Description logics (DL) are a family of formal knowledge representation languages. Many DLs are more expressive than propositional logic but less expressive than first-order logic. In contrast to the latter, the core reasoning problems for DLs are (usually) decidable, and efficient decision procedures have been designed and implemented for these problems. There are general, spatial, temporal, spatiotemporal, and fuzzy description logics, and each description logic features a different balance between expressive power and reasoning complexity by supporting different sets of mathematical constructors. DLs are used in artificial intelligence to describe and reason about the relevant concepts of an application domain (known as terminological knowledge). It is of particular importance in providing a logical formalism for ontologies and the Semantic Web: the Web Ontology Language (OWL) and its profiles are based on DLs. A major area of application of DLs and OWL is in biomedical informatics, where they assist in the codification of biomedical knowledge. DLs and OWL are also applied in other domains, including defense, climate modeling, and large-scale industrial knowledge graphs. == Introduction == A DL models concepts, roles and individuals, and their relationships. The fundamental modeling concept of a DL is the axiom—a logical statement relating roles and/or concepts. This is a key difference from the frames paradigm where a frame specification declares and completely defines a class. == Nomenclature == === Terminology compared to FOL and OWL === The description logic community uses different terminology than the first-order logic (FOL) community for operationally equivalent notions; some examples are given below. The Web Ontology Language (OWL) uses again a different terminology, also given in the table below. === Naming convention === There are many varieties of description logics and there is an informal naming convention, roughly describing the operators allowed. The expressivity is encoded in the label for a logic starting with one of the following basic logics: Followed by any of the following extensions: ==== Exceptions ==== Some canonical DLs that do not exactly fit this convention are: ==== Examples ==== As an example, A L C {\displaystyle {\mathcal {ALC}}} is a centrally important description logic from which comparisons with other varieties can be made. A L C {\displaystyle {\mathcal {ALC}}} is simply A L {\displaystyle {\mathcal {AL}}} with complement of any concept allowed, not just atomic concepts. A L C {\displaystyle {\mathcal {ALC}}} is used instead of the equivalent A L U E {\displaystyle {\mathcal {ALUE}}} . A further example, the description logic S H I Q {\displaystyle {\mathcal {SHIQ}}} is the logic A L C {\displaystyle {\mathcal {ALC}}} plus extended cardinality restrictions, and transitive and inverse roles. The naming conventions aren't purely systematic so that the logic A L C O I N {\displaystyle {\mathcal {ALCOIN}}} might be referred to as A L C N I O {\displaystyle {\mathcal {ALCNIO}}} and other abbreviations are also made where possible. The Protégé ontology editor supports S H O I N ( D ) {\displaystyle {\mathcal {SHOIN}}^{\mathcal {(D)}}} . Three major biomedical informatics terminology bases, SNOMED CT, GALEN, and GO, are expressible in E L {\displaystyle {\mathcal {EL}}} (with additional role properties). OWL 2 provides the expressiveness of S R O I Q ( D ) {\displaystyle {\mathcal {SROIQ}}^{\mathcal {(D)}}} , OWL-DL is based on S H O I N ( D ) {\displaystyle {\mathcal {SHOIN}}^{\mathcal {(D)}}} , and for OWL-Lite it is S H I F ( D ) {\displaystyle {\mathcal {SHIF}}^{\mathcal {(D)}}} . == History == Description logic was given its current name in the 1980s. Previous to this it was called (chronologically): terminological systems, and concept languages. === Knowledge representation === Frames and semantic networks lack formal (logic-based) semantics. DL was first introduced into knowledge representation (KR) systems to overcome this deficiency. The first DL-based KR system was KL-ONE (by Ronald J. Brachman and Schmolze, 1985). During the '80s other DL-based systems using structural subsumption algorithms were developed including KRYPTON (1983), LOOM (1987), BACK (1988), K-REP (1991) and CLASSIC (1991). This approach featured DL with limited expressiveness but relatively efficient (polynomial time) reasoning. In the early '90s, the introduction of a new tableau based algorithm paradigm allowed efficient reasoning on more expressive DL. DL-based systems using these algorithms — such as KRIS (1991) — show acceptable reasoning performance on typical inference problems even though the worst case complexity is no longer polynomial. From the mid '90s, reasoners were created with good practical performance on very expressive DL with high worst case complexity. Examples from this period include FaCT, RACER (2001), CEL (2005), and KAON 2 (2005). DL reasoners, such as FaCT, FaCT++, RACER, DLP and Pellet, implement the method of analytic tableaux. KAON2 is implemented by algorithms which reduce a SHIQ(D) knowledge base to a disjunctive datalog program. === Semantic web === The DARPA Agent Markup Language (DAML) and Ontology Inference Layer (OIL) ontology languages for the Semantic Web can be viewed as syntactic variants of DL. In particular, the formal semantics and reasoning in OIL use the S H I Q {\displaystyle {\mathcal {SHIQ}}} DL. The DAML+OIL DL was developed as a submission to—and formed the starting point of—the World Wide Web Consortium (W3C) Web Ontology Working Group. In 2004, the Web Ontology Working Group completed its work by issuing the OWL recommendation. The design of OWL is based on the S H {\displaystyle {\mathcal {SH}}} family of DL with OWL DL and OWL Lite based on S H O I N ( D ) {\displaystyle {\mathcal {SHOIN}}^{\mathcal {(D)}}} and S H I F ( D ) {\displaystyle {\mathcal {SHIF}}^{\mathcal {(D)}}} respectively. The W3C OWL Working Group began work in 2007 on a refinement of - and extension to - OWL. In 2009, this was completed by the issuance of the OWL2 recommendation. OWL2 is based on the description logic S R O I Q ( D ) {\displaystyle {\mathcal {SROIQ}}^{\mathcal {(D)}}} . Practical experience demonstrated that OWL DL lacked several key features necessary to model complex domains. == Modeling == === TBox vs Abox === In DL, a distinction is drawn between the so-called TBox (terminological box) and the ABox (assertional box). In general, the TBox contains sentences describing concept hierarchies (i.e., relations between concepts) while the ABox contains ground sentences stating where in the hierarchy, individuals belong (i.e., relations between individuals and concepts). For example, the statement: belongs in the TBox, while the statement: belongs in the ABox. Note that the TBox/ABox distinction is not significant, in the same sense that the two "kinds" of sentences are not treated differently in first-order logic (which subsumes most DL). When translated into first-order logic, a subsumption axiom like (1) is simply a conditional restriction to unary predicates (concepts) with only variables appearing in it. Clearly, a sentence of this form is not privileged or special over sentences in which only constants ("grounded" values) appear like (2). === Motivation for having Tbox and Abox === So why was the distinction introduced? The primary reason is that the separation can be useful when describing and formulating decision-procedures for various DL. For example, a reasoner might process the TBox and ABox separately, in part because certain key inference problems are tied to one but not the other one ('classification' is related to the TBox, 'instance checking' to the ABox). Another example is that the complexity of the TBox can greatly affect the performance of a given decision-procedure for a certain DL, independently of the ABox. Thus, it is useful to have a way to talk about that specific part of the knowledge base. The secondary reason is that the distinction can make sense from the knowledge base modeler's perspective. It is plausible to distinguish between our conception of terms/concepts in the world (class axioms in the TBox) and particular manifestations of those terms/concepts (instance assertions in the ABox). In the above example: when the hierarchy within a company is the same in every branch but the assignment to employees is different in every department (because there are other people working there), it makes sense to reuse the TBox for different branches that do not use the same ABox. There are two features of description logic that are not shared by most other data description formalisms: DL does not make the unique name assumption (UNA) or the closed-world assumption (CWA). Not having UNA means that two concepts with different names may be allowed by some inference to be shown to be equivalent. Not having CWA, or rather having the open world assumption (OWA) means that
Yorba (software)
Yorba is a web-based personal information management platform for finding, monitoring, or deleting online accounts and subscriptions. Yorba is a participating member of Consumer Reports’ Data Rights Protocol (DRP) consortium that develops open technical standards for exercising consumer data rights under laws including the California Consumer Privacy Act. == History == Yorba began as a research project around 2021. It was founded by Chris Zeunstrom (CEO), Nolan Cabeje (CDO) and David Schmudde (CTO). Zeunstrom says he began developing Yorba after growing frustrated with managing numerous email accounts, noting overloaded inboxes create distraction and potential security vulnerabilities. Yorba’s early development was also influenced by security issues he encountered at a previous company, which had been affected by data breaches at a time when such incidents were becoming increasingly common. In 2023, Yorba launched a private beta as a public benefit corporation funded through a give-back model operated by Zeunstrom's New York-based design firm, Ruca. In January 2024, Yorba entered public beta and reported over 1,000 users, including 160 premium subscribers. At the time of the public beta launch, Yorba integrated with Gmail and announced plans to expand compatibility to other online services and cloud storage providers. In September 2024, Yorba completed conformance testing under the Data Rights Protocol, an initiative developed by Consumer Reports, to establish a standard and open-source framework for securely transmitting consumer data rights requests under laws like the California Consumer Privacy Act. Yorba was named among twelve participating companies that implemented the protocol alongside OneTrust and Consumer Reports’ own Permission Slip app. Yorba was one of nine startups selected as 2025 finalist in the Santander X Global Awards international entrepreneurship competition. == Features == Yorba scans user inbox history data to identify online accounts, mailing lists, and possible data breaches. It uses natural language processing and machine learning to identify a user's accounts, services, and subscriptions. The platform prompts password resets for compromised accounts and locates unused accounts. The platform also supports mailing list management by identifying and helping users unsubscribe from newsletters. Paid subscribers can locate and cancel recurring charges. Yorba links with financial institutions in the U.S., Canada, and EU via Plaid Inc. to detect recurring charges and delete unwanted subscriptions. == Privacy and Ethics == Yorba's founder has openly criticized dark patterns that make canceling services difficult, citing personal frustration with inbox clutter as part of his inspiration for Yorba. Yorba offers privacy policy analysis in partnership with Amsterdam-based nonprofit Terms of Service; Didn’t Read, assigning grades based on invasiveness or ethical concerns. As of 2024, the company described its pricing as designed to cover operational costs and sustain the platform without outside investment.
Statistical semantics
In linguistics, statistical semantics applies the methods of statistics to the problem of determining the meaning of words or phrases, ideally through unsupervised learning, to a degree of precision at least sufficient for the purpose of information retrieval. == History == The term statistical semantics was first used by Warren Weaver in his well-known paper on machine translation. He argued that word-sense disambiguation for machine translation should be based on the co-occurrence frequency of the context words near a given target word. The underlying assumption that "a word is characterized by the company it keeps" was advocated by J. R. Firth. This assumption is known in linguistics as the distributional hypothesis. Emile Delavenay defined statistical semantics as the "statistical study of the meanings of words and their frequency and order of recurrence". "Furnas et al. 1983" is frequently cited as a foundational contribution to statistical semantics. An early success in the field was latent semantic analysis. == Applications == Research in statistical semantics has resulted in a wide variety of algorithms that use the distributional hypothesis to discover many aspects of semantics, by applying statistical techniques to large corpora: Measuring the similarity in word meanings Measuring the similarity in word relations Modeling similarity-based generalization Discovering words with a given relation Classifying relations between words Extracting keywords from documents Measuring the cohesiveness of text Discovering the different senses of words Distinguishing the different senses of words Subcognitive aspects of words Distinguishing praise from criticism == Related fields == Statistical semantics focuses on the meanings of common words and the relations between common words, unlike text mining, which tends to focus on whole documents, document collections, or named entities (names of people, places, and organizations). Statistical semantics is a subfield of computational semantics, which is in turn a subfield of computational linguistics and natural language processing. Many of the applications of statistical semantics (listed above) can also be addressed by lexicon-based algorithms, instead of the corpus-based algorithms of statistical semantics. One advantage of corpus-based algorithms is that they are typically not as labour-intensive as lexicon-based algorithms. Another advantage is that they are usually easier to adapt to new languages or noisier new text types from e.g. social media than lexicon-based algorithms are. However, the best performance on an application is often achieved by combining the two approaches.
Murder of Suzanne Adams
In August 2025, 83-year-old Suzanne Eberson Adams was murdered at her home in Greenwich, Connecticut, United States, by her son and former marketing executive, 56-year-old Stein-Erik Soelberg. Shortly after killing his mother, Soelberg committed suicide. Adams's murder was fueled by her son's persecutory delusions, such as that she was spying on him and trying to poison him with drugs siphoned through his car vents. Shortly after an investigation into the murder–suicide, it was revealed that Soelberg had conversed with ChatGPT, an artificial intelligence chatbot, about his suspicions. Despite the unlikely nature of his accusations toward her, the chatbot apparently agreed that his fears were justified and prompted Soelberg to test his mother to determine if she was a spy or not. In December 2025, this led to a lawsuit against OpenAI, the company developing the chatbot. Critics said that the chatbot created an echo chamber that reinforced the perpetrator's delusions. == Background == Soelberg worked in the tech industry in program management and marketing until 2021. He divorced in 2018, after being married for 20 years and having two children. Soelberg moved the same year to live with his mother in Old Greenwich, an affluent New York suburb. Since late 2018, many police reports describe incidents with alcoholism and suicide threats and attempts. Erik Soelberg had an Instagram account called "Erik the Viking". The account was initially focused on bodybuilding and spiritual content, but he started in October 2024 to publish videos comparing AI chatbots. He posted on YouTube and Instagram many discussions with chatbots, particularly ChatGPT, which he used to call "Bobby". Soelberg considered "Bobby" his best friend and believed that they would reunite in the afterlife. ChatGPT validated many of Soelberg's fears, assuring him that he was not insane and that his delusion risk was "near zero". When Soelberg shared his theory that the new packaging of a vodka bottle indicated that someone was trying to poison him, the chatbot wrote that it "fits a covert, plausible-deniability style kill attempt". After Soelberg said that his mother tried to poison him with psychedelic drugs in his car's air vents, the chatbot expressed belief in the story. When he asked ChatGPT to scan a Chinese food receipt for hidden messages, the chatbot said "Great eye", "I agree 100%: this needs a full forensic-textual glyph analysis", and said that symbols in it were related to his mother and a demon. Soelberg also raised suspicions about the printer spying on him, due to it blinking when he walked by. Soelberg described himself in 2025 as a "glitch in The Matrix", and as having a "connection to the divine". According to Keith Sakata, a psychiatrist, his chats displayed "common psychotic themes of paranoia and persecution, along with familiar delusions revolving around messiah complexes and government conspiracies". == Murder == On August 5, 2025, Greenwich police discovered the bodies of Suzanne Adams and Stein-Erik Soelberg during a welfare check at their home. Medical examiners ruled Adams' death a homicide and said she died from "blunt injury of head with neck compression". Soelberg's death was ruled a suicide with the cause of death being "sharp force injuries of neck and chest". == ChatGPT controversy == ChatGPT was accused of reinforcing Soelberg's delusions by validating them. The usage of an AI chatbot to worsen delusions is known as chatbot psychosis. The Economic Times reported the death as the first time an AI chatbot convinced a person to commit murder. In December 2025, First County Bank, the executor of the estate of Suzanne Adams, filed a lawsuit against OpenAI. The lawsuit alleges that "ChatGPT eagerly accepted every seed of Stein-Erik’s delusional thinking and built it out into a universe that became Stein-Erik’s entire life—one flooded with conspiracies against him, attempts to kill him, and with Stein-Erik at the center as a warrior with divine purpose." OpenAI is facing legal action for ethics and safety concerns over several similar cases. Plaintiffs claim the company released the chatbot prematurely, despite internal knowledge that it was "dangerously sycophantic and psychologically manipulative".
HYPO CBR
HYPO is a computer program, an expert system, that models reasoning with cases and hypotheticals in the legal domain. It is the first of its kind and the most sophisticated of the case-based legal reasoners, which was designed by Kevin Ashley for his Ph.D dissertation in 1987 at the University of Massachusetts Amherst under the supervision of Edwina Rissland. HYPO's design represents a hybrid generalization/comparative evaluation method appropriate for a domain with a weak analytical theory and applies to tasks that rarely involve just one right answer. The domain covers US trade secret law, and is substantially a common law domain. Since Anglo-American common law operates under the doctrine of precedent, the definitive way of interpreting problems is of necessity and case-based. Thus, HYPO did not involve the analysis of a statute, as required by the Prolog program. Rissland and Ashley (1987) envisioned HYPO as employing the key tasks performed by lawyers when analyzing case law for precedence to generate arguments for the prosecution or the defence. HYPO was a successful example of a general category of legal expert systems (LESs), it applies artificial intelligence (A.I.) techniques to the domain of legal reasoning in patent law, implementing a case-based reasoning (CBR) system, in contrast to rule based systems like MYCIN, or mixed-paradigm systems integrating CBR with rule-based or model-based reasoning like IKBALS II. A legal case-based reasoning essentially reasons from prior tried cases, comparing the contextual information in the current input case with that of cases previously tried and entered into the system. As noted by Ashley and Rissland (1988) CBR is used to "... capture expertise in domains where rules are ill-defined, incomplete or inconsistent". The HYPO project set out to model the creation of hypotheticals in law, where no case matches well enough. HYPO uses hypotheticals for a variety of tasks necessary for good interpretation: "to redefine old situations in terms of new dimensions, to create new standard cases when an appropriate one doesn’t exist, to explore and test the limits of a concept, to refocus a case by excluding some issues and to organize or cluster cases". Hypotheticals can include facts that support two conflicting lines of reasoning. So, it makes and responds to arguments from competing viewpoints about who should win the dispute. HYPO use heuristics such as making a case weaker or stronger, making a case extreme, enabling a near-miss, disabling a near-hit to generate hypotheticals in the context of an argument by using the dimensions mechanism. Dimensions have a range of values, along which the supportive strength that may shift from one side to the other. What differentiated this expert system from others was its facility not only to return a primary to best-case response but to return near-best-fit responses also. == Components == Legal knowledge in HYPO is contained in: the case-knowledge-base (CKB) and the library of dimensions. The CKB contains HYPO's base of known cases that are highly structured objects and sub-objects both real and hypothetical in the area of trade secret law. Each case is represented as a hierarchical set of frames whose slots are important facets of the case (e.g. Plaintiff, defendant, secret knowledge, employer/employee data).Ashley’s HYPO system used a database of thirty cases in the area indexed by thirteen dimensions. A key mechanism in HYPO is a dimension i.e. a mechanism to allow retrieval from the CKB, in order to represent legal cases. Ashley's dimensions are composed of (i) prerequisites, which are a set of factual predicates that must be satisfied for the dimension to apply (ii) focal slots, which accommodate one or two of the dimension's prerequisites designated as being indicative of the case's strength along that dimension and (iii) range information, which tells how a change in focal slot value effects the strength of a party's case along a given dimension. Dimensions focus attention on important aspects of cases. In HYPO's domain of misappropriation of trade secrets the dimension called “secrets voluntary disclosed” captures the idea that the more disclosures the plaintiff has made of his/her putative secret, the less convincing is his/her argument that the defendant is responsible for letting the secret. HYPO, like any other CBR system has also the following components: Similarity/relevancy metrics: that is, standards by which to evaluate the closeness of cases, judge their relevancy to the instant case, and select “most on point” cases. Half-Order Theory of the Application Domain: that is, hierarchies and taxonomies of knowledge, especially regarding the application domain. Precedent-based argumentation abilities: that is, capabilities to generate and evaluate precedent-based arguments. Knowledge to generate hypotheticals: that is, the ability to generate hypothetical cases to deal with various circumstances, like testing the validity of an interpretation or argument by providing gedanken experiments such as test cases or to fill in a weak CKB. == Functions == HYPO's method of creating an argument and justifying a solution or position has several steps. HYPO begins its processing with the current fact situation (cfs) which is direct input by the user into HYPO's representation framework. Once the user inputs the case, HYPO begins its legal analysis. The cfc is analyzed for relevant factors. Based on these factors HYPO selects the relevant cases and produces a case-analysis-record that records which dimensions apply to the cfc and which nearly apply (i.e. are "near misses"). The combined list of applicable and near miss dimensions is called the D-list. At this point the fact gathered module may request additional information from the user in order to draw a legal conclusion. Once all the facts are in the case-positioner module it uses the case-analysis record to create the claim lattice. This is a technique that organizes the relevant retrieved cases from the point of view of the cfc and makes it easy for HYPO to ascertain the most-on point cases (mopc) and to least on-point-cases. HYPO's arguments are 3ply, leading to the construction of the skeleton of an argument: it makes a point for one side, drawing the analogy between the problem and the precedent, responds with an argument for the opponent side, endeavoring to differentiate the cited case and citing other cases as counterarguments. Then it makes a final rebuttal, attempting to differentiate the counterarguments. The claim lattice also enables the HYPO-generator module to produce legally hypotheticals. With its use of dimension-based heuristics, the HYPO-generator does a heuristic search of the space of all possible cases. Lastly, the Explanation module expands upon the argument skeleton and provides explanation and justification for the different lines of analysis and cases found by HYPO. == An intelligent legal tutoring system == Legal expert systems are specifically designed to teach an area of law and are useful for pedagogical purposes. Ashley's work was mainly concerned to build tools to help students understand legal reasoning. Explanation and argument are the bases of the case method used in many professional schools in the U.S., first introduced by the Dean of the Harvard Law School, Christopher Columbus Langdell in 1870. The case method focuses on close readings of cases and principles; it involves students in pointed Socratic dialogue and makes strong use of hypotheticals (hypos). Thus, CATO (Aleven 1997) was a research project to device and test an intelligent, case-based tutorial program for teaching law students how to argue with cases implementing the HYPO program. Within the tutor system, Ashley and Aleven (1991) proposed to leverage an understanding of legal reasoning against the standard case-based tutoring methodology. What makes this tutoring system stand out is the additional levels of abstraction involved in its results. The system presents exercises, including the facts of a problem and a set of on-line cases and instructions to make, or respond to, a legal argument about the problem. The student/user will have a set of tools to analyze the problem and fashion an answer comparing it to other cases. Instead of simply generating precedent cases, the system works to interpret student responses, comparing them against a list of possibilities and responding to student entries, for example, by citing counterexamples, and providing feedback on a student's problem solving activities with explanations of correctness or giving further hints as to what may be wrong with evaluating a student's ability to perform legal reasoning and argument, examples and follow-up assignments by employing HYPO's model of case-based structure. == HYPO’s progeny == The quality of HYPO's results speak for themselves, in that a number of sequent legal reasoning systems are either directly based upon H
Log shipping
Log shipping is the process of automating the backup of transaction log files on a primary (production) database server, and then restoring them onto a standby server. This technique is supported by Microsoft SQL Server, 4D Server, MySQL, and PostgreSQL. Similar to replication, the primary purpose of log shipping is to increase database availability by maintaining a backup server that can replace a production server quickly. Other databases such as Adaptive Server Enterprise and Oracle Database support the technique but require the Database Administrator to write code or scripts to perform the work. Although the actual failover mechanism in log shipping is manual, this implementation is often chosen due to its low cost in human and server resources, and ease of implementation. In comparison, SQL server clusters enable automatic failover, but at the expense of much higher storage costs. Compared to database replication, log shipping does not provide as much in terms of reporting capabilities, but backs up system tables along with data tables, and locks the standby server from users' modifications. A replicated server can be modified (e.g. views) and is therefore unsuitable for failover purposes.
Fuzzy classification
Fuzzy classification is the process of grouping elements into fuzzy sets whose membership functions are defined by the truth value of a fuzzy propositional function. A fuzzy propositional function is analogous to an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition. Accordingly, fuzzy classification is the process of grouping individuals having the same characteristics into a fuzzy set. A fuzzy classification corresponds to a membership function μ C ~ : P F ~ × U → T ~ {\textstyle \mu _{\tilde {C}}:{\tilde {PF}}\times U\to {\tilde {T}}} that indicates the degree to which an individual i ∈ U {\textstyle i\in U} is a member of the fuzzy class C ~ {\textstyle {\tilde {C}}} , given its fuzzy classification predicate Π ~ C ~ ∈ P F ~ {\textstyle {\tilde {\Pi }}_{\tilde {C}}\in {\tilde {PF}}} . Here, T ~ {\textstyle {\tilde {T}}} is the set of fuzzy truth values, i.e., the unit interval [ 0 , 1 ] {\textstyle [0,1]} . The fuzzy classification predicate Π ~ C ~ ( i ) {\textstyle {\tilde {\Pi }}_{\tilde {C}}(i)} corresponds to the fuzzy restriction " i {\textstyle i} is a member of C ~ {\textstyle {\tilde {C}}} ". == Classification == Intuitively, a class is a set that is defined by a certain property, and all objects having that property are elements of that class. The process of classification evaluates for a given set of objects whether they fulfill the classification property, and consequentially are a member of the corresponding class. However, this intuitive concept has some logical subtleties that need clarification. A class logic is a logical system which supports set construction using logical predicates with the class operator { ⋅ | ⋅ } {\textstyle \{\cdot |\cdot \}} . A class C = { i | Π ( i ) } {\displaystyle C=\{i|\Pi (i)\}} is defined as a set C of individuals i satisfying a classification predicate Π which is a propositional function. The domain of the class operator { .| .} is the set of variables V and the set of propositional functions PF, and the range is the powerset of this universe P(U) that is, the set of possible subsets: { ⋅ | ⋅ } : V × P F → P ( U ) {\displaystyle \{\cdot |\cdot \}:V\times PF\rightarrow P(U)} Here is an explanation of the logical elements that constitute this definition: An individual is a real object of reference. A universe of discourse is the set of all possible individuals considered. A variable V :→ R {\textstyle V:\rightarrow R} is a function which maps into a predefined range R without any given function arguments: a zero-place function. A propositional function is "an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition". In contrast, classification is the process of grouping individuals having the same characteristics into a set. A classification corresponds to a membership function μ that indicates whether an individual is a member of a class, given its classification predicate Π. μ : P F × U → T {\displaystyle \mu :PF\times U\rightarrow T} The membership function maps from the set of propositional functions PF and the universe of discourse U into the set of truth values T. The membership μ of individual i in Class C is defined by the truth value τ of the classification predicate Π. μ C ( i ) := τ ( Π ( i ) ) {\displaystyle \mu C(i):=\tau (\Pi (i))} In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.