Ali and all,
Sorry I missed today, folks, just too busy these days.
I understand the notion of rule to be about the consequence relation, e.g., modus ponens. A, A -> B |- B (where |- is syntactic consequence; this is usually called entailment). This is at the meta-logic level. These are often called “inference rules”.
An axiom (typically) uses the base logical connectives, and in this case refers to (material) implication (conditional), e.g., (A /\ (A->B)) -> B. This is at the object (logic) level.
This is how I explain it to most of my students and others. Generally rule-based systems or inference engines will use generalized modus ponens, resolution, etc.
A deductive system can consist of axioms and inference rules, or axioms only, or inference rules only.
Most folks bandy these terms around, and don’t make a clear distinction between consequence and (material) implication. And you will get a lot of gruff grief from people.
Modus Ponens:
X ® Y
X
------
Y
That inference rule is actually equivalent to: ((X ® Y) Ù X) ® Y
X Y X® Y ((X® Y) Ù X) ((X ® Y) Ù X) ® Y
T T T T T
T F F F T
F T T F T
F F T F T
I have found the following to provide good descriptions (aside from the usual books introducing logic or its meta-theory):
Hunter, Geoffrey. 1973. MetaLogic: An Introduction to the Metatheory of Standard First Order Logic. University of California Press.
Ryan, Mark; Martin Sadler. 1992. Valuation Systems and Consequence Relations. In: Abramsky, S.; Gabbay, Dov; Maibaum, T.S.E. Handbook of Logic in Computer Science, Volume 1, Background: Mathematical Structures. Oxford: Clarendon Press, pp. 1-78.
Concerning the difference between ontologies and theories: a group of people will say that an ontology is just a logical theory, i.e., a theory expressed in a logic. Others, philosophically realist, will say that an ontology is a logical theory about some aspect of the real world. I prefer the latter.
Other points:
If you look at SWRL, it is really a kind of Horn Logic/Rule representation, i.e., a generalized Modus Ponens form used by logic programming. RIF contains many notions of “rules” not just this kind of “logical inference rule”, but production rules (forward-chaining rules) from expert systems, etc.
Thanks,
Leo
From: oor-forum-bounces@xxxxxxxxxxxxxxxx [mailto:oor-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Ali SH
Sent: Tuesday, October 18, 2011 2:50 PM
To: Michael Gruninger; OpenOntologyRepository-discussion
Subject: [oor-forum] Ontologies vs Theories / Axioms vs Rules
Hello Michael and all,
I noticed some confusion today during the telecon when Michael and Todd were talking about differences between Ontologies & Theories, and Rules & Axioms. I've actually encountered a number of people who make this distinction -- that an ontology doesn't really express "rules", and that these rules are somehow different from "axioms". The line is still blurry to me, but it seems to be that in this terminology, a "rule" refers to anything that OWL doesn't support, whereas in the CL (and FOL world) there is no distinction between these "rules" and "axioms".
We adhere to a rather restrictive view on what an ontology should contain: terminological knowledge, i.e. intensional definitions of concepts, represented as classes with which we interpret (model) the world (page 7)
where things that would have other types of "rules" are part of "frameworks" and not included as part of the ontology.
In these cases, it would still be possible to map from LKIF to a CL ontology, but the reverse would either lose information or require the selection LKIF and possibly one or more "frameworks" (perhaps stored as OWL + RIF or SWRL combos in OOR). That said, I'm not sure what the status of a RIF or SWRL module w/o an accompanying ontology would be in the context of OOR, and how one would represent these combinations using OMV.
Further, in their discussion of the deontics surrounding law, they state:
"Although the situation description may be expressed without problems as a framework cast in OWL-DL, but OWL-DL does not allow this deontic qualification to be expressed as a metaclass, or rather that the deontic qualification allows reified statements. There are three solutions to this problem. ... The second one is to express norms as rules, which has an intuitive appeal, as in common sense terms legal norms are synonymous with legal rules: one has to obey the rules. However, this does not immediately solve the tractability problem either as description logics and rule formalisms are not necessarily compatible (see also [Boer et al., 2007])." (page 9)
I'm not certainly how prevalent these distinction between ontology, theory, axiom and rules are, but I've come across it in (OWL heavy) literature, and it is unclear to me whether such distinctions are adequately covered by the OMV metadata. At the very least, if web services will be built around the OOR platform, then this is an issue that needs to be addressed.
Might someone help elucidate this distinction?
[1] Joost Breuker, Rinke Hoekstra, Alexander Boer, Kasper van den Berg, Rossella Rubino, Giovanni Sartor, Monica Palmirani, Adam Wyner, and Trevor Bench-Capon. OWL ontology of basic legal concepts (LKIF-Core). Deliverable 1.4, Estrella, 2007.
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