Elements of Logic for Accountants
Proficiency is the measure of capacity to use something to one’s benefit. Few accountants are literate when it comes to logic and logical reasoning. Being literate should not really be the goal though; the goal should be fluency and mastery.
Logic and logical reasoning are skills; that skill involves the selection and application of the appropriate rules of logic to specific situations.
This blog post was inspired by the article, Elements of Logic. I have brainstormed how to explain logical systems before; see this early version which grabs ideas from about 14 different inputs and this revision which starts to simplify things a bit. The Book of Proof has a decent explanation and also provides additional details. This was my last best attempt to explain the elements of logic.
There tends to be several different sources of explanations of the elements of logic: philosophy, ontology and knowledge engineers, and computer scientists. These explanations tend to be inconsistent, many are incomplete, and most are either too high level to be useful or provide too much details which becomes overwhelming for business professionals and accountants.
This explanation of the elements of logic is tuned specifically for accountants and is intended to help them understand how to think about XBRL-based digital financial reports.
This is my current best take:
The elements of logic are the fundamental building blocks of logical theories that describe the logical conceptualization of some natural or man-made logical system. An area of knowledge can describe the important logic of that system using these building blocks:
- Logical statement: A logical statement is a proposition, claim, assertion, belief, idea, or fact about or related to the area of knowledge to which the logical conceptualization relates. A logical statement is a declarative sentence. Not all sentences are statements; for example, a question such as "What is your name?", or a command such as "Stop!", are not statements. There are five broad categories of logical statements:
- Terms: Terms are important logical statements that define ideas or "things" used by a logical conceptualization. For example, “assets”, “liabilities”, “equity”, and “balance sheet” are things or ideas used in a logical conceptualization to describe the area of knowledge referred to as financial reporting.
- Associations: Associations are important logical statements that describe permissible interrelationships between the terms such as “assets is part-of the balance sheet” or “operating expenses is a type-of expense” or “an asset is a ‘debit’ and is ‘as of’ a specific point in time and is always a monetary numeric value”. Associations can be grouped into two broad types:
- "Is-a" (a.k.a. general-special, association, type-subtype, class-subclass, equivalent-class)
- "Has-a" (a.k.a. part-of, has-part, part-whole, composition, aggregation)
- Rules: Rules (a.k.a. assertions, restrictions, constraints) are important logical statements that describe what tend to be convertible into IF…THEN…ELSE types of relationships such as “IF the economic entity is a not-for-profit THEN net assets = assets - liabilities; ELSE assets = liabilities + equity”. One rule can be connected to another rule using logic gates (a.k.a. logical connective) (AND, OR, NOR, NAND, XOR, XNOR, NOT) to form complex logical statements. Rules can assert mathematical relationships or derive mathematical relationships to form new facts.
- Facts: Facts are important logical statements that are known to be true. In the context of databases and knowledge representation, facts are often used to represent known information. Facts are logical statements about the numbers and words that are provided by an economic entity within a financial report. For example, the financial report might state “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.
- Properties: Properties are important logical statements about the important qualities and traits of a model, structure, term, association, rule, and fact.
- Axioms: Axioms are foundational logical statements that are fundamentally accepted as being true per some logical system.
- Theorems: Theorems are logical statements that are determined to be true per logical steps that can be taken to arrive at a conclusion using axioms, other theorems, of facts.
- Restriction: Restrictions are a special type of axiom or theorem that is imposed by some authority which chooses to restrict, constrain, limit, or otherwise impose some range on some logical artifact.
- Classification: Classification is the grouping of logical artifacts into sets.
- Logical structure: A logical structure is as set of logical statements which describe the structure. An "infon" which is defined by Situation Theory is a unit of information. In infon is a type of logical structure. An infon is a useful, convenient unit or "set" of information.
- Logical model: A logical model is a set of specific structures that are consistent with and permissible interpretations of that model. Models add flexibility to logical conceptualizations.
- Logical conceptualization: A logical conceptualization is a set of models that are consistent with and permissible per that logical conceptualization. A logical conceptualization is made up of a set of models, structures, terms, associations, rules, and facts.
- Logical theory: A logical theory is described per some logical conceptualization forms a logical theory that explains what is permitted and what is not permitted per a logical conceptualization which is made up of a set of logical models, structures, terms, associations, rules, and facts.
- Logical system: A logical system can be explained by a logical theory. A logical theory is an abstract conceptualization of specific important details of some area of knowledge. The logical theory provides a way of thinking about an area of knowledge by means of deductive reasoning to derive logical consequences of the logical theory.
A logical system described by a logical theory and described by a logical conceptualization enables a community of stakeholders trying to achieve a specific goal or objective or a range of goals/objectives for an area of knowledge to agree on important logical statements used for capturing meaning or representing a shared understanding of and knowledge in some area of knowledge.
A logical conceptualization must be consistent (as opposed to inconsistent, making contradictory statements), complete (as opposed to incomplete, leaving a piece out), and precise (as opposed to imprecise, describing an area of knowledge incorrectly).
Propositional calculus (a.k.a. propositional logic, statement logic) which deals with statements and relations between statements is the foundation for first-order predicate calculus (a.k.a. first order logic, predicate logic) which adds the notion of variables to propositional calculus. Classical logic (a.k.a. standard logic) seems to be the basis for propositional logic and first order logic. There are many algebras of logic. By "logic" I mean first order logic.
Logic involves logical reasoning. Inference are steps in reasoning. There are three types of logical reasoning or types of steps in inference: deductive reasoning, inductive reasoning, and abductive reasoning. In logic you have logical connective (a.k.a. logical operators).
Currently, there is no standard terminology or jargon for implementing the semantics of logic or any standard physical implementation of logic. The best "boundary" of logic and implementation that I have run across is DATALOG, or Horn Clauses. Or, maybe there is ISO//IEC 24707:2018 Common Logic (CL). See this Introduction to Common Logic.
A question that you might have is how are these logical statements that describe an area of knowledge acquired? That will be the topic of another blog post.
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