On Mon, 31 Dec 2007, John F. Sowa wrote:
> I agree, but there is an issue about policy and terminology.
>
> 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.)
>
> 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.
>
> Common Logic does not support such things, and even IKL is too
> homogeneous to support them. (01)
Well, bearing in mind Pat's concerns about what a weasel word "context"
is, let's just suppose that a context is (at least) some sort of entity
relative to which sentences or propositions can be true or false. If by
"support" you mean a built apparatus, then, as we've all already
acknowledged, CL and IKL don't support contexts. But you seem to be
suggesting something stronger when you say: (02)
> With the box construct in CGs (which is not in the ISO standard
> dialect of CGIF), it is possible to represent such things. (03)
which suggests that CL and IKL are not even capable of supporting a the
notion of context you have in mind via axiomatic theories, but that CG
with your "box construct" is. If so, first, can you give an example of
the sort of thing you think CL/IKL can't represent but CG+box
can? Second, if CL/IKL can't represent this sort of example but CG+box
can, then the box construct must go beyond first-order logic, is that
right? If it doesn't, then it is surely possible to introduce a
semantically equivalent construct into CL or IKL. (04)
> CM> On the semantic side, the meanings of those constructs will
> > be *fixed* in every interpretation of the language.
>
> That depends on what you mean by "the language". (05)
I don't see how. I am just stating a very standard conception of what
it means for something to be a *logic* for a certain notion, viz., a
language+semantics where the semantics fixes the interpretations of
certain "reserved" syntactic constructs of the language. Propositional
logic is a logic of the boolean operators, and fixes the interpretation
of corresponding syntactic operators in terms of truth functions.
First-order logic extends propositional logic by adding quantifiers and
fixing their interpretations by any of several (equivalent) means.
Modal logic extends FOL by adding by adding modal operators and fixing
their interpretation in terms of quantification and possible worlds (or
similar notions like those of Hintikka and Dunn). A context logic would
have to do the same for the primitive syntactic apparatus it uses to
represent contexts. (06)
> Tarski considered every metalevel in his hierarchy to be a different
> "language". (07)
Well, they *are* all different, aren't they? Each metalevel contains
truth predicates and other appropriate semantic apparatus for all the
preceding levels. (08)
> I have no quarrel with that terminology for logic, but it creates
> confusion for people who think of natural languages. (09)
Well, so what? The word "mass" in physics causes confusion for people
who think of, say, the Catholic church or a physical pathology. Seems
like an opportunity to educate to me. :-) (010)
> 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. (011)
Again, so what? If they do, then you disabuse them of their confusion. (012)
> 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.
>
> 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'. (013)
Well, seems to me you get exactly the same thing at the limit of the
Tarskian hierarchy. You get a single language that encompasses all the
finite 'levels'. (014)
> 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. (015)
Sure thing, fine. I just don't see the relevance to my point about what
it means for a system to be an X logic for a given notion X. (016)
> 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.
>
> 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. (017)
I'll be interested in seeing it what it looks like when it's ready to
come out of the oven. :-) (018)
-chris (019)
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