Logic, Computability and Grammar

Given a natural language, we can ask what the logical and computational prerequisites for generating and checking and analyzing valid expressions of that language ? And can we define the minimal, optimal way in which expressions are generated and checked and analyzed ? We are looking here at meta-grammar,  the formal language of grammar itself.  How rich are formal languages in themselves compared to standard algebraic systems ! How much remains to be said about the relationship between (specially imperative) programing languages and logic, between logic and recursion theory, between recursion theory and combinatorics and formal languages. These are all topics of our Analyticity and the A Priori.

We view a descriptive grammar as a function which takes a series of choices, semantic categories, and generates the corresponding expression. We should formalize grammar using functional programming. 

Take the rules of Sandhi in Sanskrit. To express them we need the same logical framework as in Aristotle's Topics, in particular as Galen's Relational Syllogism. 

But we can also think of an abstract mathematical framework for Sandhi.  Suppose we have for instance finite sets $A,B,...,G$ and let $P \subset A \times B \times...\times G$.  Then we can define an operation $\otimes: P \times P \rightarrow P \times P$ such that for instance a certain component is right-dominant:

if for $p \in P$ we have $p_C = c_1$ then $q\otimes p = (q', p)$ with $q'_C = c_1$ and $q'_X = q_X$ for $X \neq C$.

For binary sets $A = \{a,b\}$ we can define transposition $p^A$ as $(p^A)_A = a$ if $p_A = b$ and $(p^A)_A = b$ if $p_A = a$.  Then the above example becomes $q \otimes p = (q,p)$ if $q_C = p_C$ and $q \otimes p = (q^C, p)$ otherwise.

In order to represent Sanskrit sounds more precisely we can include in each finite set a 'null' element $0$ expressing that the classification of this set does not apply. For instance $p_N \neq 0$ means $p$ is a nasal and then $p_A = 0$ means for instance that the aspiration  Boolean does not apply.

The algebraic aspect of Sandhi can be illustrated by the existence of left and right (quasi-) absorbing elements or identity elements. $\dot{n}$ is a left identity element. 

While external Sandhi can be given relatively simple well-defined rules, even Max Müller shies away from giving a complete list of rules and exceptions for internal Sandhi.

We will also give  representation of external Sandhi using permutation groups (inspired by the Rubik cube).