Glossary for Representing Knowledge

Definitions of terms as this web site uses them.

Data

Lowest level abstract concept with only a simple meaning from which information and then knowledge are derived.

Information

Data considered from the perspective of a system.

Knowledge

Incorporated information that may be used. Justified true belief (Plato). What Epistemology considers:

• What is knowledge?
• How is knowledge acquired?
• To what extent is it possible for a given subject or entity to be known?

Intelligence

Combining sensed data with knowledge to maximize the chances of success.

Probability

A function from event sets taken from a sample space to the unit interval [0,1] obeying the rules:

• The value for the empty set is 0.
• The value for the entire sample space is 1.
• The sum of the values of two disjoint sets is the value of their union.

Probability models reality by assigning larger values to more likely events.

Higher-order probability

Probability estimate of a probability, which, for orders above two (2), may in turn be an estimate of a probability. Irving John Good described them for Bayesian methods. These methods may subsume other methods for estimating probabilities such as Dempster-Shafer theory.

Conditional probability

Probability of some event set of outcomes given the occurrence of some other event set, a condition. The condition and outcomes may each be described with a logical expression. Members of an event are included if and only if their logical expression is true.

Condition

The part of a conditional probability statement that describes event sets that are assumed. Some conditions could describe empty event sets but then their conditional probability statements would be undefined. In the implementation, conditions model events with context logic expressions. Although a condition could be assigned a global, a priori probability, the implementation does not manage them.

Outcomes

Although an outcome is more typically used as the result of an experiment, here outcomes are the part of a conditional probability statement that describes the potential event set if the condition holds. Typically, that part of a probability statement would be the conditioned event but the word "outcomes" is more distinctive. (In retrospect, something else like "result" may have been better.) In the implementation, outcomes model an event set with a context logic expression. Sometimes, outcomes are displayed without their condition. As seen from an expansion of the definition of conditional probability, with logic expressions written as propositions, P(A|B) = P(A∧B|B) = P(A∧B)/P(B) where P(B)≠0, the outcomes logic expression A∧B (A and B) formally includes the condition B, even if the condition is not displayed.

Conditionee

The part of a conditional probability statement that describes outcome sets, events, that are measured to occur within a conditional probability statement. In some cases, the term outcome is loosely used instead.

Logical expression

A combination of constraints with logical connectives of negation, disjunction, and conjunction. The combination may be written linearly with balanced parentheses as lists in Polish prefix notation with a logical operator as the first item and the remaining items either constraints or recursively as logical expressions. Equivalently, a combination may also be depicted as a graph.

Constraint

A call of a predicate with a value, which may be null, and variables representing aspects of the delimited situation. Within higher-order logic, the variables may represent either simply entities or situations, which have embedded constraints of their own.

Predicate

As called within first and higher-order logic expressions, functions to truth values, true or false, from a value, which may be null, and other arguments that are modeled entities. Within higher-order logic expressions, some modeled entities are situations that contain predicates themselves.

Primitive

A collection of predicates with a common description. The description includes the domain, which may be null, of the values of predicates and the arity, which limits the acceptable quantity of predicate parameters. One aspect of the hypothesis states that a limited set of primitives can describe all predicates of logical expressions, which, in turn, can describe anything, although possibly inefficiently.

Property

A predicate that expresses an firstness, independent, in the terminology of Charles Sanders Peirce as John Florian Sowa elaborates, which describes an entity without referring to other entities. In the representation logical expressions of this site, each predicate has an initial variable representing the situation for which the predicate is relevant.

Relation

A predicate that expresses a relative, secondness, in the terminology of Charles Sanders Peirce as John Florian Sowa elaborates, which describes two entities that refer to one another. The relationship cannot be described by either entity alone and does not immediately involve further entities. In the representation logical expressions of this site, each predicate has an initial variable representing the situation for which the predicate is relevant.

Mediation

A predicate that expresses a mediating, thirdness, in the terminology of Charles Sanders Peirce as John Florian Sowa elaborates, which describes more than two entities that refer to each another. The relationship cannot be described without all of the entities involved. Although predicates with three entities alone could be sufficient, it may be more convenient to use predicates with more than three variables or an arbitrary positive quantity of variables. In the representation logical expressions of this site, each predicate has an initial variable representing the context situation for which the predicate is relevant.

Continuant

A predicate saying that an entity does not depend on time for its definition even though the entity may change with time. The identity of the entity can be determined at any point in its lifetime without time being an essential differentiator. John Florian Sowa elaborates on the difference between continuant and occurrent with slightly different definitions.

Occurrent

A predicate saying that an entity depends on time for its definition. even though the entity may stay the same during portions of its lifetime. John Florian Sowa elaborates on the difference between continuant and occurrent with slightly different definitions.

Entity

A name for the top of the ontology. Anything can be an entity. In logic expressions, variables reference entities and constraints define entities by using their variables as arguments.

Terry closure

A closure of constraints in a logic expression. A Terry closure is both an entity and a set, which is defined both by the constraints within it and by the constraints of its enclosing environment. Like closures of functional programming languages, such as Lisp, this logic closure captures the entities (variables) and predicates, which are analogous to functions, of its defining environment. Such closures serve as quantifiers of sets in closure logic. To evaluate each model for the containing logic expression, a member of the set is chosen. To evaluate the outcomes of events in the containing conditional probability expression, a probability measure considers measurable subsets of the closure. The Terry closure is named for Allan Terry, a friend who suggested using terminology from computer programming.

Rieger predicate

A macro of defining constraints and entities for use in a logic expressions to improve understanding and perhaps help guide the search for subexpressions of conditions. Diagrams show Rieger predicates as larger rounded boxes with bigger fonts and arrows to their externalized parts, which stand in for combinations of predicates and entities. Rieger predicates are named for Charles J. Rieger, III, who pointed out that people want to deal with larger groupings of meaning.

Condition group

A data structure of an implementation approach. A condition group combines a condition with its conditionees, conditional probabilities, and metadata such as an identifier and comments.

Declarative representation

A model that explicitly represents knowledge through depictions such as logical expressions, without directly executable steps. Such models facilitate meta-level reasoning about the model.

Procedural representation

A model that includes encoded steps, in languages such as Prolog or algorithmic computer languages, as part of representing knowledge. Such models may improve computational efficiency but may obscure relationships.

Propositional calculus

Zeroth-order logic, in which predicates are variables with a true or false assignment in a logic model and without arguments.

First-order logic (FOL)

A formal system with quantified variables and logical expressions with the logical connectives of negation, disjunction, and conjunction, which maps a domain into truth values, true or false. The quantified variables only represent entities and exclude situations where a predicate could be a variable.

Higher-order logic (HOL)

A formal system with variables, which are not quantified except through constraints (predicates), and logical expressions on constraints (no other functions are used here) with the logical connectives of negation, conjunction, and disjunction (which is actually represented through extensions of DeMorgans Laws with negations and conjunctions), which maps a domain into truth values, true or false. The variables may model both simple entities and situations that have embedded constraints of their own. Called closure logic rather than context logic since quantification occurs through Terry closures rather than over constraints, which should be expressively equivalent nonetheless, and to connote the lack of bounds on nesting Terry closures. In this formalism, the maximum depth of nesting would be the order of an expression, unlike in higher-order logic where conflicting definitions of order exist. Closure logic subsumes modal logic.

Conceptual model

An abstract entity representing intentions or semantics.

Representation

An abstraction mapping some features of an entity.

Graph

A collection of nodes and arcs that collect two nodes each.

Directed acyclic graph (DAG)

A graph with direction where each arc has a distinguished first node and any sequence of arcs, where the first node or any arc but the first is the second node of the preceding arc, cannot have the second node of the last arc be the first node of the first arc. There cannot be cycles.

Well-founded graph

In a closure logic graph of predicate nodes, each of which is contained in exactly one Terry closure, with directed edges to variables or entities, which may be Terry closures, there exists no sequence from any starting Terry closure or subsequent one to a predicate contained within those Terry closure that refers to a Terry closure beyond the one that contains the predicate that either refers to the starting Terry closure or to some eventual Terry closure containing a predicate refering to the starting one. The arguments of predicates cannot directly refer to their containing Terry closure either. More formally: let R be the relation through explicit arguments of predicates Pi to Terry closures other than the implicit single Terry closure Ci that contains each predicate Pi. There is no finite set S of predicates {Pi} such that R relates every Pi in the set S to a Terry closure Cj of another predicate Pj, which could be the same predicate, in the set S. Such a finite set S would contain a cycle.

¬∃S={Pi} ∀Pi∈S ∃Pj∈S PiRCj

There cannot be cycles of Terry closures, which implies any Terry closure has a unique outermost Terry closure that contains or is equal to it. In this implementation, the unique outermost Terry closure is the same for all predicates.

Edition

Organization of information that allows collaborators to create an experimental working environment for primitives, value types, and condition groups. Editions can be private or shared either with all collaborators or specified collaborators. Editions can include layers from other editions of information that may be shadowed by the current edition. After a collaborator has registered and been authenticated, the collaborator may be granted permission to create editions and dynamically change the layers and sharing of their own editions. Editions have a display name, which may be changed, and a more permanent internal name, which may appear in a URL query component. Some editions are reserved. The base edition (♥), which has the empty string as its internal name, includes some of the strongest commitments to statements. The system edition, which has a single hyphen (-) as its internal name, includes defaults that the web server uses.

Leibniz system

An implementation of the research that this site describes. The implementation is named for Gottfried Wilhelm von Leibniz, an important mathematician, probabilist, philosopher, and logician, who, among many foundation contributions, proposed his characteristica universalis, which are similar to the collection of primitives that this site advocates.

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