[FOM] Shapiro on natural and formal languages
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sat Nov 27 23:01:45 EST 2004
I found myself largely endorsing Arnon Avron's comments on S.
Shapiro's book, "Foundations without Foundationalism." (Including his
opening point that-- ignoring Shapiro's philosophical tic about
"natural language" and the bee he has in his bonnet about
foundationalism-- the book is a beautifully clear exposition of the
basic ideas and metamathematical results concerning Second Order
Logic: it is definitely a book that any first-year graduate student
in logic-- whether in a philosophy department or a mathematics one--
should take home from the library over a Christmas or Easter break
and read!)
One slightly tangential remark. Avron says:
>4) One final note: there are plenty of geometrical ways of reasoning
>that can easily be visualized and understood, but are extremely
>difficult to be translated into any natural language. In my opinion
>this fact strongly supports my belief that reasoning comes before
>languages
This is something that has received a bit of attention from
logicians-- the late Jon Barwise was a leader here-- over the past 15
years or so. One of the ill effects that the fascination with
"natural" language has had on logical pedagogy in philosophy
departments is that, if you think of the formal language of First
Order Logic as a model* of natural language, you will tend to
emphasize "translation" between FOL and English (or whatever) in your
teaching and exercises. You'd be laughed out any teacher-training
college if you proposed teaching a second "natural" language that
way! Barwise and Etchemendy's textbook "The Language of First Order
Logic" (earlier versions and the included software are titled
"Tarski's World") came with software that provided another sort of
exercise: using FOL to describe pictures, and drawing pictures to
illustrate FOL sentences. If Arnon's (plausible) belief "that
reasoning comes before languages" is correct, it should be possible
to learn to use FOL (and other "formal" ways of representing things)
DIRECTLY as a vehicle for reasoning about real-life (well, computer
imaged...) situations rather than as something essentially derivative
from "natural" language: B & E allow us to perform the experiment.
Similarly for proofs. Their next textbook, "Hyperproof," has
a software package allowing the student to work with "hybrid
proofs"-- the simple ones are conventional natural deduction, but it
is also possible to have diagrams as "lines" of the proof, and to
infer a formula from a diagram ("read off from the picture"). This
is a primitive beginning, but I think at least a beginning for a
view of logic-- both pedagogical and theoretical-- that is less
"glottocentric" than the one expressed in some of Shapiro's dicta.
Second Order Logic, like FOL, is "artificial" in the sense that we
can trace its historical origins. Rather than thinking of it as a
model* of natural language, we can see it as one more of a family of
ways of representing things (on paper and in the imagination): a
family that multiplies over time, and no one of whose members
exercises patriarchal authority!
---
* "Model" is a slightly unfortunate word to use with the FoM
community, given its technical meaning in model theory. This is NOT
what is meant when formal languages are called models of natural
language. Think of logicians as like naval architects, but
interested in the seaworthiness of arguments: ONE use of formal
languages is to allow "towing tank tests" of arguments.
--
Allen Hazen
Philosophy Department
University of Melbourne
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