On Model Theory
So, I've started a part-time Ph.D program at the EECS department of
Case Western University. I'm hoping to write a thesis on higher
education information science as patient advocacy. I've taken a
course on Machine Learning, database systems, and now on model theory.
Luckily Case has a class on model theory in the philosophy
department. But it also counts as credit from the graduate
mathematics department.
The class on model theory is about Kurt Godel's (I know I'm
misspelling it) theory on the incompleteness theory of peano's
arithmetic ( an axiomatization of number theory ). The book for the
class, It is supposed to be written by mathematical geniuses, our
professor proclaims. The book is ".."
Models are states of absolute (binary) logical certainty: things that
are or things that aren't. Statements in this language are 'sound'
(they follow from our understanding of logic). The consist of
domains: sets of 'things' denoted by terms in a written language (a
first-order language). The model also consists of 'interpretations'
of phrases (or formulas) in this language, some of which are 'closed'
(i.e., all variables refer to constants interpreted through this
model) or open. The axiomatic nature of basic science presumes
logical certainty in its canon. The language consists of constants
that can be interpreted 'against' the domain of a model, sets of
constants composed form constants interpreted against the domain of
the model, functions that map sets of members of the domain to members
of the domain, 'relations' over the domains, and a determination of
'equality' over members of the domain. Model theory is the basis of
first-order theory, logic-based knowledge representation, numeric
theory, etc..
Godel's theory says that any axiomatic system that is as expressive as
some computable representation of number theory is not complete (i.e.,
there is at least one question you can formulate in the language for
which you cannot say with certainty that it is so or it isn't so).
Models 'entail' sets of formulas when they entail all the members of
the set. Models satisfy sentences and 'terms' in languages. Some
formulas can be said to be valid in every model (i.e., they are
satisfied by every model of a language) or they specifically are
entailed by a particular model. Axiomatic machinations (finite state
automata, etc..) are systems about languages and a set of axioms that
can be used to 'derive' expressions in the language via a finite set
of valid theories, specifically given sentences in a theory, or
sentences that follow from modus ponens (common sense if / then
conditionals) and sentences in a theory (a scientific theory).
As our prof puts it, all of logic programming (and database theory) is
spawned by this known limitation to model theory and first-order
logic. They are restricted forms of it that are complete (unlike
number theoretic languages), and sound and thus 'decidable' by a
finite state automaton or turing machine.
Systems that describe their 'formal semantics' - the meaning of
statements made in their languages often describe them using model
theory (all of semantic web theory does this: RDF, RDFS, OWL-DL,
etc..). More on this later
