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Re: [ontolog-forum] CL, CG, IKL and the relationship between symbols in

To: "John F. Sowa" <sowa@xxxxxxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Tue, 1 Jan 2008 12:24:03 -0800
Message-id: <p0623090bc3a03ac28516@[]>
>Pat and Chris,
>I sympathize with your negative views about various proposals
>for versions of context logic.
>PH> Two points. First, it is absolutely not clear what counts
>  > as a 'kind of context'. IMO the very idea of "context" is
>  > so ill-defined as to be meaningless as an theoretical tool
>  > of analysis.
>Yes.  That is why I use the word 'context' in a very precise
>way:  a context is a box (or other enclosure) that delimits
>some statement or conjunction of statements that make some
>assertion about the many different kinds of things that people
>have called contexts.    (01)

Well, that is a syntactic definition: you have defined "context" here 
to be a syntactic construction which is intended to denote or 
represent what are more commonly (and I think more correctly) called 
'contexts'. This rather begs the question about what contexts (in the 
usual sense, i.e. the things your constructions denote or represent) 
actually *are*.    (02)

>CM> ... can you give an example of the sort of thing you think
>  > CL/IKL can't represent but CG+box can?
>I'll give an example, and I'd  be very happy if you can show
>that IKL can represent it.    (03)

Be delighted. Read on.    (04)

>  Consider the following sentence:
>     Tom convinced Sam that it's impossible for
>     a cow to jump over the moon.
>To avoid getting into formal details, I'll put brackets around
>parts of that sentence to show where the context boxes would go:
>     [Tom convinced Sam that [it's impossible for
>          [a cow to jump over the moon] ] ].    (05)

You are here assuming then that the 'convinced' in this phrase is 
unanalysed, simple a three-place relation, which is how I will 
represent it also.    (06)

(Convinced Tom Sam (that (not (possible (that (exists ((x 
Cow))(JumpOver x Moon))) ))) )    (07)

>The outer box delimits the clause that says
>     "Tom convinced Sam that p"
>where p is a statement in a nested box that can be treated
>as an unanalyzed unit when examining the outer clause of the
>sentence.    (08)

Quite. IKL has the construct (that p) which denotes the proposition 
expressed by p, which seems to play a similar role. And which takes 
IKL well beyond the expressive power of FOL, by the way.    (09)

>  The verb 'convince' is an intensional verb that
>introduces some axioms (preconditions and postconditions)
>about how p is related to Tom's beliefs before and after the
>convincing.    (010)

Ditto, above.    (011)

>  And the beliefs would be in another context box,
>which would be updated in the next time step (more context
>boxes) as a result of the convincing.    (012)

Im not sure what you mean by 'the next time step' here, as we are 
talking about a logic rather than a  process specification. But 
certainly it should be possible to write axioms allowing one to infer 
things about Sam's beliefs after the convincing is done. My own 
preferred way to do this would be to introduce an explicit temporal 
parameter to the various time-sensitive relations, put conventionally 
as the last argument:    (013)

(Convinced Tom Sam (that (not (possible (that (exists ((x 
Cow))(JumpOver x Moon))) ))) T)    (014)

and then one might have for example    (015)

(forall (A (B agent) p t t')(if
        (and (Convinced A B p t)(later t t'))
        (Believes B p t')
))    (016)

 From which it would for example follow that Sam doesn't believe in 
bovine moon-jumpers any time after T.    (017)

>The statement p has the form "not possible q" where q is
>another statement "a cow jumps over the moon".  The box that
>contains p requires different axioms about alethic modalities.
>CM> A context logic would have to do the same [state axioms] for
>  > the primitive syntactic apparatus it uses to represent contexts.
>I agree, but I do not envision a single "context logic", but
>a mechanism that allows open-ended sets of axioms for various
>kinds of contexts.    (018)

Sounds like IKL, in fact :-)    (019)

>  Every intensional verb, such as 'convince',
>'seek', or 'fear' (and there are thousands of such verbs), would
>have a different collection of axioms that would specify how the
>statements in the nested box were interpreted.    (020)

Exactly. One would use multiple ontologies, rather than a special 
logic, to represent the complexities of the various kinds of context.    (021)

>CM> Well, they *are* all different [languages], aren't they?
>  > Each metalevel contains truth predicates and other appropriate
>  > semantic apparatus for all the preceding levels.
>Look at the English sentence above.  Each of the three bracketed
>clauses is treated differently, but nobody says that they are
>different languages.    (022)

You are both right. The language that John is talking about is the 
union of all the languages that Chris is talking about.    (023)

>CM> Seems like an opportunity to educate to me.
>I agree.  The main question is whose terminology is more misleading.
>My recommendation is to avoid the word "language" as too overloaded,
>especially when you start using logics to represent NL semantics.
>I prefer to talk about different metalevels with different axioms
>for interpreting the nested levels.    (024)

But that is just as confusing, and has the added disadvantage of 
being wrong. The 'that'-constructions in IKL are not "meta" at all. 
They are not *about* the sentences they contain. Tarski's theories 
were metatheories, but IKL is not an omega-limit of metatheories, it 
is an untyped logic of propositions.    (025)

>CM> If they do, then you disabuse them of their confusion.
>I realize that Tarski was not confused, but his unfortunate
>choice of terminology creates confusion.  As I said, I would
>replace his word 'language' with 'level' or 'metalevel'.
>CM> 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'.
>In that sense, I would agree.  But the simplest thing to do is
>to talk about levels instead of languages.    (026)

I see no pressing reason to replace a terminology that has been in 
widespread use for half a century. All that the layman has to get 
used to is the idea (surely not wholly new to any adult) that there 
are many languages.    (027)

Pat    (028)

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