A reflexive use of a language A is the use of the language A to talk
about itself, for example to analyze its own structure.
Instead, one can use another language B to talk about A. Then B is a
metalanguage for A.
For example, if you define a mathematical formalism, say BNF, to
express the syntax of languages, then BNF can be used (to some extent)
to define the syntax of English, analyze it, talk about it, and do a
variety of things. BNF is then a metalanguage for English, or more
precisely for the syntax of English.
But you could also use English to describe the syntax of English, as
many grammarians did in the past. Then it is a reflexive use of
English to describe its own syntax.
An interesting note, not considered in the text you copied, is that a
meta language like BNF can be used to define its own syntax. Then, it
is a reflexive use of the metalanguage.
What the part in red says is that, even when we do mathematics, we
conduct the discussions in English, or whatever language we happen to
speak (some people believe there is only one). This is only partly
true as computer scientists and mathematicians are developping systems
that can conduct mathematical work in a purely formal way. But we tend
not to do that when exchanging between human beings, and rather use
natural language.
Applied to formal metalanguage, this means that even though we can have
a specialized language B to express precisely some aspects of English,
it is only used as a scaffolding to actually express our thinking more
precisely in English, as we would do (with less precision) if we
simply used English reflexively. So even the use of metalanguage ends
up in reflexive use of the language, at least where natural language
is concerned.
This said, you can talk perfectly well about English in German or other natural languages. In
this case, German becomes a metalanguage for English. And native
German speakers do not have to resort to English to say whatever they
have to say, thus contradicting in a way the statement in red.
Nevertheless, this remark does not really weaken the argument of the author, since we use metalanguages for mathematical precision. German as a metalanguage is probably not much more precise than English used reflexively. The
author is really using the word "English" to mean natural language (his
responsibility). By the way, this was a reflexive use of English, as I was discussing the meaning of an English word in a sentence, and
the current sentence is a reflexive sentence talking about itself.
A problem is that metalanguages need also to be understood and
precisely defined. So we could build a further metalanguage for the
purpose, ... But this would not end. So at some point we cannot escape
using reflection.
This is precisely what is done when defining the syntax of BNF in BNF.
(BNF is just a specific notation for context-free syntax).
Further remarks
I cannot explain this text much more, as he does not seem to me that
the author justifies all his assertions in the available text
fragment. And furthermore, I do not always agree with them, short of
more developed arguments.
To begin with the OP's question, I would definitely consider that a
reflexive statement is a meta-linguitics statement about the
language (here English) expressed in the language itself. The language
is then its own metalanguage. If a language B has and expressive power
that goes beyond the expressive power of a language A, while fully
covering it, then one may expect that B would permit more general
statements, or simply more statements about A than the reflexive use
of A for itself. But that is somewhat a trivial remark.
The point of view of the author seems however to be that reflexive
statements in English can cover more ground than the use of a formal
language, since, for lack of a constraining formalization, it will be
adaptable to a wider range of concepts and situation. This may well be
true, at the expense of precision, univocality of expressions, and
possibly accuracy of reasonning and avoidance of paradoxes.
Hence, it may indeed be undesirable to identify metalinguistic
statements and reflexive statement as they do not serve the same
purposes. Except maybe when the metalanguage is itself another natural
language, hence having pretty much the same properties as English used
reflexively. Still, it seems that the paragraph in red states that
this identification necessarily occurs when using a formal metalanguage.
I disagree with the last sentence of the first paragraph (which should
have a question mark added). My view is that there is substance
(semantics) and names (syntax) used to physically denote that
substance. Whatever we do is syntactic. If I write "Socrates has
eight letters", I am certainly using the word Socrates, but I am
using it as substance, not as a name for another substance. And one
never accesses substance directly, but only through a name, if only to
always proceed consistently. But, the
author says it is unreasonable. He may have reasons to think that, but
I do not see that he is giving them. Furthermore, we remarked that
using English reflexively, or German as a metalanguage for English, is
about the same. Using German as metalanguage, would he oppose
considering "'sesquipedalian'" as a name for an English word. I
do not know. But that is what it is in formal metalanguages
(programming languages for example). Would that say something about
a possible difference between formal and natural languages, rather than about
the concept of reflexivity.
Regarding the sentence in red. I do agree with it on a first level.
Usual discourse on English, even when based on a formal metalanguage,
will make use of English itself (let's forget other languages), and
will hence fall back on reflexive use of English. But, as I said, the
formal metalanguage is still there as a scaffolding, possibly not very
visible. And this scaffolding does structure discourse so that it will
not wander beyond the limits set by the formal metalanguage. Hence
there is not much that is changed by reflexive use of English,
whathever "its richness, complexity and alleged inconsistencies". At
best, you can get analogies and support for personnal intuition, but
without really changing the end result. English is used reflexively in
a formal sense, but it has little consequence ... unless the
discourse starts bearing on the adequacy of the metalanguage. But this
is yet another game, as English is then used as a metalanguage for the
formal metalanguage.
I would rather not comment the rest of the text, which encompasses too
many things, And I do not understand the reason for preferring to talk
"in terms of use and mention" rather than substance and name as I
did. I made up this terminology to avoid the more formal syntax and
semantics. My opinion is that the philosophers'view (according to the
author) is a lot more limitative because it is absolute rather than
relative. Recall that higher up I wrote «"'sesquipedalian'"» so has to
have a name for the notation of the name of a concept. This allows
recursive use of the concept of naming, while use and mention do not
allow it, if I am not mistaken in my understanding of their intended
meaning.
If you reached this line without skipping, you won a coffee next time
we meet.
PS I thought "sesquipedalian" qualified a bicycle with a broken
pedal. I was apparently wrong.
