[FOM] Re: Shapiro on natural and formal languages

Timothy Y. Chow tchow at alum.mit.edu
Fri Nov 26 16:42:17 EST 2004

Arnon Avron <aa at tau.ac.il> wrote:
>  There is however one central thesis in this book which Shapiro repeats 
> several times in it, and he seems to take it for granted, as no 
> justification of this thesis is given. The thesis (as I understand it) 
> is that natural languages have priority over any formal language, and 
> real proofs and arguments are only those that are done in natural 
> languages.
> This thesis has surprised me. I have always taken for granted the 
> complete opposite: that both natural languages and formal languages are 
> used to model valid reasoning and arguments, and that formal languages 
> are designed precisely because natural languages fail to do it 
> adequately.

I don't know exactly what Shapiro's views are.  I am guessing, though, 
that he may be saying no more than that informal concepts of "proof," 
"integer," and so forth necessarily precede their formal counterparts. 
When we write down a formal language, it is not created in a vacuum but
is supposed to capture certain key aspects of pre-existing mathematical 

I can see someone not bothering to draw a sharp distinction between 
natural language itself and the kinds of reasoning that we carry out using 
natural language, especially when making a somewhat casual statement that 
is not going to be explicitly defended in detail.  So if Shapiro uses
the term "natural language" to refer to what I have called "informal 
concepts," then maybe his view is not as "completely opposite" to yours
as it would be if one took his use of the term "natural language" more 


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