How do you know when you have a complete formal theory?

Alternative question, "Are there standard theory formats for different types of theories?"

I think you need to be a little more specific - there are different kinds of theory, for example, theories in mathematics are different from theories in logic (the latter are more formally and restrictively defined) - in any case, i would expect a formal theory to include definitions of a collection of sets of entities and rules for combining them to produce new entities, and some non-trivial provable results about those definitions - without the results, you might want to call it a theory, but it is not very interesting, and i'm not sure what an informal theory could or should be

Christopher:

Your are correct in suggesting that this is an open ended question.

My interest is focused on the semantics of the term "formal theory."

Which, as you point out, implies "informal theory."

Theory is a rational type of generalized thinking or the formalized outcome of a rational thinking process.

One definition of a formal theory is "a theory expressed in a formal language."

Therefore, one might expect the an informal theory is a theory expressed in an informal language.

Natural language is considered informal, so any theory expressed in terms of a natrual informal language may be (?) considered an informal theory. That is one question.

Another question relates to theroy, truth and types of truth. Steve Wallis addresses the concept of theory validation in "Validation of Theory: Exploring and Reframing Popper's Worlds." The worlds are: World One (Facts and data), World Two (Meaning and dmotions), and World Three (Theory).

These three worlds are closely aligned with A.D. Halls types of truth: formal truth, factual truth and value truth.

The use of the term theory appears to be different in each of the three worlds.

Take care, be good to yourself and have fun,

Joe

An incomplete formal theory may be associated with a theory statement in a formal language where the constraints of the formal language are not completed.

There may be other ways of addressing this topic, but I wanted to start the conversation.

Christopher:

You said:

"formal theory to include definitions of a collection of sets of entities and rules for combining them to produce new entities, and some non-trivial provable results about those definitions - without the results, you might want to call it a theory, but it is not very interesting,..."

The "non-trival provable results about the definitions" is a very important part. Provable in the sense of formal language, like logic or mathematics. What about relationship to the real world? It seems like a theory should also be verifiable by experience.

Take care and have fun,

Joe

i think that the fundamental issue is the notion of models, the formal and informal, computational and evocative models we seem to make of the "real world" more or less continuously - then we use these models to reason about what will or might happen next

we have been making these models for a very long time - in fact, there is some evidence that many other mammals make such models, of varying sophistication

the question is related to Eugene Wigner's "Unreasonable effectiveness of Mathematics in the Physical Sciences" - i have heard it argued that the nature of reality has shaped our own notions of consequences so that simple consequences derived from observations generally match the processes that naturally occur very well - in other words, our experience of the physical world has shaped what we consider to be valid reasoning - as a mathematician, i don't like that idea very much (that our notions of formal logic are derived from experience of the real world), but i have no clear counter-arguments, even though our notions of logical consequence can be formally defined without reference to anything in the real world

the same question also shows up in the "model map" that has puzzled us at least since bertrand russell, which has formal models and formal reasoning on one side, and some physical phenomenon and whatever it does on the other

in that sense validation is about whether or not you have a good model of the phenomenon you are interested in, not about whether or not the logical reasoning itself is sound

anyway, i'm rambling, but i think the question is important, since i have seen too many occasions where the model was not understood to be a construct whose behavior does not depend on reality (that is an old problem "the map is not the territory" Plato)

Christopher:

Models are of central importance.

Understanding the scope, type and application of different model types is also central to effective integration of different types of knowledge and truth.

Take care, be good to yourself and have fun,

Joe