This web site is concerned with the theory and practice of Concept Relationship Lattice Logic, CRL Logic, or CRLL, pronounced 'cril logic' or just 'crill'.
You may join as registered user of CRLL Prototype.
CRL Logic is the underlying logic of the Relationship Lattice Formalism, which was defined and explored in a ph.d. project [1993a] [1993b] [1993c] [1994a] [1994b] [1994c] [1995], on a background of lattice theory [1990], concept algebra [1992b] [1993e], conceptual graphs [1976b] [1984a], mereology [1940] [1987] [1991], entity-relationship modelling [1976a], object-oriented software development and databases [1982] [1993d], relational databases [1970] [1975], and bibliographic databases [1992a].
The ph.d. project was directed and supervised by Jens Langeland-Knudsen, COO at Computer Resources International A/S, by Professor Hans Siggaard Jensen, Copenhagen Business School, by Professor Jørgen Fischer Nilsson, Technical University of Denmark, and by Professor Peter Ingwersen, Royal School of Library and Information Science.
Conceptual modelling has roots in the 1960's and 1970's, where Chen's entity-relationship diagrams [1976a] and Sowa's conceptual graphs [1976b] [1984a] [2000] were developed for software and database designs and for knowledge representation. CRL Logic owes much to Sowa's oeuvre. Ontological conceptual modelling [2008a] [2022a] is an important recent development.
Conceptual modelling is an abstraction process. It is like drawing a geographical map. Only a small fraction of the features of the real thing is expressed in the model, only those, which serve the purpose.
The use of the term "conceptual modelling" became common in computing science during the 1970's alongside such terms as "semantic data modelling" in the database area, "knowledge representation" in the artificial intelligence area, and "abstraction" in the programming language area [1982]. The contributions from these three areas were assembled by Brodie et al. [1984b]. In their view, conceptual modelling needs can be met only by a leap to a higher, more abstract level of system descriptions, like the leap from assembly languages to high level programming languages. The fundamental characteristic of the new level of system descriptions is that it is closer to the human conceptualization of a problem domain. Descriptions at this level can enhance communication between system designers, domain experts and ultimately, system end-users. CRL Logic offers a style of conceptual modelling, which is applicable in all three areas.
Plural Logic [2016a] [2018] [2020] [2021a] [2022b] is First Order Logic with plural terms, plural variables, plural predication, and plural quantification, and it has the expressive and deductive power of monadic Second Order Logic.
Plural Logic has roots in mathematical logic and metaphysics. Recently many references discuss further aspects of plural logic, e.g. superplurals, plural higher order logic, critical plural logic, multigrade phenomena, etc. These discussions indicate that First Order Logic and Plural Logic are stepping stones to a more complete logic. We investigate CRL Logic as such a more complete logic, including considerations about Second-order and Higher-order Logic [2024d], but based on both plurals/pluralities and sets, as discussed in chapter 4 of [2021a]. Our example meta-level CRLL model shows that properties may be quantified in CRL Logic.
Concepts are all the thoughts and ideas you have about physical or immaterial things, factual or imagined, countable or measurable, propositions, situations, events, time, space, stuff, measurements, numbers, texts, data structures, URIs, predicates, functions.
Relationships are dual properties, relating two concepts, such that each concept gets a property with the other concept as the value. Relationships may be potential or actual.
Potential relationships have minimum and maximum cardinalities as in entity-relationship modelling, they are like schema declarations in a database system. CRL Logic uses relationship modifiers to incorporate both cardinalities and logical quantification, e.g.:
Actual relationships are like the data in a database system.
CRL Logic relationships owe much to relational database systems, one of the most successful software technologies.
Lattices consist of ordering links between a superconcept and a subconcept, giving a partial order. The partial order is a "conceptual part-of", not a "physical part-of" as in physical mereology. A subconcept inherits properties from its superconcepts. So properties are like the differentiae of ontologies.
CRL Logic lattices with relationships owe much to object-oriented programming with class hierarchies, also one of the most successful software technologies.
A Concept Relationship Lattice, or CRLL model, or CRLL model module, is a boolean lattice, where all sums and products are implicitly present on all defined concepts, although most sums are not very meaningful, like the lattice sum of a trout and a turkey, or the lattice sum of the Eiffel Tower and president de Gaulle (examples from the mereology literature).
The meaning of a concept c is given by its intension and its extension. The intension consists of all the properties of c, including inherited properties. The extension consists of c itself and all the concepts below c in the lattice, a sublattice. Thus the intension of a superconcept is part-of the intension of its subconcepts, and the extension of a subconcept is part-of the extension of its superconcepts. So the lattice sum of a trout and a turkey has very little intension, just properties like weight and length.
Considerations of both intension and extension are necessary, when modelling what something is, preventing erroneous conceptual modelling, such as "the committee is the group of persons", or "the gold ring is the mass of gold".
A CRLL model represents a conception of a domain of interest. CRLL models may be merged in order to represent more complex domains. So CRLL models are modules that may build up, endlessly. Sometimes, contradictions may be detected among modules, when merged. Detecting contradictions between conceptual models may be a way to resolve disputes about misinformation. How to detect contradictions is a possible subject for scientist and student projects.
A concept called TOP represents everything in the domain, it has no superconcepts, nor properties.
A concept called BOTTOM represents nothing in the domain, it has no subconcepts.
A concept with a declared ordering link to BOTTOM is called a unit or unit concept (previously called an atom). A unit has a cardinality of 1.
A concept above units in the ordering is called a plural or plural concept. A plural has a cardinality, which is either the number of units below it in the ordering or a declared cardinality as a shorthand for the sum of unspecified units, e.g. a subconcept of Person labelled Trio with a cardinality of 3. The cardinality may be 0, while it has no units declared below it, in which case the plural has an implicit ordering link to BOTTOM.
In model-theoretic terms, the plural concepts and their potential relationships form a theory, and the unit concepts and their sums and actual relationships form a model.
A query, or query concept, is like a plural concept, except its extension is evaluated when required. Thus inferences may be performed on CRLL models in order to reason and deduce new insights, as by queries in a database system, or by formulas in First Order Logic.
Philosophy and linguistics have a distinction between countables, e.g. persons and physical things, and mass terms, e.g. water and gold. This has a long history [2024a]. CRL Logic as such does not make the distinction between countable concepts and mass concepts. In our example CRLL models of masses, mass concepts have measures to define the least value of their unit concepts. So when merging two model modules, where one measures water in litres, the other in millilitres, then an implementation shall transform to the most specific, millilitres.
CRL Logic is Plural Logic without the distinction between an element and the singleton set consisting of that element, whereby the Russell paradox of set theory is avoided, like in mereology. Is the expressive power of CRL Logic therefore reduced compared to Plural Logic? No, on the contrary, it brings clarity to our models, and the singleton sets reappear in plural form from chains of relationships, as we will see in many examples, e.g. Person and Committee. Or phrased differently, a unit concept has properties inherited from its plural superconcepts (e.g. Person), while related plural concepts (e.g. Committee) have their particular properties, just like elements and sets have their separate properties.
Modal logic
[2023] has modal operators, where ◊ and ◻ represent possibility and necessity.
It is likely and debatable, that they are equivalent to the potential and actual relationships of CRL Logic.
Other modal operators may be included in CRL Logic after further considerations.
Among our examples, we propose a CRLL model of the sentence
"Tom believes Mary wants to marry a sailor."
as discussed by John Sowa
[2002].
There, we have modal aspects of beliefs and desires to be modelled.
CRLL models are abstract creatures in your mind. You need software tools in order to build, visualize, communicate, merge, utilize, and manage them, and to exchange them with others.
In [1995] a very incomplete tool was developed, called RLL for Relationship Lattice Laboratory, programmed in Smalltalk, an object-oriented programming language, on a portable Mac computer.
Diagrams was used to visualize CRLL models, inspired by Hasse diagrams of lattices [1990] and by entity-relationship diagrams in a variant of the original form by Chen [1976a].
Here is an example of the use of RLL:
Here, family relationships are modelled between the plural concepts labelled person, man, and woman.
Further, three query concepts with labels in italics, grandmother, sonAndFather, and grandchild, model the rule that a woman is grandmother of a person, the grandchild, if she is the mother of a man, her son, who is the father of the grandchild.
When three unit concepts labelled Corinne,
Edward and B, are defined,
then the relationship
Corinne • isGrandmother0f ► ◄ hasGrandmother • B
may be derived.
Subwindows for each of the query concepts show their intensions and evaluated extensions.
The current tool development, CRLL Prototype, is used in the following example.
The term regimentation is used in philosophy for the transformation of natural language sentences into a formal language, similar to our use of the term conceptual modelling.
The most used example of a natural language sentence regimented into Plural Logic is the Geach-Kaplan sentence:
"Some critics admire only one another".
The sentence cannot be transformed into First Order Logic, but it can be transformed into Plural Logic.
According to [2021a] at point (2.27), the Plural Logic formula for the sentence is
∃xx (∀x(x ≺ xx → C(x)) ∧ ∀x∀y[(x≺xx ∧ A(x,y))→(y≺xx ∧ x≠y)])
meaning that the evaluated answer xx (a plural variable) shall include all the x that are critics (C(x)) and that admires (A(x,y)) a y that is also in the result and is not x.
Formulated as a query, the sentence is: "Which critics admire only one another?".
The Geach-Kaplan sentence as a CRLL model in CRLLP is shown to the right.
It assumes that the query concept labelled
?which-Critics-admire-only-one-another
can be evaluated as the transitive closure of the relationship
• admires►◄admiredBy •.
It is open for discussion, whether this model is a correct or satisfactory interpretation/regimentation of the sentence.
The plural concept Critic is a subconcept of the plural concept
Person, and it has potential relationship
• admires(0-n)►◄admiredBy(0-n) •
to itself.
The unit concepts are examples with actual relationships • admires►◄admiredBy •.
The answer evaluated as the transitive closure of the relationship is shown by the white lines from ?which-Critics-admire-only-one-another to the plural sums of Critic units that satisfy it: ⊕c21⊕c22, ⊕c23⊕c24, ⊕c31⊕c32⊕c33, and ⊕c34⊕c35⊕c36 ('⊕' is used in CRLLP for lattice sum).
Thus, the answer is a plural of plurals, also called a superplural [2008b].
You may experiment with the model as a registered user of CRLLP.
RLL figure 10 - RLL model of the Author/Document/Genre universe
The interpretation of the example is that
RLL figure 12 - Evaluating a query concept
Figure 12 shows a Boolean sublattice implicitly constructed over three documents, LittleMermaid, SnowQueen, and EitherOr, showing three plurals, unit sums, of two documents.
We have a classification of documents according to their Genre, induced by the relationship between Document and Genre.
Then we may talk about the documents, which has the Genre of Fairytale labelled FairytaleDocument.
The query concept FairytaleDocument is a subconcept of Document, its extension is some of the documents, and its intension is the intension inherited from Document with the addition of Fairytale as Genre.
When evaluated, FairytaleDocument evaluates to one of the plurals, LittleMermaid+SnowQueen.
RLL figure 25 - Two Boolean sublattices generated by a potential relationship
RLL figure 40 - An event with absolute time
RLL figure 41 - Events and relative time
RLL figure 43 - Person, Committee, and Location
RLL figure 46 - Products with various properties
RLL figures 76-77 - Mass concepts Water and Gold
The CRL Logic Prototype (CRLLP) enables registered users to experiment with CRL Logic.
You are invited to register as a user of the CRL Logic Prototype here. We appreciate your feedback, advice, and insight in the discussion forum.
If you prefer, you may browse CRLLP as anonymous guest.
See a brief guide to CRLLP here.
We hope that scientists and students will be inspired to develop systematic presentations of CRL Logic and its uses.
The screenshot shows the selected CRLL model module in the graphical form inspired by Hasse diagrams and entity-relationship diagrams. The selected CRL model module is also displayed in a table format and in a tree-structured text format, showing the intension and extension of concepts.
Users may cooperate, copying from each other and merging CRLL model modules to give more comprehensive models.
The implementation has a web interface to the core logic module that opens for application dependent user interfaces.
The core logic module is implemented in php with a mysql database running on an apache web server. The CRLLP user interface is implemented in html, css, javascript, and php.
Investors with teams of relevant experts are invited to join the CRL Logic software community, where the CRL Logic Prototype will develop into production-ready CRL Logic Management Systems, CRLLMSs. We encourage innovation, cooperation and competition.
CRLLMSs will act like Relational DBMSs and Knowlege Base Management Systems, KBMSs. There is potential to take over the DBMS and KBMS markets.
Implementations may lead to other product types, e.g. CRL Logic spreadsheets, where users may set up connected sheets ("crllsheets") with interface to CRLL modules in interoperating CRLLMSs with clickable query concepts for dynamic evaluation of queries. A CRL Logic software product will have CRL Logic in a backend core module and will have frontend interaction modules for users and web services.
Qualified investor teams with certified legal identity may acquire a licence to the source code of the CRLL Prototype for their own further development and for reimplementations on other platforms. The teams will report regularly on their plans and progress.
Licenses will be free of charge for a limited period of time. For more information on licenses, please write secure email to info@crllsoft.dk.
The implementation of CRLLMSs shall be evaluated by test sets, so that all implementations may be evaluated by the same criteria:
Some decisions taken for the implementation of CRLLP:
Example CRLLP--1 Two related plural concepts, 5 unit concepts.
Example CRLLP--2 Three related plural concepts and a deriving relationship.
Example CRLLP--3 Two related plural concepts and quantifiers
Example CRLLP--4 The Boolean Lattice
Example CRLLP--5 Mass concepts
Example CRLLP--6 Which units are transitively related?
Example CRLLP--7 A superconcept F MINUS a subconcept FB of F
Example CRLLP--8 A chain of queries with MINUS
Example CRLLP--9 A chain of queries with various quantifiers
Example CRLLP-11 Meta-level CRLL model of CRLL models, example AB
Example CRLLP-21 "Tom believes Mary wants to marry a sailor."
Example CRLLP-31 FOUST Case 1
Example CRLLP-32 FOUST Case 2
