Logic and Language
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Smullyan’s Paradoxes

Raymond Smullyan has written several popular books that delve into various permutations of informal self-referencing, many of them very similar to those expounded by Willard Quine, see Quine’s Paradox, and the Liar paradox, and which use the inherent ambiguity of natural languages such as English to construct statements that appear paradoxical. He has also written a book which begins in the same populist vein as his other books, and then delves into a more formal treatment of the material. Here we will deal only with the informal parts of that book.

 

Note: There is another page dealing with a paradox attributed to Smullyan, see The Drinker Paradox.

 

Smullyan’s Diagonalization and Self-Reference

The book in question is Smullyan’s book Diagonalization and Self-Reference. (Footnote: Raymond Smullyan, Diagonalization and Self-Reference, Oxford University Press, 1994 Diagonalization and Self-Reference: Details. ) 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 variable when enclosed within quotation marks, otherwise it could not be what Smullyan calls “true”. ), 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’s claim is that the diagonalization of expression (3) is expression (4), that is, that:

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

is, and hence is equivalent to:

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

 

But if that is the case, then where the expression:

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

occurs in the expression:

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

by the previously established equivalence we have that the expression (4) is equivalent to:

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

which is equivalent to:

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

and so on. These are not grammatically valid expressions, which means that either the supposition of equivalence of:

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

and:

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

is incorrect, or that diagonalization is not clearly defined.

 

One might try to get around this difficulty by supposing that what Smullyan actually meant was that

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

is

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

 

But if that were the case, that latter expression is not syntax of the language but is merely an object, and is not a sentence, hence it cannot be equivalent to the expression (4).

 

Either way, Smullyan’s claims are nonsensical. The trick that Smullyan tries to pull here is to allow an expression to be part of the syntax (and not an object) when it suits his purpose on one occasion, and not allowing it to be part of the syntax (and simply an object) on another occasion if it would not produce the result that he wants. 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. Smullyan’s fudge is that he wants to allow an expression to be part of the syntax and not part of the syntax 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 = 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

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” = “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 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 different 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.

 

Other paradoxes

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Footnotes:

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