I am giving it a try, but I fear that, whatever the
definitions chosen, there will be unsatisfied people. Let's discuss it.
A formal grammar may be defined as a set of string rewriting
rules that are used to specify a set of strings, in a mathematically
precise way. In some cases they may associate structural information
with the string.
There are also tree grammars, used to define sets of trees, and graph
grammars used to define sets of graph, both by mean of rewriting
rules. There may be others.
There may actually be many way to read a formal grammar:
- Typically, a
context-free grammar may be read as a way to generate strings by
rewriting them according to some rules (I am simplifying a bit).
All the strings that can be generated form the language defined by
the grammar. This is the generative reading of a formal grammar.
- It may also be read as a system of equation over sets of strings,
the language being the smallest solution to the equation for one of
the variables. You may call it the denotational reading of the grammar.
- It may also be read as a tree grammar.
- And, it can even be read as the specification of an algorithm to check
whether a string is in the defined language.
The techniques for attaching other mathematical meaning to the strings
(or trees, or graphs) may be considered as outside the mathematical
domain of formal grammars.
But basically, formal grammars are mathematical entities.
Generative grammars is more a linguistic concept. In some looser sense
(I guess) it is a set of rules that specify how linguistic elements
can be combined to form linguistically acceptable construction, such
as sentences (syntax) or simple words (morphology).
As formal grammars usually have a generative reading, they are often
used by linguists to specify formally generative grammars, often in
conjonction with other mathematical tools attached to them.
In a larger sense, generative grammar can also refer to a description
of mechanisms by which the structure of syntactically correct
linguistic constructions will emerge.
Generative grammars are concerned with syntax only, with sentences
that will respect structural rules of the language. According to such
rules, and to Noam Chomsky, "Colorless green ideas sleep furiously" is
syntactically an English sentence. Whether it means anything is for
Salvador Dali to say.
Further thoughts on the issue.
As I said, I do not expect to satisfy everyone.
Not being very knowledgeable in either semantics or pragmatics, I
gladly take in @jlawler's remark that "lots of generative grammars
deal with semantics and pragmatics [and] only the ones who accept
Chomsky's assumptions insist of segmenting them."
This point of view can be vindicated in the following way.
Ignoring the metaphysical issues on human thought and language (and I
include here all forms of linguistic expression in the word language), the
human brain and language system (including all senses) may be seen a
physical device that receives input data, manipulates it in
various ways, including memorization, and produces other data
through various means (speech, writing, gesture, ...). In a nutshell, hoping
not to hurt anyone's feelings, this is pretty much the description
of a general purpose computing device. Note that I used the word
"data" rather than "symbol".
Can the working of such a machine be theorized formally with a formal
grammar, viewed as a generative formal system? For example, could
Chomsky's type 0 grammars be such a generative grammar formalizing
the human language machine?
The first point to note is that the human language machine is a physical
computational device. Hence we may wonder whether physical
computational devices can be fully described by a generative formal grammar. This
raises two open questions, one in mathematics that may not be
answerable, and one in physics.
The first question is the Church-Turing thesis. It is a conjecture
that any kind of algorithmic computability is equivalent to Turing
computability. Turing computability is itself proved to be equivalent
to "computation" by type 0 grammars. Note again the multiple readings
of a grammar.
This conjecture is generally considered not to be provable. I suspect
it relies on an understanding of computability that assumes
denumerability of all things computational. Computational devices are
considered to be symbol manipulation devices, operating along an open
time-line (there may be other assumptions).
Whether that covers all possible physical devices remains an open
problem in physics. There is some research work on these issues. Some
is attempting to understand what kind of constraints in the physical
world would imply the denumerability and time open-linearity
hypothesis. Other research is trying to imagine more powerful
computational models that could be allowed if some fundamental hypotheses were changed.
For the time being, short of answering these questions, we have to
assume the correctness of the Church-Turing thesis for all
computational devices, including the human language machine. Hence,
human language, together with the knowledge, memorization and
interaction that goes with it, must necessarily have a description as
a type 0 grammar, which can be read as a generative description.
Thus, considering that the whole language process may be described by a
generative grammar is in this sense an appropriate point of view.
However, my vindication of it is not fully satisfactory for at least
two somewhat related reasons:
this analysis only states that the language abilities of each human
being can be modeled by a generative formal grammar. But it says
nothing about a general model that could be used for all human
beings, with maybe only part of the rules to account for
differences between people, the other rules being a common core
that could be taken as representing the language ability of
type 0 grammars can be very unstructured, and hard to
understand. They generally do not have perspicuity, unless very
carefully constructed, organized, and commented. A type 0 grammar,
however generative, is not necessarily expressing a linguistic
theory, an organized and structured body of knowledge "explaining"
the inner working of language so that linguistics phenomena of all
kinds can be accounted for and even predicted.
Even if some magic event gave us a generative grammar for English
(assuming the existence of a precise definition of English), we might
not be able to extract from it any useful knowledge about the
This is a bit intended to debunk the magic of words, and of formal
systems, however important they may also be. It is not intended to
disparage the work that would extend the idea of generative grammars
to semantics or pragmatics. I often felt the distinction
between syntax and semantics to be a somewhat arbitrary convenience,
and informally took syntax to be what has tractable mathematical
formalization ... which is not too precise either. The distinction is
a bit more precise in logic with model theory.