SBVR ~ Ground Facts and Fact Types in First-Order Logic
Summary
After Frege introduced first-order logic in 1879, at least a century passed without successful attempts to make it understood by the masses. SBVR (in OMG) and CL (Common Logic in ISO) are recently approved standards that indicate that there is now enough support to start using logic on a much larger scale. This has many substantial advantages. In practice many people employ the essence of logic, without ever thinking of the underlying theory. Historically, introductions to logic have often failed to give adequate attention to the fact that semantics is of prime importance. This article is the first in a series of articles to introduce first-order logic to a wider audience, with a strong focus on semantics, using the CogNIAM (Cognition enhanced NIAM) approach. This article concentrates on what are called ground facts and fact types in SBVR.
1 Introduction
First-order logic has many advantages over other formal languages.[8] SBVR[6] is a standard based on first-order logic. CL (Common Logic)[3] is a recently-accepted ISO standard based on first-order logic. If first-order logic is effectively taught and expressed in Controlled or Structured English or any other controlled natural language, the advantages will be enormous.[5] By using everyday natural language, people are already thoroughly accustomed to using logic, even though most have never been introduced to "the formidable notation, which doesn't bother mathematicians, but [...] is an instant turn-off for many other people."[9] [1] Hence it is a matter of awareness and integration, instead of applying a traditional stovepipe approach.
In addition, instead of using the arcane notation of first-order logic, people are invited to use the natural language based notation of CogNIAM, which "is as precise as any formal notation, but as readable as their familiar (natural) language."[9] CogNIAM is a speech community, utilizing as much diagramming as possible, within the semantic SBVR community. In this way, we can expect a large number of advantages in many different areas.
2 The components of a ground fact
In Figure 1 we see the shape of a certain country. That country is referred to by the name "France".
Figure 1. Country
The verbalization of Figure 1 could be: "France is the name of an individual country."
In Figure 2 the shape of a well-known symbol is used as reference to a well-known city. That city is referred to by the name "Paris".
Figure 2. City
The verbalization of Figure 2 could be: "Paris is the name of an individual city."
The contents of Figure 3 can be verbalized as follows:
"The city Paris is the capital of the country France."
Figure 3. Country and its capital
We could give a second example fact with the same fact structure. This example is represented in Figure 4.
Figure 4. Country and its capital
The fact represented there is:
"The city Brussels is the capital of the country Belgium."
The contents of Figure 5 can be verbalized as:
"Rome is the name of an individual city."
Figure 5. City
The contents of Figure 6 can be verbalized as:
"Spain is the name of an individual country."
Figure 6. Country
We will now make the first bridge between these well-understood concepts and parts of first-order logic.
In first-order logic we would say:
The term "France" is an individual constant.
The term "Belgium" is an individual constant.
The term "Spain" is an individual constant.
The term "Paris" is an individual constant.
The term "Brussels" is an individual constant.
The term "Rome" is an individual constant.
In first-order logic one would say:
"Is the capital of"is a predicate constant.
Ground fact
Having established the bridge, we now give a concept definition of a ground fact in SBVR:
P is a predicate constant
AND
t1 , ..., tn are individual constants
IMPLIES
P(t1 , ..., tn ) is a ground fact.
Example:
P = is the capital of
t1 = Paris
t2 = France"Paris is the capital of France" is an example of a ground fact in SBVR.
"Brussels is the capital of Belgium" is another example.
Individual constant in first-order logic
An individual constant in first-order logic is a reference to an object that has no grammatical function, i.e., is not part of the domain-specific nor generic conceptual schema. Examples: "Paris", "Brussels", "France", "Belgium". Conversely, "city", "country", "fact type", "role" are not examples of an individual constant.
Predicate constant in first-order logic
A predicate constant in first-order logic corresponds to a verb in natural language with a fixed number of roles. Examples: "is the capital of", "is born in", "works for", "likes".
"Paris is the name of an individual city" is an example of a ground fact in SBVR.
"France is the name of an individual country" is an example of a ground fact in SBVR.
Having described ground facts we will now proceed to define a domain-specific fact type of SBVR.
3 Fact types at the domain-specific conceptual schema level
At the domain-specific conceptual schema level, the basis of SBVR consists of concept definitions and fact types.[7] "An important asset of business rules is that they are based on clear terms and facts."[10] In this article we will concentrate on the fact types and will discuss concept definitions in a future article.
Let us repeat the two binary ground facts (i.e., facts with two variable positions) and represent them in a way that clearly shows the variable and constant parts in the facts.
Brussels is the capital of Belgium.
Paris " " " " France.
The position taken by "Brussels" and "Paris" in the ground facts could be generalized to the variable "city"; the position taken by "Belgium" and "France" could be generalized to the variable "country".
In Figure 7 we use a well-known representation technique[4] for illustration purposes to represent the ground facts discussed above. Please note that we have classified elements (or objects) into 4 classes: "city", "city name", "country", and "country name". There are furthermore 3 classes of ground facts. The first and third classes contain non-lexical objects (the things themselves), and the second and fourth classes contain lexical objects. Remember that we need a reference or name to an element or object to have it represented in a ground fact. Hence line 5 – a specific city is the capital of a specific country – can only be used as a ground fact if the two non-lexical elements are replaced by their linguistic representations.
city is the capital of country
Figure 7. The world of things and designations
The combination of connection 1, 5, and 11 represents the ground fact:
The city with city name Paris is the capital of the country with country name France.
Of course many people prefer to abbreviate this fact to:
The city Paris is the capital of country France,
or even shorter:
Paris is the capital of France.
In Figure 7 we see that there are 3 individual cities. If we want to use a specific city in a fact we use the specific city name to refer to the specific city. The same holds for a specific country.
If we want to express the ground fact that a specific city is the capital of a specific country, e.g., Paris is the capital of France, we use the combination of 1, 5, and 11. Hence in the fact type "Is the capital of", the variable or object type "city" needs a linguistic representation as well as the variable "country".
In SBVR: city is the capital of country
The linguistic representations or naming conventions (e.g., between "city name" and "city" as well as between "country name" and "country") can fruitfully be considered as unary fact types, as will be further illustrated in the next paragraph. Hence instead of using the binary SBVR fact type for the naming convention we recommend to apply a formal transformation and use a unary fact type.
Hence instead of the binary fact type in SBVR
city has city name
I recommend a formal transformation with the following result, but first a few concrete ground facts
Paris is a designation of an individual city.
Brussels " " " " " " " .
with the unary fact type
city name is a designation of an individual city
In CogNIAM we prefer to use a diagram technique in which ground fact instances and fact types can be represented together. This technique is 1:1 mappable to the ground facts and fact types of SBVR. It is recommended to have the fact type reading included in the diagram as well as the starting population. This is represented in Figure 8.
Figure 8. The world of things and designations in CogNIAM
In Figure 7 we count 2 binary ground facts and 6 unary ground facts. Why? A fact needs a linguistic representation. To talk about the leftmost city in Figure 7 (i.e., a non-lexical object of the class "city"), we use the linguistic representation "Paris". In Figure 8 we count the same 2 binary and 6 unary ground facts.
The unary fact type will be described in the next paragraph.
In traditional predicate logic notation:
Q is a predicate constant
AND
s1 , ..., sm are individual variables
AND
s1 , ..., sm are domain-specific
IMPLIES
Q(s1 , ..., sm ) is a fact type in a domain-specific component of the conceptual schema.
Example:
Q = is the capital of
s1 = city
s2 = country
4 Best practice recommendation for transforming a binary naming fact
type into a unary
If we look at diagram 7 we could formulate three binaries as is done in SBVR Structured English (SBVR-SE). The three binaries in SBVR-SE are:
city has city name
country has country name
city is the capital of country
However, extensive experience with fact orientation in business practice has demonstrated that it is much more understandable to transform each naming convention binary into a unary as specified below:
city name is a designation of an individual city
country name is a designation of an individual country
city is the capital of country
a name is a designation of an individual a
b name is a designation of an individual b
The "a" and "b" at the end of both fact types can be transformed into an element of the constant part as is given below:
a name is a designation of an individual a
b name is a designation of an individual b
Q is a predicate constant
AND
s1 , ..., sm are individual variables
AND
s1 , ..., sm are generic
IMPLIES
Q(s1 , ..., sm ) is a fact type in the generic component of the conceptual schema
5 Further connection between SBVR and first-order logic
In first-order logic we have in this case a set of individual constants. Suppose we want to represent this set in a figure -- we get the following:
Figure 9. Eight individual constants
Please note that traditional first-order logic puts all constants into one big container or class. Such a class needs a very generic name, e.g., "individual constant". In business, of course, we categorize this class into more meaningful classes such as "city name" and "country name".
This can be represented in a CogNIAM diagram as shown in Figure 10.
Figure 10. Eight individual constants in CogNIAM
If we now connect this to the diagrams developed before, we obtain:
Figure 11. Typing of individual constants
In Figure 11 we see that an individual constant of first order logic can play a certain role in more than one naming convention. "Each subworld must admit only certain types of individual."[2] E.g., we see that the individual constant "Luxemburg" acts in the role of city name as well as country name. Fact 7 can be expressed as follows:
The city
with city name
Luxemburg
is the capital of
the country
with country name
Luxemburg
The corresponding CogNIAM diagram is given in Figure 12.
Figure 12. Typing of individual constants in CogNIAM
In Figure 12 we count 16 unary facts and 3 binary facts. It can be seen that the fact type diagram "individual constant" has a subset relationship with fact type diagram "country" as well as fact type diagram "city". This implies that the individual constant can have a role in more than one naming convention. E.g., an individual constant can play a role in the naming convention "city name" or "country name", in both, or in none.
6 Conclusion
In this paper we argue that using first-order logic on a larger scale can provide substantial advantages over other formal languages. Since new standards like SBVR as well as CL are based on first-order logic, it is evident that expressing and using first-order logic in common education on a much larger scale can bring substantial and tangible advantages. However, thus far classical methods to introduce first-order logic using a somewhat esoteric notation have largely failed in bringing the essence of this theoretical framework to the masses. SBVR is a big step forward to making first-order logic as common as arithmetic.
We recommend using a controlled natural language based approach in order to accomplish this. Due to extensive experience with fact orientation in various business environments, we further recommend to formulate naming convention binaries, as is done in SBVR-SE, into unaries. It was argued that naming convention binaries can be considered as unaries and therefore a formal transformation and the use of the unary is recommended.
I would like to take the opportunity to gracefully thank Terry Halpin, Ron Ross, and John Sowa for their valued comments on earlier versions of this article.
References
[1] Frege, Gottlob, Begriffsschrift, eine der arithmetischen nachgebildete Formalsprache des reinen Denkens, Halle a. S., 1879.
[2] Halpin, Terry, A Logical Analysis of Information Systems: Static Aspects of the Data-oriented Perspective, 1989.
[3] ISO/IEC 24707:2007, Common Logic, Information technology - Common Logic (CL): A Framework for a Family of Logic-based Languages, First Edition, Geneva, Switzerland: International Organization of Standards (October 1, 2007).
[4] Nijssen, G. M., An Architecture for Knowledge Base Software, Paper presented to the Australian Computer Society, Published in: Proceedings SPOT-2 conference Stockholm, Brussels (July 1981).
[5] Nijssen, Sjir, "SBVR: Semantics for Business," Business Rules Journal, Vol. 8, No. 10 (Oct. 2007), URL: http://www.BRCommunity.com/a2007/b367.html
[6] OMG, Semantics of Business Vocabulary and Business Rules (SBVR). Second
Interim Specification, September 2006. Available as dtc/06-08-05 (Convenience
Document without change bars) at http://www.omg.org
[7] Ross, Ronald, Principles of the Business Rules Approach, Addison-Wesley (2003 ).
[8] Sowa, John F., "Fads and Fallacies about Logic," IEEE Intelligent Systems, 22:2, pp. 84-87, (March 2007). URL: http://www.jfsowa.com/pubs/fflogic.htm
[9] Sowa, John F., review correspondence by e-mail, November 2007.
[10] Vanthienen, Jan, "What Business Rules and Tables Can Do for Regulations," Business Rules Journal, Vol. 8, No. 7 (July 2007), URL: http://www.brcommunity.com/a2007/b355.html
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