The SubProS and ProChainS compatibility services for OWL ontologies to check for good and ‘safe’ OWL object property expression  may be considered ontological reasoning services by some, but according others, they are/ought to be plain logical reasoning services. I discussed this issue with Alessandro Artale back in 2007 when we came up with the RBox Compatibility service —which, in the end, we called an ontological reasoning service—and it came up again during EKAW’12 and the Ontologies and Conceptual Modelling Workshop (OCM) in Pretoria in November. Moreover, in all three settings, the conversation was generalized to the following questions:
- Is there a difference between a logical and an ontological reasoning service (be that ‘onto’-logical or ‘extra’-logical)? If so,
- Why, and what, then, is an ontological reasoning service?
- Are there any that can serve at least as prototypical example of an ontological reasoning service?
There’s still no conclusive answer on either of the questions. So, I present here some data and arguments I had and that I’ve heard so far, and I invite you to have your say on the matter. I will first introduce a few notions, terms, tools, and implicit assumptions informally, then list the three positions and their arguments I am aware of.
Some aspects about standard, non-standard, and ontological reasoning services
Let me first introduce a few ideas informally. Within Description Logics and the Semantic Web, a distinction is made between so-called ‘standard’ and ‘non-standard’ reasoning services. The standard reasoning services—which most of the DL-based reasoners support—are subsumption reasoning, satisfiability, consistency of the knowledge base, instance checking, and instance retrieval (see, e.g., [2,3] for explanations). Non-standard reasoning services include, e.g., glass-box reasoning and computing the least common subsumer, they are typically designed with the aim to facilitate ontology development, and tend to have their own plugin or extension to an existing reasoner. What these standard and non-standard reasoners have in common, is that they all focus on the (subset of first order predicate logic) logical theory only.
Take, on the other hand, OntoClean , which assigns meta-properties (such as rigidity and unity) to classes, and then, according to some rules involving those meta-properties, computes the class taxonomy. Those meta-properties are borrowed from Ontology in philosophy and the rules do not use the standard way of computing subsumption (where every instance of the subclass is also an instance of its super class and, thus, practically, the subclass has more or features or has the same features but with more constrained values/ranges). Moreover, OntoClean helps to distinguish between alternative logical formalisations of some piece of knowledge so as to choose the one that is better with respect to the reality we want to represent; e.g., why it is better to have the class Apple that has as quality a color green, versus the option of a class GreenObject that has shape apple-shaped. This being the case, OntoClean may be considered an ontological reasoning service. My SubProS and ProChainS  put constraints on OWL object property expressions so as to have safe and good hierarchies of object properties and property chains, based on the same notion of class subsumption, but then applied to role inclusion axioms: the OWL object sub-property (relationship, DL role) must be more constrained than its super-property and the two reasoning services check if that holds. But some of the flawed object property expressions do not cause a logical inconsistency (merely an undesirable deduction), so one might argue that the compatibility services are ontological.
The arguments so far
The descriptions in the previous paragraph contain implicit assumptions about the logical vs ontological reasoning, which I will spell out here. They are a synthesis from mine as well as other people’s voiced opinions about it (the other people being, among others and in alphabetical order, Alessandro Artale, Arina Britz, Giovanni Casini, Enrico Franconi, Aldo Gangemi, Chiara Ghidini, Tommie Meyer, Valentina Presutti, and Michael Uschold). It goes without saying they are my renderings of the arguments, and sometimes I state the things a little more bluntly to make the point.
1. If it is not entailed by the (standard, DL/other logic) reasoning service, then it is something ontological.
Logic is not about the study of the truth, but about the relationship of the truth of one statement and that of another. Effectively, it doesn’t matter what terms you have in the theory’s vocabulary—be this simply A, B, C, etc. or an attempt to represent Apple, Banana, Citrus, etc. conformant to what those entities are in reality—as it uses truth assignments and the usual rules of inference. If you want some reasoning that helps making a distinction between a good and a bad formalisation of what you aim to represent (where both theories are consistent), then that’s not the logician’s business but instead is relegated to the domain of whatever it is that ontologists get excited about. A counter-argument raised to that was that the early logicians were, in fact, concerned with finding a way to formalize reality in the best way; hence, not only syntax and semantics of the logic language, but also the semantics/meaning of the subject domain. A practical counter-example is that both Glimm et al  and Welty  managed to ‘hack’ OntoClean into OWL and use standard DL reasoners for it to obtain de desired inferences, so, presumably, then even OntoClean cannot be considered an ontological reasoning service after all?
2. Something ‘meta’ like OntoClean can/might be considered really ontological, but SubProS and ProChainS are ‘extra-logical’ and can be embedded like the extra-logical understanding of class subsumption, so they are logical reasoning services (for it is the analogue to class subsumption but then for role inclusion axioms).
This argument has to do with the notion of ‘standard way’ versus ‘alternative approach’ to compute something and the idea of having borrowed something from Ontology recently versus from mathematics and Aristotle somewhat longer ago. (note: the notion of subsumption in computing was still discussed in the 1980s, where the debate got settled in what is now considered the established understanding of class subsumption.) We simply can apply the underlying principles for class-subclass to one for relationships (/object properties/roles). DL/OWL reasoners and the standard view assume that the role box/object property expressions are correct and merely used to compute the class taxonomy only. But why should I assume the role box is fine, even when I know this is not always the case? And why do I have to put up with a classification of some class elsewhere in the taxonomy (or be inconsistent) when the real mistake is in the role box, not the class expression? Differently, some distinction seems to have been drawn between ‘meta’ (second order?), ‘extra’ to indicate the assumptions built into the algorithms/procedures, and ‘other, regular’ like satisfiability checking that we have for all logical theories. Another argument raised was that the ‘meta’ stuff has to do with second order logics, for which there are no good (read: sound and complete) reasoners.
3. Essentially, everything is logical, and services like OntoClean, SubProS, ProChainS can be represented formally with some clearly, precisely, formally, defined inferencing rules, so then there is no ontological reasoning, but there are only logical reasoning services.
This argument made me think of the “logic is everywhere” mug I still have (a goodie from the ICCL 2005 summer school in Dresden). More seriously, though, this argument raises some old philosophical debates whether everything can indeed be formalized, and provided any logic is fine and computation doesn’t matter. Further, it conflates the distinction, if any, between plain logical entailment, the notion of undesirable deductions (e.g., that a CarChassis is-a Perdurant [some kind of a process]), and the modeling choices and preferences (recall the apple with a colour vs. green object that has an apple-shape). But maybe that conflation is fine and there is no real distinction (if so: why?).
In my paper  and in the two presentations of it, I had stressed that SubProS and ProChainS were ontological reasoning services, because before that, I had tried but failed to convince logicians of the Type-I position that there’s something useful to those compatibility services and that they ought to be computed (currently, they are mostly not computed by the standard reasoners). Type-II adherents were plentiful at EKAW’12 and some at the OCM workshop. I encountered the most vocal Type-III adherent (mathematician) at the OCM workshop. Then there were the indecisive ones and people who switched and/or became indecisive. At the moment of writing this, I still lean toward Type-II, but I’m open to better arguments.
 Keet, C.M., Artale, A.: Representing and reasoning over a taxonomy of part-whole relations. Applied Ontology, 2008, 3(1-2), 91–110.
 F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, and P. F. Patel-Schneider (Eds). The Description Logics Handbook. Cambridge University Press, 2009.
 Pascal Hitzler, Markus Kroetzsch, Sebastian Rudolph. Foundations of Semantic Web Technologies. Chapman & Hall/CRC, 2009,
 Guarino, N. and Welty, C. An Overview of OntoClean. In S. Staab, R. Studer (eds.), Handbook on Ontologies, Springer Verlag 2009, pp. 201-220.
 Keet, C.M. Detecting and Revising Flaws in OWL Object Property Expressions. Proc. of EKAW’12. Springer LNAI vol 7603, pp2 52-266.
 Birte Glimm, Sebastian Rudolph, and Johanna Volker. Integrated metamodeling and diagnosis in OWL 2. In Peter F. Patel-Schneider, Yue Pan, Pascal Hitzler, Peter Mika, Lei Zhang, Jeff Z. Pan, Ian Horrocks, and Birte Glimm, editors, Proceedings of the 9th International Semantic Web Conference, volume 6496 of LNCS, pages 257-272. Springer, November 2010.
 Chris Welty. OntOWLclean: cleaning OWL ontologies with OWL. In B. Bennet and C. Fellbaum, editors, Proceedings of Formal Ontologies in Information Systems (FOIS’06), pages 347-359. IOS Press, 2006.