[FOM] Shapiro on natural and formal languages

mjmurphy 4mjmu at rogers.com
Sun Nov 28 07:30:54 EST 2004

Mr. Avron wrote, concerning natural and formal languages, and quoting

"As a mathematical model, there is always a gap between the language of
a logic and its natural language counterpart. The fit between model and
modeled can be good or bad, useful or misleading, for whatever purpose
at hand"

And then:

"Many of the considerations of this book concern the match, or lack of
match, between natural languages and the formal languages of various



I suspect that Shapiro won't get a lot of sympathy for this kind of position
on a
list like FOM.  Generally speaking, people that do sympathize (like me)
tend to think that logic (or at least semantics) should be, broadly,
empirical in nature and natural language should provide the empirical
facts that constrain our logical theories.

The issue of the relative merits of formal vs. natural languages is
extremely complex.  In some obvious sense of "clarity", for example, natural
language is obviously clearer than any formal counterpart.  For example,
next time you are given a parking ticket, try to argue your way out of it
using the FOL
translation of the natural language sentences you would normally employ
to the same end.  It is difficult enough to formulate a single sentence
with a couple of def. descs. In it, let alone conduct a debate.

It is also useful to point out that the vast majority of reasoning/arguing,
scientific reasoning, is conducted in natural language (even allowing
the examples you give later).  It would be almost certainly false to argue
that it would all go along a little faster or better if it were done in some
formal language.

Avron goes on:

1) Let me start with the question: what does Shapiro exactly mean
by saying that formal languages model ordinary natural languages?
What natural language? There is English (the example given by Shapiro),
Hebrew (my own natural language) and several hundreds other "natural
languages". Does Shapiro mean their union? intersection?"


This argument may be fine in the abstract, but in practice it is not
relevant.  The points where divergences between natural and
formal languages occur are not particularly controversial.  For example, if
I were to
theorize that:

1) If dinosaurs wear wigs, the moon is green cheese.

...it would be bizarre to say that my theory has been proven correct if
there are no dinosaurs.

Other non-controversial examples are the kinds of cases where first year
logic students
are known to balk.  "All the authors in the room are English." where there
are no authors in the room, and so on.

In general, I would say the philosophical consensus since these kinds of
divergences were pointed out by Strawson et al, is that they are real
and that standard FOL has to be "tweaked" to make them go away.  The
development of a supplementary notation and machinery of Pragmatics has,
been one way of doing this.  Roughly, on such accounts, sentences like 1)
remain true, but are uttered
inappropriately when there are no authors in the room.

Avron goes on:

2) After discussing the doubtful meaning of a "natural language" in
Shapiro's thesis, it is time to note that whatever counts
as a "natural language", no present text of mathematics is written
in such a "natural language". Shapiro is aware of this, of course, and
he corrected his thesis by referring ("perhaps") to `natural languages
of mathematics', i.e. `ordinary languages augmented with expressions
used in mathematics'.


This criticism, though,  I have more sympathy for.   Where our natural
language intuitions are weak or non-existant, naturally an appeal to them
cannot carry much weight.   And, for my own part, I have no idea what our
natural language intuitions concerning mathematics would be.



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