In discussions about logic and ontology, people have raised many
questions about uncertainty, probability, fuzzy logic, modal logic,
and other kinds of reasoning methods that are not "supported" by
the usual first-order logic or by the more general Common Logic.
My response has been that such extensions can be handled by
metalevel reasoning about formulas in first-order logic. (01)
Tarski made the point that Truth and Falsity are metalevel notions
*about* FOL, not notions that are expressed in the same language.
But Tarski added that you can have a hierarchy of metalevels, each
of which is classical FOL, but each of which formalizes the notion
of truth or falsity for the level beneath it. (02)
Proof theory is also a metalevel theory, and you state FOL axioms
at the metalevel that formalize the proof theory for the level
beneath it. You can even write FOL axioms for fuzzy logic that
formalize the operations on the fuzzy uncertainty values of the
formulas at the bottom level. Logic programming languages, such
as Prolog, are routinely used to implement such methods. (03)
Whenever I make such comments, some people object by saying
that such two-level methods are inefficient. And I agree that
is sometimes true. But the metalevel methods don't have to
use exactly the same inference engine that is used at the
object level. In fact, you can "compile" the metalevel theory
into a more streamlined program that performs the equivalent
operations in a more efficient manner. But for the theoretical
analysis, you can still think of it as two levels of FOL. (04)
As an example, there are some interesting papers and software
available for *Markov Logic Networks* (MLNs). In a very short
summary, an MLN is a graph of first-order formulas, each with
an associated numerical weight. Such weights could be used
to do fuzzy kinds of reasoning, for example. (05)
It's a simple idea that brings together many useful techniques.
With Common Logic, for example, you can think of an MLN as
a metalevel conceptual graph, in which each node contains one
or more nested CGs (or statements in any other CL dialect). (06)
A group at the U. of Washington has been developing the theory
and writing some open-source software to support it: (07)
http://alchemy.cs.washington.edu/ (08)
They call it a unifying framework, and it does unify many ideas.
For a brief overview, see (09)
http://www.cs.washington.edu/homes/pedrod/papers/aaai06c.pdf (010)
See the end of this note for the abstract of this overview.
Following is a more detailed 44-page paper: (011)
http://www.cs.washington.edu/homes/pedrod/kbmn.pdf (012)
This is just one example among many of the way that FOL at the
metalevel can be used to define many novel kinds of reasoning
about FOL statements used at the bottom level. (013)
John
_____________________________________________________________ (014)
Unifying Logical and Statistical AI (015)
Pedro Domingos, Stanley Kok, Hoifung Poon, Matthew Richardson,
Parag Singla (016)
Intelligent agents must be able to handle the complexity and
uncertainty of the real world. Logical AI has focused mainly
on the former, and statistical AI on the latter. Markov logic
combines the two by attaching weights to First-order formulas
and viewing them as templates for features of Markov networks.
Inference algorithms for Markov logic draw on ideas from
satisfiability, Markov chain, Monte Carlo, and knowledge-based
model construction. Learning algorithms are based on the
voted perceptron, pseudo-likelihood and inductive logic
programming. Markov logic has been successfully applied to
problems in entity resolution, link prediction, information
extraction and others and is the basis of the open-source
Alchemy system. (017)
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