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Re: [ontolog-forum] Context [was: Truth]

To: Pat Hayes <phayes@xxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Thu, 26 Jul 2012 15:13:16 -0400
Message-id: <501196CC.4040805@xxxxxxxxxxx>
Pat,    (01)

Semiotics establishes a framework of patterns for analyzing the
mechanisms underlying perception, representation, reasoning,
and communication.  The basic principles were laid down Plato,
Aristotle, and their buddies -- especially Hippocrates, Galen,
and other physicians who analyzed those signs called symptoms.    (02)

The medieval logicians developed semiotics in great detail.
One of the most detailed was Ockham's Theory of Terms, which
analyzed metalevel relationships in great detail.  His Theory
of Propositions included a model-theoretic analysis of Latin.
He pursued it about as far as possible without having a modern
symbolic notation.    (03)

>>   1. A*mark*  X is anything observable that has not been interpreted.    (04)

> Which is absolutely anything, right?    (05)

No.  It does not include mathematical entities such as numbers or sets,
and it does not include logical entities such as the propositions
expressed by sentences.    (06)

> but if Peirce had phrased this honestly everyone would have seen
> immediately how vacuous and content-free it was...    (07)

Any detailed theory must start with some simple primitives.  That first
step, by itself, doesn't get you very far.  But with a starting point,
a few basic patterns, and some combining methods, you can build a
very useful framework for analysis and reasoning.    (08)

Please look at Ockham for an example of the level of sophistication
that semiotics had attained 600 years ago.  (See Refs at end.)    (09)

Peirce's primary contribution was to adopt and extend the framework
developed by the medieval logicians with modern logic.  For example:    (010)

  1. CSP adopted the medieval terms 'first intentions' for language about
     external things and 'second intentions' for metalanguage (language
     about language).    (011)

  2. He coined the term 'first-intentional logic' for ordinary FOL,
     and 'second intentional logic' for statements that let the
     quantifiers range over relations.    (012)

  3. Ernst Schroeder translated those terms to German as 'erste ordnung'
     and 'zweite ordnung', which Russell translated back into English
     as 'first order' and 'second order'    (013)

  4. CSP also developed a model theory for existential graphs by
     adapting Ockham's semantics to his EGs.  Risto Hilpinen showed
     that Peirce's method was a variant of Hintikka's game theoretical
     semantics.  For further info with references, see    (014)

     http://www.jfsowa.com/pubs/egtut.pdf    (015)

> A Tarskian model (of a set of sentences) is an interpretation of the
> set of sentences which makes the sentences true.    (016)

Yes.  We all agree on that.  But note that in his original paper,
Tarski did *not* use the word 'model'.  Why did other people chose
that word?  They must have seen some similarity to the way the word
'model' was used.  Petri did, and he said so.    (017)

> For models, I suggested the following definition.  I was inspired
> by Petri, who suggested a commonality among the various meanings of
> the word 'model'.  I stated the commonality in a Peircean triad:
>  1. An *object* X is any physical thing, event, situation...
>  2. A *model* Y of X is a physical or mathematical construction that
>     resembles X according to some specification Z.
>  3. A *specification* Z is a statement of the way X resembles Y.
>     Z may be single word, such as "shape", or it may be a detailed
>     list of statements in some language, linear or graphic.
> An engineering model has all three parts.  A Tarski-style model has
> parts #2 and #3,    (018)

> I still do not understand how this is supposed to relate to the idea
> of a mathematical structure "resembling" something, or what the role
> of the specification is supposed to be.    (019)

For an engineering model, the model can be physical (traditional),
on paper (eg. blueprints), or mathematical (as in computer models).
In all cases, there is a mapping from the model to some aspect of
the thing that is being modeled.    (020)

For a Tarski-style model, the specification is the set of axioms,
and the model is a mathematical structure that makes the axioms true.
Tarski did not talk about the real world, as the engineers do, but
some people can and do relate the mathematical structure to some
aspect of the world.    (021)

>> Given this preamble, I would use the notion of model defined
>> above to state a general definition of a context for a text:
>>  1. A *text* X is a statement in some language, natural or artificial.
>>  2. A *context* Y for X is a model (X', Y', Z') plus a mapping of
>>     of every referring expression in X to something in Y'.
>>  3. An *interpretation* of X in terms of Y is the truth value of
>>     X as determined by the model Y and the mapping in point #2.    (022)

> How does this even remotely resemble the ideas that context theorists mention?    (023)

As you have said, they have a bewilderingly large number of views.
That is why you dismissed all of them.  What I have been trying to do
is to apply a Peircean framework that includes all of them as special
cases.  I believe that all the aspects that anyone has included in
their theories of contexts can be included.  But some adjustments
of terminology are necessary.    (024)

> Take a time interval as being an archetypical context, for example.
> In what sense can a time interval be seen as a model of a set of sentences?    (025)

I don't know anyone who claimed that a time interval is a context.
More likely, they said that the context under discussion had a duration
specified by a certain time interval.  Their context probably also had
some physical coordinates as well.    (026)

To relate that to my definition of model above, their context was
some physical object X.  But the complete model also has components
Y and Z, which they may or may not have mentioned.    (027)

> More generally, surely the whole point of invoking a "context" in the
> first place is that it is supposed to be something that influences
> the meaning of a sentence but is NOT described directly by the sentence,
> or even mentioned in the sentence's syntax. And given this, how can it
> be seen as 'resembling' the sentence?    (028)

The sentence is the text X in my proposed definition.  The context is
a model with three components (X', Y', and Z').  None of those three
components is the text X, nor are the described by the text X.    (029)

But note that point 2 of the definition says "A *context* Y for X is
a model (X', Y', Z') plus a mapping of of every referring expression
in X to something in Y'."    (030)

That definition specifies a large amount of information (sometimes
called "background knowledge").  The sentence X definitely does not
describe the entire model (X',Y',Z').  But note that the context
includes "a mapping of of every referring expression in X to something
in Y'."  That mapping is critical for determining the meaning of the
sentence.    (031)

> An interpretation is a truth value? That seems just plain wrong.    (032)

I admit that I first wrote that definition this morning and that I
should revise it.  Instead of writing    (033)

> An *interpretation* of X in terms of Y is the truth value of
> X as determined by the model Y and the mapping in point #2.    (034)

Revised version:
An *interpretation* of the text X in terms of the model Y is the
mapping of all referring expressions in X to elements of the domain
of the model Y and the truth value that is determined by that mapping.    (035)

I admit that there have been so many conflicting theories of context
that it would undoubtedly be impossible to find a consistent
generalization of all of them.  I also admit that some further
revisions would probably be needed to work out the details for
a significant number of the most important theories of contexts.    (036)

But I believe that the exercise in this note shows that this
general approach -- if not this particular definition -- has
some promise of relating a wide range of theories of context.    (037)

In any case, I am about to leave on a short trip, and I'll be
out of touch of email until next week.    (038)

______________________________    (039)

References:    (040)

Ockham, William of (1323) Summa Logicae, Johannes Higman, Paris, 1488; 
the edition owned by C. S. Peirce. Also volume 1 of Opera Philosophica, 
ed. by P. Boehner, G. Gál, & S. Brown, Franciscan Institute, St. 
Bonaventure, NY, 1974.    (041)

Ockham, William of (T) Ockham's Theory of Terms translation of Part I of 
Ockham (1323) by M. J. Loux, University of Notre Dame Press, Notre Dame, 
IN, 1974.    (042)

Ockham, William of (P) Ockham's Theory of Propositions translation of 
Part II of Ockham (1323) by A. J. Freddoso & H. Schuurman, University of 
Notre Dame Press, Notre Dame, IN, 1980.    (043)

Ockham, William of (W) Philosophical Writings, ed. and translated by 
Philotheus Boehner, revised by S. F. Brown, Hackett Publishing Co., 
Indianapolis. Includes Latin and English selections from Ockham (1323) 
and other works.    (044)

See also many web sites about Ockham on the WWW.    (045)

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