=Paper=
{{Paper
|id=Vol-3155/short3
|storemode=property
|title=Inferences: Integrals and Derivatives (short paper)
|pdfUrl=https://ceur-ws.org/Vol-3155/short3.pdf
|volume=Vol-3155
|authors=Graham Gal
|dblpUrl=https://dblp.org/rec/conf/vmbo/Gal22a
}}
==Inferences: Integrals and Derivatives (short paper)==
https://ceur-ws.org/Vol-3155/short3.pdf
Inferences: Integrals and Derivatives
Graham Gal
Isenberg School of Management, University of Massachusetts, Amherst, MA 01003 USA
Abstract
Orbst’s [1] paper on ontological interoperability presents a continuum from descriptions with very
week semantics to those with strong semantics. Those ontologies represented in terms of axioms
in first order logic have the strongest semantics and therefore the highest level of interoperability.
These representations also allow for inferences about the system described by the axioms. The
mathematicians of the early 20th century were focused on the possibility of determining which
inferences were possible from the axioms. The problem for otologists working in describing
services and business models is reversed. We know what inferences must be possible within the
model, and it is therefore incumbent on the developers of a particular ontology to show whether
these inferences are possible. For example, it is critical for a set of axioms that purport to describe
the REA ontology to make inferences about not only accounting artifacts, but also other auditing
conclusions. This paper attempts to describe the types of inferences that are required of any
ontology.
Keywords
Inferences, Derivatives, Integrals, Attribute Bundles, Naming versus Defining
1. Introduction
The development of a set of axioms which describe a particular domain establishes the essential ontological
categories and relationship within the domain. Conclusions or inferences from these axioms is also a critical
aspect of understanding and validating the particular domain ontology. The results of these inferences (or
formulae) are obtained from initial formulae through a series of symbolic manipulations [2]. Within
mathematics, Hilbert [3]1 gave the study of mathematical proofs and symbolic logic (the symbolic
manipulations) the name metamathematics. Hilbert argued that the formulae used to represent the axioms
and theorems of a particular system also describe the chain of deductions in the system, i.e. the conclusions
obtainable within the system. This can also be viewed in reverse as any theorem which can be deduced
from the formulae is also an essential component of the system’s axioms. Thus, a theorem that results from
the application of rules of inference is equivalent to the axioms contained in the original system. The goal
of metamathematics was to provide an understanding of whole classes of theorems which could be proved
[2]. This goal was to determine a priori the potential for a particular theorem to be derivable from the given
axioms. Another implication of the equivalence of the axioms to the derivable theorems or formulae is that
if a demonstrability false formula can be obtained, then it can also be concluded that the axioms are incorrect
or inconsistent. For example, if the following axioms are presented:
Proceedings of the 16th International Workshop on Value Modelling and Business Ontologies (VMBO 2022), held in conjunction with the 34th
International Conference on Advanced Information Systems Engineering (CAiSE 2022), June 06–10, 2022, Leuven, Belgium
Email: gfgal@isenberg.umass.edu
ORCID: 0000-0001-6526-9367
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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1
Hilbert original work [4]
Mammals give live birth
Humans are mammals
Men are Humans
Women are Humans
Then it can be concluded that Men can give live birth. This is a conclusion that is not true in our world,
so therefore these axioms, and their ontological validity are in question. This would be the same as if the
axioms of Principia Mathematica[5] could be used to show that the sum of two even numbers would result
in an odd number.
Another issue that can arise when trying to infer theorems within a particular system is that some
propositions may be undecidable. This conundrum is the result of Gӧdel’s [6]2 proof that concluded that
within certain systems described by what deemed to be fundamental axioms, such as those described in
Principia Mathematica [5], there are theorems that would exist that were undecidable, i.e. it would not be
possible to prove that they were either true or false. In business ontologies the problem faced by
mathematicians, trying to determine whether certain theorems are contained, derivable, is actually reversed.
In the business domain certain formulae, using axioms which are purported to describe the fundamentals
of a business ontology, must be derivable using standard rules of inference. For example, accounting
statements are fundamental descriptions of a state of a business. Therefore, if it is suggested that a set of
axioms present a model of a business enterprise, then through rules of inference accounting statements must
be obtainable. The demonstration that the axioms presented by Gal [7] can be used to derive the accounting
statements is critical to accepting the ontology described by the axioms. That the paper goes beyond the
development of accounting statements to include inferences related to auditing conclusions is a further
indication of the completeness of the representations contained in the axioms. Certainly, Gӧdel’s proof
suggests that there are propositions which will not be derivable, and this possibility cannot be ignored. The
purpose of this paper is to explore some issues related to inferences that go beyond the demonstration of
accounting and auditing propositions and to suggest some concerns. The following sections will explore
some of the potential issues.
The next section will look types of inferences related to integrals, i.e. different ways to integrate a set
of events to make conclusions about domain. The third section examines propositions that use derivatives
to make inferences about the possibility of future events. Section four looks at some of the implications for
using these different approaches to inferences about the nature of organizations. These sections will use
views of events corresponding to a soccer match from the perspective to three individuals as an example to
demonstrate certain issues. Person A considers the events from the perspective of a fan. Person B views
the events from the perspective of the Head Referee. Finally, Person C is a player.
2. Inferences Related to Integrating Events
Bubenko’s [8] temporal data cube is a conceptualization of an approach to information modeling. While,
its value in modeling data for an organization is established, it is also possible to use this model to
conceptualize the set of all possible events. Figure 1shows this conceptualization as a Universal Relation
(one with all attributes in the intension) and that extends from time 0 (t0). At each point in time a new layer
is added to the front of the cube with values for the attributes that describe this event at the new time (tα).
A corporate database includes the subset of event instances considered relevant for management and the
attributes management considers most appropriate to describe these events.3
2
Gӧdel’s original work published in 1931 and is translated in this reference.
3
Denna et al. [9] argues that the decision about the events and attributes to be included is based on managements’ need to plan, control, and evaluate
the organization.
t0
tn
tp
tα
Attributes {a1,a2,a3, … , an }
Instances
Figure 1: Universal Relation (From Gal [6])
From the larger event data cube, the three individuals can select events and the attributes which they
consider most salient to their view of the soccer match. However, this idiosyncratic bundling of events
implies a difficulty is defining any particular bundle as the “soccer match”. For instance, when asked about
the match the fan, Person A, responds that it was important to leave their house early in the morning so his
mates could get to the stadium before it got crowded. However, when asked about the match the Head
Referee speaks about the meeting the day before when all the officials and the League’s commissioner
discuss the previous match between the two teams when fisticuffs broke out after the final whistle. Finally,
the player, Person C, indicates that when he met with his agent two days ago they agreed that this match is
important as they go into contract negotiations. Each of these individuals sees the match as “beginning” at
a different point in time and including different attributes (and their values). Each person may also have a
different “ending” point in time. This means that the bundle of events corresponding to their match will be
different. Grupp [10] discusses this issue and the problem of bundling when attempting to construct a
theory of objects. These different bundles of the match indicate different definitions of what are the salient
events to consider when making inferences about that match. The distinction between definitions results
in assigning or naming different bundles as the match is critical if the goal is to develop an ontology of a
soccer match. This is an example of the problem with naming versus defining objects [11]. Conceptually
this implies that each person will use different integrals when making inferences concerning the events (and
attributes) in their individual bundles. So, when the fan concludes that it was a good game, they are making
an inference based on the integration of the events in their bundle. The referee’s inference that the game
was good was based on the lack of any fisticuff events in their bundle. Finally, the player’s inference that
is was a bad game is based on his event bundle lacking personal scoring events. Each of these inferences
can also be influenced by each observer’s comparison to historical comparison (or inferences) to events
which comprised past “matches” [12].
There are well-established inferences that integrate events to form views of a corporate data cube. As
mentioned in the introduction, Gal [6] derives certain conclusions using axioms that are integrals
(aggregations) of the corporate data cube. That these conclusions are defined in the accounting literature
makes it easier to validate their correctness [13]. Another set of inferences that are necessary for an
ontological representation of an organization includes those about future events. These are the set of
inferences based on derivatives of functions which use different events.
3. Inferences Related to Derivatives
When someone faces a question about an expectation of some future event(s) they must create a function
based on the events (and attributes) that have occurred, which they view as relevant for forecasting the new
event. For example, Barbour [14, p. 10] says that astronomers observe events related to changes the shape
of the universe and deduce (infer) that it is expanding. In a similar fashion the attendees at the soccer match
use observations of the events relevant to their forecasts and use them to make inferences about future
events. For example, the fan, observer A, makes the following forecast, “I was sure that X was going to
score.” In contrast the referee, observer B, forecasts that, “X was going to shoot wide.” While, the player,
observer C, forecasts “the goalie was going to make the save.” So, each observer has the potential to see
the same events, and yet they all make different forecasts. As with the observers’ conclusions about the
integrals of the events, the data cube must include the events and attributes of these events which allow
observers to make their idiosyncratic forecasts. To be clear at some point in the future there will be an
event where the player either scores or doesn’t. Then this event can be used by the various observers in
their integrals or aggregations. As the fan makes the inference that the player who did not score is worthless,
and the teammate concludes that he should not have passed him the ball. Any ontology which seeks to
model soccer matches, business organizations, airline travel, services, etc. must support the forecasts
(inferences) of the observers. The next section looks at some issues that need to be considered.
4. Some Concluding Issues
The different inferences made by the three observers raise a few questions. Perhaps the most relevant
question is whether it could be concluded that they were all at the same game. Is the only criterion that
matters their similar spatiotemporal location attributes? Is, the lack of any agreed upon view of the match
irrelevant to the conclusion (inference) that they all attended the same match. Clearly, they did not occupy
the exact same spatiotemporal coordinates, so any observer must also make an inference that they were at
the same match. Again, we are faced with the issue of naming versus defining. The question may be
phrased as whether the three observers’ spatiotemporal coordinates were similar enough to allow an
observer to make such a conclusion. Must all observers come to the same conclusion or are they free to
name their observations, “attended or did not attend the match?”
There is also an issue with the events related to forecasts of future events. If a pen were held 1meter
above the floor and then let go, it would be a rare observer indeed to forecast that the pen would not hit the
floor. So, why did each observer make a different forecast about the scoring play? What is the difference
between scoring a goal and dropping the pen? Is it simply that there are more events needed to forecast
goal scoring as opposed dropping the pen? As the player gets closer to the goal is the inference of the two
events of similar certainty? In either case the problem for the otologist is the inclusion of sufficient detail
to allow different forecasts to be done with the same certainty.
Finally, a comment about what events are not. Events only have values for time and their other
attributes. There are no events that are late, early, fast, slow, etc. These are all inferences about an event.
Guarino [15] discusses the issue of identifiers for future events. If they are to be the result of a forecast,
then they are put in the derivative of a function. They are when they are. The future events will be
integrated into the data cube as they occur. A future event cannot be late. Someone may wish the event
had happened at an earlier time. The wishing that the event had already taken place is itself an event, but
it has no bearing on the existence of an event that has yet to occur.
Developers of ontologies must be aware of the inferences that are central to the domain of interest.
Ontologies that seek to describe organizations, must allow for inferences of the types discussed in this
paper. If, for example, an ontology of soccer matches, does not allow for a user to infer what the final score
was, then it has a fundamental flaw. However, it must also allow for conclusions about the quality of a
match; as idiosyncratic as they may be. To evaluate the completeness of veracity of a particular ontology
based solely on the axioms describing without also considering the inferences is to avoid a central
requirement of an ontology. This paper has suggested two classes or sets of inferences, those that deal with
integrals of events and those that deal with the forecasting of changes to objects represented by the domain.
5. References
[1] L. Obrst, “Ontologies for Semantically Interoperable Systems,” in Conference on Information and
Knowledge Management, New Orleans, LA, USA, 2003, pp. 366–369.
[2] R. B. Braithwaite, “Introduction,” in On Formally Undecidable Propositions of Principia
Mathematica and Related Systems, New York, NY, USA: Dover Publications, 1992, pp. 1–32.
[3] D. Hilbert, “The Foundations of Geometry,” 1950.
[4] D. Hilbert, Grundlagen der Geometrie. 1899.
[5] A. N. Whitehead and B. Russell, Principia Mathematica, 2nd ed. Cambridge, UK: Cambridge
University Press, 1910.
[6] K. Gӧdel, On Formally Undecidable Propositions on Principia Mathematica and Related Systems.
New York, NY: Dover Publications, 1992.
[7] G. Gal, “The Metaphysics of Internal Controls.” 2022.
[8] J. A. Bubenko, The Temporal Dimension in Information Modelling. Amsterdam: North-Holland,
1977.
[9] E. Denna, J. O. Cherrington, D. P. Andros, and A. S. Hollander, Event-Driven Business
Solutions:Today’s Revolution in Business and Information Technology. Homewood, IL: Irwin, 1993.
[10] J. Grupp, “Compresence is a Bundle: A problem for the Bundle Theory of Objects,” Metaphys. Int.
J. Ontol. Metaphys., vol. 5, no. 2, pp. 63–72, 2004.
[11] S. Kripke, Naming and Necessity. Princeton University Press, 1970.
[12] R. Schank, “Memory Organization Packets,” in Dynamic Memory Revisited, Cambridge, UK:
Cambridge University Press, 1999, pp. 123–136.
[13] Financial Accounting Standards Board, “Statement of Financial Accounting Concepts No. 6 -
Elements of Financial Statements,” Financial Accounting Standards Board, Norwalk, CT, Accounting
Standard Concept Statement No 6, 1985.
[14] J. Barbour, “Time and its Arrows,” in The Janus Point, Basic Books, 2020, pp. 1–22.
[15] N. Guarino, “On the Semantics of Ongoing and Future Occurrence Identifiers,” in Conceptual
Modeling, vol. 10650, H. C. Mayr, G. Guizzardi, H. Ma, and O. Pastor, Eds. Cham: Springer
International Publishing, 2017, pp. 477–490. doi: 10.1007/978-3-319-69904-2_36.