[FOM] Shapiro on natural and formal languages

[A.P. Hazen]

   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