Jerry, Stephen, and Helmut,
In his later philosophy, Wittgenstein defined a natural language
as the totality of all the language games that can be played with
a given syntax and vocabulary.
He did not state that point in those terms because he died several
years before Chomsky made an outrageous and hopelessly misguided
claim: A natural language is the totality of all the grammatical
sentences that can be expressed with a given syntax and vocabulary.
If Wittgenstein had heard that claim, he would have been livid with
rage. I believe that the linguist Michael Halliday, whose career
spans the same extent as Chomsky's, had a much more accurate view:
JFS: every artificial language, which includes all the artificial
notations of mathematics, logic, chemistry, computer programming...
JLRC: I find this phrase to be very confusing, John. In today’s
terminology, Symbol systems are not the same as “artificial notations”,
but most formal notations are artificial symbols created by humans to
express human thought or intent or meaning.
Every symbol system or formal notation begins as a language game
that the developers or designers use to discuss the subject matter
among themselves. When designing that notation, they discuss every
definition in some NL, and they use exactly the same definitions
for the corresponding words in their preferred NL.
Secondly, a critical distinction is whether or not the terms originate
within a discipline and flow into the spoken language with time, or
incorporated into a different technical language or otherwise.
A PARTICULARLY INTERESTING CASE IS “DNA”.
DNA is an excellent example. The language game *originated* with the
first use of the term 'desoxyribonucleic acid' and its abbreviation.
The scientists who study DNA and talk with their colleagues about it
express every word, symbol, and phrase in their preferred NL with
exactly the same precision as they do when they use the symbols and
notations of chemistry.
Very few authors choose to use common spoken language formally.
There is no such thing as "common spoken language". Every sentence
anybody says from infancy to death is in some language game, which is
as vague or precise as appropriate for their purpose at the moment.
It's true that people who don't understand the science may pick up
and repeat parts of the scientists' precise language game and use it
in very loose analogies. I believe that's what you mean by "flow
into the spoken language with time". But the scientists themselves
still talk about DNA with exactly the same precision as ever.
Units must be defined! The meaning of the “+” sign / symbol varies
with the purpose of author and the logical notation (sybol system)
the author is communicating with.
Yes. When precision is required for some language game, the speakers
express exactly the same precision in their NL and in other notations.
And the symbol '+' varies with different language games for different
kinds of numbers. See Figure 2 of "What is the source of fuzziness?":
Logic is in my view good... Words are a sort of utility by which we
can perform everything from mundane to exalted feats. But to give them
more than their due is an error.
When a logician, mathematician, or scientist in any field uses special
symbols in any formal notation, those symbols have *exactly* the same
meaning as the NL words that they use in talking with their colleagues
or students when they're explaining those symbols.
Logic is definitely prior to words through words are the instruments
for expressing it.
No! Every version of logic or any other artificial notation is
nothing more nor less than some NL language game expressed in
a notation that is specially designed just for that purpose.
graphs, as most mathematic symbol language too, does not symbolize
time (continuity)? But: Might it not be possible to do that, by
inventing symbols for time and its flow?
Scientists use the symbol 't' and predicates spelled T-I-M-E in
mathematics. They also use equivalent words when they talk about
the same subjects in their preferred NLs.
But the discrete words and symbols of any language, natural or
artificial, can't express the full continuity of their experience.
A photograph or movie is better. And systems of virtual reality
are even better. But nothing expresses the full continuity.
What is the "natural language"? Chomsky´s "universal grammar"?
Is it the same as logic?
According to my interpretation of Peirce and Wittgenstein, I would
say that all our language experience, from infancy on up, involves
learning language games for expressing our perceptions and actions.
But language is not the same as logic. It's more accurate to say
that every version of logic, from Aristotle to the present, is
a special-purpose notation for some language game.
The primary advantage of using some logic notation is that many errors
can be detected just by checking the syntax. But syntactic checks
alone cannot prevent all errors in semantics and pragmatics.
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