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[Fwd: Time, Causality and Demand-Pull]

I'd like to pick out three ideas that arise from Chris Partridge's message from June 5:

1. Nonstandard interpretations and relative consistency

Chris Partridge: 

I think your parallel postulate point only holds for standard
interpretations. I seem to remember a seminar where it was shown that if, in
the geometry of a surface of a sphere, you interpret the lines of latitude
as parallel (a non-standard interpretation) then you have a model of the
axioms of Euclidiean geometry.
This is not an example of nonstandard interpretations; rather, it is what is known as relative
consistency. To show that a theory is consistent, we simply need to find a structure that satisfies all
axioms of the theory. In the case of geometry, both Hilbert and Tarski used real ordered fields as their
structures; in this approach, points are interpreted as numbers, lines are interpreted as linear
equations, etc. This is NOT a nonstandard interpretation of numbers; rather, the numbers and
linear equations satisfy the axioms of the theories. In the case of noneuclidean geometries, it turns
out that one can construct models of noneuclidean theories (i.e. theories inconsistent with the
parallel postulate) using models of Euclidean geometry (which is the example you mentioned).
This approach is called relative consistency, since the existence of the class of structures in turn
depends on the consistency of another theory (e.g. real ordered fields).
2. How do we validate these interpretations with respect to our intuitions?
 
  
It seems to me odd that you cannot give at least one example of an
interpretation of these timepoints. It seems to me an eminently practical
matter to know how to interpret a primitive like this.


There seem to be some conflicting views on the nature and role of interpretations.
When  asked for an interpretation of timepoints in PSL-Core, as logicians/mathematicians
we point to the structure of a linear ordering with endpoints.  Such a structure satisfies all of
the (temporal) axioms of PSL-Core, and further, any structure that satisfies the temporal
axioms of PSL-Core must be isomorphic to such a structure. In fact, this is where people typically
refer to nonstandard interpretations. An interpretation is nonstandard if it satisfies the axioms
but is not in the class of structures that we initially specified; in the case of PSL-Core, such a
nonstandard interpretation would be a structure that satisfied the axioms but which is not
a linear ordering with endpoints.

Of course, we are not simply dealing with axiomatic theories, since we want these theories to
capture our intuitions about some domain. Thus we must also give a physical interpretation
of our axioms and structures, just as physicists must give a physical interpretation of the
differential equations they use in their theories. I suspect this is the motivation behind your
comments -- what is the physical interpretation of timepoints in models of PSL-Core?
However, it is perhaps better to consider the physical interpretation of sentences that can
be expressed using timepoints, rather than the interpretation of timepoints themselves.

Physical theories are meant to be predictive -- based on the solutions to the equations, and
the physical interpretation of these equations and their solutions, we can make predictions about
the ways in which the world behaves. If the theory makes the wrong predictions, then either the
equations or their physical interpretations are wrong.
I think that we need to take a similar approach with ontologies.

One disadvantage that we have compared with physicists is that they get to work with
a single universe in which to carry out their experiments. On the other hand, for ontological
engineers who are developing ontologies to support interoperability, every software application
is its own universe. Each application has its own terminology, with its own `natural' interpretation.
We cannot mandate that everyone must use the same ontology; rather, we must
assist the vendors in identifying their ontological commitments, specify their semantics as classes
of structures, and finally to axiomatize these structures. Since we are supporting interoperability,
we need to axiomatize any concept that is shared by at least two vendor applications.

What does it mean for a theory to be predictive in this case? This depends on the type of application.
For example, consider scheduling; if the ontology of a scheduling application is equivalent to some
part of a  process ontology, then any theory that uses that part of the process ontology should
correspond to a consistent schedule produced by the application. If there are scheduling problems
which can be solved by the scheduling application, but which cannnot be represented correctly by
the process ontology, then the process ontology must be revised.

3. Should time be a primitive in a process ontology?

Chris Partridge: 

The issue is Liebnitz's - about whether it is practically better
to 'construct' timepoints and time-intervals out of more 'fundamental'
relations.


By saying that there are more "fundamental" relations, I still think that you are changing a
design decision into a design requirement. This is the reason that I brought up the Hilbert/Tarski example.
Our current discussion is no different from a mailing list where the Tarski camp is claiming that
lines are not primitives, and that it is practically better to `construct' lines out of more `fundamental'
relations. Such a discussion is not as productive as explicitly comparing the axiomatizations and
models of the theories.
 

- michael