Logical System Explained in Simple Terms (Another Try)

A logical system provides a formal structured mechanism to help us clearly think, argue, and discover truth. Both humans can understand such logical systems and machines can effectively interpret such logical systems if those systems are represented effectively.

Over the years I have tried to explain the "moving parts" of a logical system many times.  See under "Additional Information" below for the list of my best attempts. This is another shot.  What I am trying to achieve is to explain the moving pieces of a logical  system in concise terms such that a business professional with a liberal arts degree can understand what I am trying to explain.

To start, consider the Triangle of Meaning.  The Triangle of Meaning explains that meaning exists simultaneously in three forms:

  • The actual real world thing which is being explained/described. (Thing)
  • The conceptualization of that actual real world thing in the form that a human can understand (Conceptualization; Model; Theory; Blueprint)
  • The representation of the conceptualization of the actual real world thing in machine interpretable form (Implementation)

Now, the "actual real world thing" just is. It just exists.  It seems to me that both the conceptualization and the representation could take a number of different forms (i.e. knowledge representation approaches). Today, the term "ontology" is defined in many, many different ways.  I try and avoid that term, that is one of the reasons I use the term "theory". When you describe a theory, you use logic which is a formal language within the discipline of philosophy.

Logic is a formal communications tool that defines the rules of correct reasoning. A logical system is a structured method for representing information and reasoning about that information, using a defined language, semantics, and inference rules. A logical system is a formal framework that defines:

  • A language: the symbols, expressions, and rules for forming well‑structured statements.
  • A semantics: the meaning of those statements, typically expressed through models, interpretations, or truth conditions.
  • A set of valid inferences: the theorems or consequences that follow from the axioms under the proof rules.
  • A proof theory: the rules for deriving conclusions from premises (inference rules).

The "language" is like the legend of a map.  The "semantics" is like the territory or landscape the map is referring to.  The "inferences" are like the navigation rules. The "proof theory" makes sure everything is working as you would have expected.

A theory provides statement based meaning. A declarative statement is a modular semantic unit of meaning. Declarative statements are independent of any technology. All implementations of the declarative statements using different technologies must be consistent. 

The objective is a shared understanding of the meaning of of the conceptualization such that a proper representation can be created using computer software which properly reflects how the actual real world thing works so that the software works as would be expected to work. This would mean that the Triangle of Meaning is satisfied. I am leveraging the ideas of Atomic Design Methodology.  

Here is my most current shot at explaining/describing the components of a logical system independent of any implementation of the logical system:

* * *

A logical theory is a coherent set of logical statements that expresses a shared conceptualization of a domain of understanding to which a community of stakeholders is committed. A theory specifies the elements of the domain, the connections among them, the conditions that must always hold, and the facts and properties that characterize that domain of understanding.

  • Logical conceptualization: A logical conceptualization is the worldview of a domain expressed in logical form. It defines the set of permissible models; all models that are consistent with the assumptions, distinctions, and rules of the conceptualization.
  • Model: A model (a.k.a. assembly) is a permissible interpretation of the conceptualization. It consists of a set of structures that satisfy the constraints of the conceptualization and instantiate its elements, connections, and conditions.
  • Structure: A structure (a.k.a. organism, subassembly, compound element, complex element, artifact, module) is a compound element composed of logical statements that describe how a portion of the domain is organized. Structures assemble elements and connections into meaningful configurations.
  • Logical statement: A logical statement is a declarative proposition about the domain of understanding; a claim, belief, idea, notion, or fact. Logical statements are the units of meaning of the conceptualization and there are five broad categories (a.k.a. molecules) of logical statements:
    • Element (a.k.a. term, report element, thing, entity, node): An element is a logical statement that defines an idea or notion used by the logical conceptualization. An element may be privative and therefore not decomposable or an element can be compound and decomposable into a set of primitive elements.  An example of primitive elements might be "assets”, “liabilities”, and “equity”. An example of a compound element might be “balance sheet”. Elements tend to be nouns.
    • Connections: (a.k.a. associations, relations, interrelationships, edge): Connections are logical statements that describe permissible relations between elements.  Connections assemble elements into structures and structures into models. An example of a connection is the statement "assets is part of the balance sheet".  Connections tend to be verbs.  The  following are common types of connections:
      • Categorization (is-a, type of, general-special, class-of)
      • Compositional (has-a, part of, has part, whole part, instant-inflow, instant-outflow)
      • Aggregational (summation, mathematical)
      • Navigational (parent-child but where ordering does not matter)
      • Presentational (parent-child where ordering matters)
      • Simile (elements that are similar but not identical)
      • Equivalent (elements that are exactly the same)
      • Disjointed (elements that are explicitly not part of)
    • Conditions (a.k.a. assertions, restrictions, constraints, axioms, rules): A condition is a logical statement that always must be satisfied within any valid model. Conditions can be connected using logical connectors (e.g. AND, OR, NOT, NOR, IF) and make use of logical operators (e.g. +, =, /, *. <, >, ^). An example of a condition is "Assets = Liabilities + Equity".
    • Facts: A fact is a logical statement representing a measurement or observation typically expressed with numbers and words. For example, a fact might be “assets for the consolidated legal entity Microsoft as of June 20, 2017 was $241,086,000,000 expressed in US dollars and rounded to the nearest millions of dollars". Dimensions (a.k.a. aspects, axis, facet) can be used to distinguish and differentiate the context of facts.
    • Properties (a.k.a. quality, trait, attribute):  A property is a logical statement describing the important characteristics of a model, structure, element, connection, condition, or fact. An example of a property is "assets is a debit".
The following world view is assumed. Unless otherwise stated a closed world assumption is assumed. Unless otherwise stated Horn-clause logic is assumed.  Unless otherwise stated negation as failure is assumed.

A logical system is a formally governed arrangement of elements, connections, conditions, and facts that together define how a domain is represented, interpreted, and reasoned about by humans and machines. It includes a conceptualization of the domain, a world view, a set of permissible models, a formal language for expressing statements, and rules for determining which statements follow from others.

* * *
Here is a set of reference logical systems represented using the global open industry standard language XBRL. Those logical systems could just as well be represented using RDF+OWL+SHACL or Graph Query Language (GQL) or Structured Query Language (SQL) or some other implementation approach.

The core aspects of the logic of a digital financial statement are shown in the red box below (chick here for a  larger image): (this image was inspired by Semantic Arts periodic table of GIST)


Or maybe


Accounting, reporting, auditing, and analysis for the next 1000 years should be elegant; not a kludge. To make this happen, accountants, auditors, and analysists need to improve their digital proficiency.

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