifies. The other understanding of predicates is inspired from work in predicate calculus...
, as for example "My sentence is false" does.
The Pinocchio paradox has nothing to do with Pinocchio
Pinocchio
The Adventures of Pinocchio is a novel for children by Italian author Carlo Collodi, written in Florence. The first half was originally a serial between 1881 and 1883, and then later completed as a book for children in February 1883. It is about the mischievous adventures of Pinocchio , an...
being a known liar. If Pinocchio is to say: "I am getting sick", then, it could be either true or false, but Pinocchio's sentence "My nose grows now" can be neither truth nor false hence this and only this sentence creates the Pinocchio's Liar paradox.
History
The Pinocchio paradox was proposed in February 2001 by 11 year old Veronique Eldridge-Smith. Veronique is the daughter of Dr. Peter Eldridge-Smith, who works for Australian National UniversityAustralian National University
The Australian National University is a teaching and research university located in the Australian capital, Canberra.As of 2009, the ANU employs 3,945 administrative staff who teach approximately 10,000 undergraduates, and 7,500 postgraduate students...
and specializes in Logic and the Philosophy of Logic. Peter Eldridge-Smith explained the liar paradox to Veronique and Veronique's older brother, and asked the children to come up with their own versions of the famous paradox. In a few minutes Veronique suggested: "Pinocchio
Pinocchio
The Adventures of Pinocchio is a novel for children by Italian author Carlo Collodi, written in Florence. The first half was originally a serial between 1881 and 1883, and then later completed as a book for children in February 1883. It is about the mischievous adventures of Pinocchio , an...
says ‘My nose will be growing’". Dr. Eldridge-Smith liked the formulation of the paradox suggested by his daughter and wrote an article on the subject. The article was published in the journal Analysis
Analysis (journal)
Analysis is a peer-reviewed academic journal of philosophy established in 1933 that is published quarterly by Oxford University Press on behalf of the Analysis Trust. Prior to January 2009, the journal was published by Blackwell Publishing. Electronic access to this journal is available via JSTOR ,...
, and the Pinocchio paradox became popularized on the web
Internet
The Internet is a global system of interconnected computer networks that use the standard Internet protocol suite to serve billions of users worldwide...
.
The story behind the paradox
PinocchioPinocchio
The Adventures of Pinocchio is a novel for children by Italian author Carlo Collodi, written in Florence. The first half was originally a serial between 1881 and 1883, and then later completed as a book for children in February 1883. It is about the mischievous adventures of Pinocchio , an...
is a hero of the children's novel The Adventures of Pinocchio by Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
author Carlo Collodi
Carlo Collodi
Carlo Lorenzini , better known by the pen name Carlo Collodi, was an Italian children's writer known for the world-renowned fairy tale novel, The Adventures of Pinocchio.-Biography:...
. Pinocchio, an animated puppet, is punished for each lie that he tells by undergoing further growth of his nose. There are no restrictions on the length of Pinocchio's nose. It grows as he tells lies and at one point grows so long that he can not even get his nose "through the door of the room".
The paradox
The paradox suggested by Veronique, "My nose grows now", or in future tense: "will be growing", leaves room for different interpretations. In the novel Pinocchio's nose continues to grow as he lies: "As he spoke, his nose, long though it was, became at least two inches longer." So logicians question if the sentence "My nose will be growing" was the only sentence that Pinocchio spoke, did he tell a lie before he said "My nose will be growing", or was he going to tell a lie—and how long would it take for his nose to start growing?The present tense of the same sentence "My nose is growing now" or "My nose grows", appears to provide a better opportunity to generate the Liar paradox.
The sentence "My nose grows" could be either true or false.
Assume the sentence: "My nose grows now" is true:
- Which means that Pinocchio's nose grows now because he truthfully says it is, but then
- Pinocchio's nose does not grow now because according to the novel it grows only as Pinocchio lies, but then
- Pinocchio's nose grows now because Pinocchio's nose does not grow now, and Pinocchio trustfully says it grows now, and it is false, that makes Pinocchio's sentence to be false, but then
- Pinocchio's nose does not grow now because Pinocchio's nose grows now, and Pinocchio trustfully says it grows now, and it is true that makes Pinocchio's sentence to be true, but then
- And so on ad infinitum.
Assume the sentence: "My nose grows now" is false:
- Which means that Pinocchio's nose does not grow now because he falsely says it is, but then
- Pinocchio's nose grows now because according to the novel it grows only as Pinocchio lies, but then
- Pinocchio's nose does not grow now because Pinocchio's nose grows now, and Pinocchio falsely says it grows now, and it is false that makes Pinocchio's sentence to be true, but then
- Pinocchio's nose grows now because Pinocchio's nose does not grow now, and Pinocchio falsely says it grows now, and it is true, that makes Pinocchio's sentence to be false, but then
- And so on ad infinitum.
And just to make it easier, as Dr. Eldridge-Smith states, "Pinocchio’s nose is growing if and only if it is not growing", which makes Pinocchio's sentence to be "a version of the Liar."
Dr. Eldridge-Smith argues that because the phrases "is not true" and "is growing" are not synonyms, the Pinocchio paradox is not a semantic paradox:
Dr. Eldridge-Smith believes that Alfred Tarski's
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
theory, in which he states that Liar paradoxes should be diagnosed as arising only in languages that are "semantically closed". By this he means a language in which it is possible for one sentence to predicate the truth (or falsehood) of a sentence in the same language should not be applied to the Pinocchio paradox:
In his next article, "Pinocchio against the dialetheists", Dr. Eldridge-Smith states: "If it is a true contradiction that Pinocchio's nose grows and does not grow, then such a world is metaphysically impossible, not merely semantically impossible." He then reminds the readers that, when, (in Buridan's bridge
Buridan's bridge
Buridan's Bridge is described by Jean Buridan, one of the most famous and influential philosophers of the Late Middle Ages, in his book Sophismata...
) Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
asked if he may cross a bridge, Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
responded that he may cross the bridge only "if in the first proposition that you would utter you speak the truth. But surely, if you speak falsely I shall throw you into water." Socrates replied, "You are going to throw me into the water". Socrates's response is a sophism
Sophism
Sophism in the modern definition is a specious argument used for deceiving someone. In ancient Greece, sophists were a category of teachers who specialized in using the tools of philosophy and rhetoric for the purpose of teaching aretê — excellence, or virtue — predominantly to young statesmen and...
that puts Plato in a difficult situation. He could not throw Socrates into water, because by doing this Plato would have violated his promise to let Socrates to cross the bridge, if he speaks the truth. On the other hand, if Plato would have allowed Socrates to cross the bridge, it would have meant that Socrates said untruth, when he replied "You are going to throw me into the water", and therefore should have been thrown into the water. In other words Socrates could be allowed to cross the bridge if and only if he could be not.
Criticism
Dr. William F. VallicellaWilliam F. Vallicella
-Biography:Vallicella has a Ph.D. , taught for a number of years at University of Dayton and Case Western Reserve University , and retired to Gold Canyon, Arizona from where he now contributes to philosophy mainly online...
, while admitting that he has not read the articles published in Analysis, says that he does not see a paradox in the future tense of the sentence "My nose will grow now", or in the present tense of the sentence "My nose grows now".
Dr. Vallicella argues that the future tense sentence cannot generate the Liar paradox because this sentence cannot be ever treated as a falsity. He explains his point with this example: "Suppose I predict that tomorrow morning, at 6 AM, my blood pressure will be 125/75, but my prediction turns out false: my blood pressure the next morning is 135/85. No one who heard my prediction could claim that I lied when I made it even if I had the intention of deceiving my hearers. For although I made (what turned out to be) a false statement with the intention to deceive, I had no way of knowing exactly what my blood pressure would be the next day." The same explanation could be used to explain Pinocchio's sentence. Even if his prediction that his nose will grow turns out to be false, it is impossible to claim that he has lied.
Then Dr. Vallicella explains why he does not see the Liar paradox in Pinocchio's sentence if the present tense is used:
However, Dr. Vallicella's argument can be criticized in the following way: Unlike Pinocchio, Dr. Vallicella's blood pressure does not respond to the veracity of his own statements. However, Pinocchio, operating within the framework of having observed that his nose grows when and only when he lies would be making an inductively reasoned statement which he believes to be true based on his past experiences.
A Possible Solution
A possible solution to the paradox, informally formulated by Brazilian professor Fabio Miranda Rodrigues, is to simply examine what is really axiomatic in the story of Pinocchio and what is mere consequence, combined with analysis of the truth table of the possibilities for events arising from the sentence said by Pinocchio and by eliminating the potential inconsistences.According to Fabio, what is known and taken premised on the story of Pinocchio is simply:
"If Pinocchio lies, then his nose grows."
Thus, as Pinocchio lies, it follows that his nose grows. We can formulate the following set of possibilities and, based on the premise, eliminate impossible cases:
- Pinocchio lies and his nose does not grow: this possibility is discarded because whenever he lies, his nose grows.
- Pinocchio lies and his nose grows: this possibility is discarded, because if he lied in the sentence "my nose will grow now," it means that his nose should not grow.
- Pinocchio tells the truth and his nose does not grow: this possibility is discarded, because if he told the truth in the sentence "my nose will grow now," it means that his nose should grow.
- Pinocchio tells the truth and his nose grows: nothing prevents such a possibility from occurring, based on the premise.
- Pinocchio's nose shrink.
So the only viable possibility is that the sentence is true and that his nose grows.
According to Fabio, the premise "if Pinocchio lies, then his nose grows," says nothing about what happens to Pinocchio's nose when he tells the truth, meaning that his nose might grow even as he told the truth.
Unfortunately, this stance becomes problematic, and the paradox still remains in tact, if we consider the following premise: "if, and only if, Pinnochio lies, then his nose grows".