Chris, (01)
I agree, but there is an issue about policy and terminology. (02)
CM> What makes a logic a context logic is that a notion of context
> is taken as a logical *primitive*. What this means syntactically
> is that appropriate constructs (designed intuitively to express
> a notion of context) will be required elements of the language
> of the logic. (An example is McCarthy's "ist" operator in his
> context logic.) (03)
In natural languages, different kinds of contexts tend to have
very different axiomatizations, and people frequently make
statements that relate different contexts in different clauses
of the same sentence. Any kind of context logic that is going
to be used for analyzing and reasoning about NLs has to support
something like that. (04)
Common Logic does not support such things, and even IKL is too
homogeneous to support them. With the box construct in CGs
(which is not in the ISO standard dialect of CGIF), it is
possible to represent such things. (05)
CM> On the semantic side, the meanings of those constructs will
> be *fixed* in every interpretation of the language. (06)
That depends on what you mean by "the language". Tarski considered
every metalevel in his hierarchy to be a different "language".
I have no quarrel with that terminology for logic, but it creates
confusion for people who think of natural languages. People who
know English, for example, would consider it weird to say that
they are speaking a different language whenever they learn a new
word or talk about a different subject. (07)
When you add the box construct to CGs (in an enhanced CGIF notation,
for example), it can provide a mechanism for defining metalevels,
as in Tarski's hierarchy. At each level N, it is possible to state
axioms that define truth for level N-1 and to define the rules of
inference that preserve truth at that level. (08)
You can call each level a different language, but I prefer to
avoid that terminology by saying that there is a single 'language'
with different 'levels'. Instead of talking about different
kinds of truth for each level, I prefer to treat "truth"
as an indexical, which may be evaluated in different ways
in different contexts at different levels. (09)
I have no quarrel with people who want a more homogeneous
system, but I believe that NLs require different axioms for
defining truth and rules of inference in different contexts. (010)
This is, of course, still a research project, and I would
not propose it for a standard -- but I think that a mechanism
that allows such things to be defined is actually simpler and
more general than IKL. (011)
John (012)
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