How’s this for a solution to Grelling’s paradox?
(The paradox concerns the distinction between expressions
which are true of themselves, and expressions which are not
true of themselves. There can be no expression which is true
of all and only the latter kind of expressions. Say there was
such an expression, is this expression true of itself? From
each answer the opposite follows. This is paradoxical
because it apparently contradicts the natural assumption that
for any plurality of things, it is possible to define an
expression as being true of all, and only, those things.)
Consider the following expression:
expression which yields a true sentence
when substituted for "X" in "`X´ is not an X"
Substituting this expression for "X" in "`X´ is not an X"
yields the expression:
`expression which yields a true sentence when substituted
for "X" in "`X´ is not an X"´ is not an expression which yields
a true sentence when substituted for "X" in "`X´ is not an X"
At first sight this second expression seems to be a sentence
which asserts its own falsehood. However the contradiction
may only be derived if a given pair of quotation marks is able
to quote an expression in which they appear. If they may not,
the first expression may not be quoted using the quotation
marks "`" and "´", and so the second expression is
ungrammatical.
As far as I know, the question of whether quotation marks
may quote themselves was first subjected to logical analysis
in 1995, by the late George Boolos (`Quotational
Ambiguity´, reprinted in his Logic, Logic & Logic, 1998).
This paper did not involve an explicit discussion of self-
reference, but rather was written in reaction to a problem
pointed out to Boolos by a student of his, Michael Ernst.
Consider the following ungrammatical expression:
a´ appended to `b
This may not be quoted using the quotation marks it contains.
Rather the expression:
`a´ appended to `b´
clearly just refers to the result of appending `a´ to `b´, i.e. the
expression `ba´.
Boolos concluded that an expression containing a given type
of quotation marks, should always be quoted using a different
type of marks. Given this assumption, it is not necessary to
distinguish between opening and closing marks. For this reason
Boolos suggested a notation involving an infinite series of
expressions such as:
* ´* ´´* ´´´* …
which he called `q-marks´. An expression is quoted by placing
it between a pair of q-marks of the shortest type which does not
appear in that expression. This allows a single language to form
quotation names of all its own expressions.
Returning to Grelling’s paradox, the expression:
expression which yields a true sentence
when substituted for *X* in ´**X* is not an X´*
is thus true of all expressions which may be quoted using the
first q-mark, and which are not true of themselves. It may not
itself be quoted using the first q-mark, and so is not true of
itself.
Similarly the following expression:
expression which yields a true sentence
when substituted for *X* in ´´´*´´*X´´* is not an X´´´*
is true of all expressions which may be quoted using the third
q-mark, and which are not true of themselves. It is true of the
previous expression, but not itself. And so forth.
While Boolos’s single infinite series of quotation marks
allows a language to name all its expressions, introducing
further quotation marks allows new definitions to be stated,
and so this process may be iterated transfinitely. For
example, consider the following infinite series of infinite
series of marks:
* ´* ´´* ´´´* …
^* ´^* ´´^* ´´´^* …
^^* ´^^* ´´^^* ´´´^^* …
…
Using these it is possible to construct an expression which is
true of all expressions which may be quoted using the first
infinite series of q-marks, and which are not true of
themselves:
expression which yields a true sentence
when substituted for *X* in ´^*^*X^* is not an X´^*
This leads to a notation involving an infinite series of infinite
series of infinite series of q-marks, and so forth. And so forth!
I have been working on a formal analysis of this hierarchy of
languages, and a related approach to the paradoxes of set
theory. Anybody interested?
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