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Quine’s Paradox

Quine’s paradox, originated by Willard Quine is the following statement (Note: there are several variations on the exact presentation of the statement):

“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

The supposed paradox is that the expression - Yields falsehood when preceded by its quotation - when preceded by itself in quotation marks, gives a new expression. If this new expression is true, then it says of itself that it must be false. But if it is false, then what it says of itself is true.

 

In the supposed paradox, the term

quotation

appears to be a variable term, since, in the above expression, the value of the variable ‘quotation’ is supposedly specified by the use of ‘its’, where one is invited to assume that it refers to

“Yields falsehood when preceded by its quotation”

 

But an invitation to assume something is not acceptable for rigorous logical analysis. Without making any such assumptions, we can simply take the phrase:

yields falsehood when preceded by its quotation

and precede it by:

its quotation

which gives:

its quotation yields falsehood when preceded by its quotation

 

Now, supporters of the supposed paradox will then tell you, “No, that is not what is meant” and that it means that you precede the phrase:

yields falsehood when preceded by its quotation

by the phrase itself enclosed in quotation marks. Well, then, if that is the case, why isn’t it so defined at the outset rather than the ambiguous phrase originally given? So let’s define it this way to avoid any ambiguity. But does this resolve the problem? This would give:

yields falsehood when preceded by this phrase itself enclosed in quotation marks

So we take this phrase:

yields falsehood when preceded by this phrase itself enclosed in quotation marks

and precede it by:

this phrase itself enclosed in quotation marks

to give:

this phrase itself enclosed in quotation marks yields falsehood when preceded by this phrase itself enclosed in quotation marks

 

But we see that we are no further forward. So while the claim is that there is no direct self-reference, for a paradoxical contradiction to arise, either it has to be explicitly asserted (as in the Wikipedia article, see below) or it has to be implicitly assumed that:

its quotation = “yields falsehood when preceded by its quotation”

But now it is patently obvious that

its quotation

is defined in terms of itself, so there is a direct self-reference.

 

Of course, the problem is that in order to generate a paradox one has to read the phrase

yields falsehood when preceded by its quotation

both as syntax of the language, and at the same time, as an object of the language. Given the statement:

“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

we are being asked on the one hand to read the latter half of the expression as syntax in order for the expression to be meaningful at all, whereas we are also being asked at the same time to read the latter half of the expression as an object of the language. At first glance this may not appear to be the case, and that it is only the first part of the expression that we are to treat as an object (the part enclosed by quotation marks). But to generate the “paradox”, one has to assert a particular correspondence between certain terms that are objects of the language. That correspondence is between the:

it

in the term:

its quotation

and the second part of the expression itself, which is:

yields falsehood when preceded by its quotation

 

Clearly, it is an object of the language, so to assert the above correspondence requires that one must also read the second half of the expression as an object (a correspondence between an object and a non-object within the same language is an absurdity) so that one can confirm that the it in its quotation refers to the actual sequence of symbols that is:

yields falsehood when preceded by its quotation

and which occurs in the latter part of the overall expression.

 

Logical analysis reveals that the apparent complexity of the formulation of the “paradox” merely serves to obfuscate what is happening. And we can also analyze it in another way. Since the expression:

yields falsehood when preceded by its quotation

begins with a verb “yields”, if it is to generate a valid sentence, then it must be preceded by an object. If preceded by something that is not an object, then the result is not a valid sentence. So if we want to remove any ambiguity, then the obvious way to do so is to formulate the expression:

x yields falsehood when preceded by “x

where x is a variable whose domain is terms that are objects of the language in question. This asserts that for any such x, the expression:

xx

yields falsehood. But obviously, since all such expressions consists only of consecutive objects of the language, no such expression can be a valid sentence of the language, and cannot have a truth value. Again, logical analysis reveals that the supposed paradox is merely the result of conflation of syntax and objects of the language in question - in order for a paradox to arise, one must take an expression to be at the same time syntax of the language and an object of the language.

 

This shows that Quine’s paradox is simply a convoluted version of the Liar paradox, where the use of quotation marks as delimiters fails to prevent the ambiguity inherent in natural languages such as English.

 

Wikipedia version

In the Wikipedia article on Quine’s paradox (as of 12th November 2016) the claim is that the paradox can be clarified by:

  1. it = yields falsehood when preceded by its quotation
  2. its quotation = “yields falsehood when preceded by its quotation”
  3. it preceded by its quotation = “yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

If these equalities are correct then we can take the expression:

“yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

and substitute according to equality 2 to give:

its quotation yields falsehood when preceded by its quotation,

and substitute again by equality 1 to give:

its quotation it

which is nonsensical, yet if the equalities given are correct, that is an entirely logical application of the given equalities with the inevitably absurd result.

 

Smullyan and self-reference

Raymond Smullyan’s book Diagonalization and Self-Reference (Footnote: Raymond Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994 Diagonalization and Self-Reference: Details.) contains similar self-referential expressions.

 

It is notable that at the very start of the book Smullyan refers to the difference between an expression enclosed by quotation marks, and an expression not enclosed by quotation marks. He says:

 

When one talks about a word, rather than that which is denoted by the word, one encloses the word in quotation marks. There is a difference between using a word and mentioning the word (which is talking about the word, instead of the denotation of the word).

 

In other words, what Smullyan is saying is that enclosing a word in quotation marks is confirming that that which is within the quotation marks is an object of the language, and is not be taken as syntax of the language, that is, its meaning in the language is to be ignored. Unfortunately, Smullyan fails to follow his own advice. Smullyan continues:

 

… We use the symbol “x” as a variable ranging over expressions of the English language. By the diagonalization of an expression, we mean the result of substituting the quotation of the expression for every occurrence of the variable “x” in the expression. For example, consider the following expression.

 

(1) John is reading x

 

The expression (1) is not a sentence, true or false, but becomes a sentence (true or false) upon substituting the quotation of any expression for “x”. If we substitute the quotation of (1) itself for “x”, we obtain the diagonalization of (1), which is

 

(2) John is reading “John is reading x

 

Now, (2) is a sentence (Footnote: Note that this confirms that x is not a varaible when enclosed within quotation marks), and it asserts that John is reading (1). However, (2) is not self-referential; it does not assert that John is reading (2); it asserts that John is reading (1).

 

Let us consider the following expression.

 

(3) John is reading the diagonalization of x.

 

The diagonalization of (3) is the following

 

(4) John is reading the diagonalization of “John is reading the diagonalization of x.”

 

Sentence (4) asserts that John is reading the diagonalization of (3), but the diagonalization of (3) is (4) itself. Thus (4) asserts that John is reading the very sentence (4). Thus sentence (4) is self-referential.

 

This is arrant nonsense. Smullyan had previously defined that enclosing an expression by quotation marks indicates that any reference to what is enclosed is directly to that enclosed as an object without syntactical meaning, and not to what that expression refers to, which requires extracting meaning from the syntax of the expression. The mistake that Smullyan makes is the failure to ensure the distinction between the case when x is a variable and when it is not. When it is within quotation marks, it is not a variable that can be substituted, since whatever phrase is within quotation marks is always referred to by the rest of the expression as an object, i.e., as exactly that phrase. In Smullyan’s (4) above, the x cannot be substituted, since it is not a variable, but part of the object that is within quotation marks, so that there can be no diagonalization of the phrase:

 

“John is reading the diagonalization of x.”

 

as it occurs within the sentence (4) - since it is enclosed by quotation marks.

 

This means that Smullyan’s assertion that the sentence (4), which is:

John is reading the diagonalization of “John is reading the diagonalization of x.”

is a sentence that expresses the concept that John is actually reading the sentence (4) itself has no logical basis. All it says is that John is reading the string of symbols that follows the phrase John is reading, which is:

the diagonalization of “John is reading the diagonalization of x.”

 

Now, if we want to know what the diagonalization above is, we must be aware that the x within the quotation marks is not a variable and so cannot be substituted; this means that Smullyan’s defintion, as in the above:

the result of substituting the quotation of the expression for every occurrence of the variable “x” in the expression

when applied to:

the diagonalization of “John is reading the diagonalization of x.”

is simply:

John is reading the diagonalization of x.

which, of course, is not Smullyan’s sentence (4) as above.

 

Of course, if x was a variable there, then if you substitute it by any expression that has an x in it, then you have a new expression that also has x as a variable, and hence no matter how many times you substitute, you never end up with an expression that is what Smullyan calls a sentence, and hence cannot be either true or false. Note that Smullyan said above, “The expression (1) is not a sentence, true or false, but becomes a sentence (true or false) upon substituting … for “x”.

 

Smullyan’s fudge is that he wants to have x both as a variable and not as a variable at the same time, which is nonsensical.

 

Smullyan’s “norm”

Smullyan continues by erroneously claiming that he can demonstrate another valid method of self-reference that does not involve substitution, nor variables, saying:

By the norm of an expression we shall mean the expression followed by its own quotation. For example, consider the following expression.

 

(1) John is reading

 

The norm of (1) is the following.

 

(2) John is reading “John is reading”

 

… But now consider the following.

 

(3) John is reading the norm of

 

Its norm is the following sentence.

 

(4) John is reading the norm of “John is reading the norm of”

 

Sentence (4) asserts that John is reading the norm of (3), but the norm of (3) is (4) itself. And so (4) asserts that John is reading (4). Thus (4) is self-referential.

 

When Smullyan says that the norm of an expression is that expression followed by its own quotation, he isn’t very clear as to precisely what he means by this. However, we know something of his intention of meaning by his statement that the norm of (3) is (4). That is, we have either that the rule is:

(a) the norm of  x =  x x, where x = John is reading the norm of

which gives:

the norm of John is reading the norm of = John is reading the norm ofJohn is reading the norm of

or the rule is:

(b) the norm of “x” = “ xx” ”, where x = John is reading the norm of

which gives:

the norm of John is reading the norm of = John is reading the norm ofJohn is reading the norm of” ”

 

We can analyze these options (note that in the following, we use red text simply to make it easier to see what is happening).

First we suppose that rule (a) above applies. Take the sentence (4) above.

(4) John is reading the norm of John is reading the norm of

 

Now, either what John is reading is literally precisely:

(A) the norm of John is reading the norm of

or it is what the rule (a) gives for the above symbol sequence, which is:

(B) John is reading the norm of John is reading the norm of

 

So which is it? You can’t be ambiguous here if you are asserting that you have a logical argument - you must have a single definitive rule and abide by it. (A) cannot apply, since if that was that case, what John is reading as described by sentence (4) is literally precisely:

(A) the norm of John is reading the norm of

which is not the sentence (4) itself, as required by Smullyan, so it must be the case that (B) applies. So, given that that is the case, then

(4) John is reading the norm of John is reading the norm of

states that

(C) John is reading John is reading the norm ofJohn is reading the norm of

 

The problem with the above sentence (C), of course (and which is Smullyan’s huge fudge factor) (Footnote: Also note that Smullyan omits the end of sentence full stops as part of his fudge factor.) is that there is no unambiguous delineation of what is to be defined as objects in the language, and what is to be defined as syntax of the language. Smullyan’s description of “the norm of” implies that the symbol sequence to which it applies to is to be taken as an object of the language - that is, that it is not to be read as part of the syntax of the language. So (4) is syntax of language where:

John

is the subject noun,

is reading

is the verb, and

the norm of John is reading the norm of

is the object. And, in (B) above, since:

the norm of John is reading the norm of

is an object, then that which it refers to:

John is reading the norm of John is reading the norm of

is also an object, not syntax of the language.

 

But the problem is that, taken on its own, the sentence:

(C) John is reading John is reading the norm of John is reading the norm of

is ambiguous, because there is no unambiguous rule as to how this should be read. What is required is clearly a delineation of objects and syntax, such as by the use of delimiters such as quotation marks, as in:

(D) John is reading John is reading the norm of John is reading the norm of” ”

where what John is actually reading is completely unambiguous.

 

So, clearly, the rule (b) is the unambiguous rule that must apply, giving

(b) the norm of “x” = “ xx” ”, where x = John is reading the norm of

which gives:

the norm of “John is reading the norm of” =

John is reading the norm of John is reading the norm of” ”

 

This gives, for sentence (4), which is:

(4) John is reading the norm of John is reading the norm of

that John is reading

(E) John is reading the norm of John is reading the norm of” ”

which clearly and unambiguously shows that the sentence (4) is not actually precisely the same as the sentence (E) to which it refers. Sentence (4) is syntax of the language, whereas sentence (E) is merely an object of the language, which is unambiguously indicated by the fact that it is enclosed by quotation marks.

 

Logical analysis shows that Smullyan’s attempt to slip in a fudge factor is easily demonstrated for what it is. It can be seen from the above that Smullyan’s ‘norm’ is simply another attempt using vague natural language to create a valid self-reference, but which merely relies on the same tired old confusion of what constitutes syntax of the language and what constitutes objects of the language. It is evident that it is simply the nature of natural languages that syntax and objects are easily confused, and one must examine any claims of self-reference made in natural language with logical rigor, and not take them at face value.

 

By applying rigorous logical analysis, Smullyan’s claims are all seen to be elementary logical errors, where he allows the confusion of expressions that are syntax of language with expressions that are objects of language. In terms of levels of language, expressions of a sub-language , while they constitute valid syntax in that language, are seen as objects by a meta-language and which have no syntactical meaning in that meta-language. In the case of syntax of language, it is possible for two expressions to be equivalent, but in the case of objects that are simply sequences of symbols without any attached meaning, it is not possible for two different sequences of symbols to be precisely equal.

 

Of course, natural languages such as English permit ambiguities where levels of language are intermingled, and where objects and syntax can be confused. The problem with Smullyan’s claims are that he claims that his methods of self-reference are free of such ambiguities and are entirely logical when this is clearly not the case.

 

 

Footnotes:

 

 

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The Lighter Side

 

NEWS

Lebesgue Measure

There is now a new page on Lebesgue measure theory and how it is contradictory.

 

 

Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which demonstrates the illogical assumptions used by Halbach and Zhang.

 

 

Peter Smith’s ‘Proof’

It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.

 

 

Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

 

 

O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

 

 

New page on Chaitin’s Constant

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

 

Previous Blog Posts  

 

Links  

 

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

 

– a page with general links:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

 

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