Avid DS

Avid DS (which was called Avid DS Nitris until early 2008) is a high-end offline and finishing system comprising a non-linear editing system and visual effects software. It was developed by Softimage (this company was owned by Microsoft at the time of DS v1.0's launch before being acquired from Microsoft by Avid Technology, Inc. shortly thereafter) in Montreal. DS was discontinued on September 30, 2013 with support ending on the same date the following year. == Software == DS was called ‘Digital Studio’ in development. It was envisioned to be a complete platform for video/audio work. The first previews of the system were on the SGI platform, but this version was never released. The system was rewritten on Windows NT with different video hardware platforms (Matrox DigiSuite or Play Trinity running on a NetPower system) before the final system was released on Intergraph/StudioZ hardware in January 1998. After its acquisition by Avid, DS was always positioned as a high end video finishing tool. However, many users found it to be uniquely soup-to-nuts in its capabilities. From version 1.0 of the product, it competed with products like Autodesk Smoke, Quantel and Avid Symphony. The toolset in DS offered video timeline editing, an object-oriented vector-based paint tool, 2D layer compositing, sample based audio and starting with version 3.01 of the product, a 3D environment. Originally, a subset of the Softimage|XSI 3D software was planned to become part of the DS toolset, both were built on the same software foundation, but over time the code bases divided between the applications and the integration never happened. While the first version of the DS still lacked a few key features (no 3D, poor keying, no real-time effects), it had some significant features compared to the competing products at the time. It offered a large number of built in effects. Avid OMF import was available, positioning Softimage DS as a strong finishing tool for then typical off-line Avid systems. Lastly the integration of the toolset of Softimage DS was beyond what other product offered. A Softimage DS user could quickly go from editing, to paint, to compositing with a few mouse clicks all inside the same interface. Some of the lacking features were quickly resolved, within months of version 1.0 a new chroma keyer was released. Early versions of the software (up thru 4.0) added additional key features. Development continued with one of the first uncompressed HD editing systems (version 4.01) and an attempt to make the system more friendly to Media Composer editors in version 6. In later versions (v7.5 on beyond) DS was criticized for slow development of compositing tools, mainly lack of a new 3D environment and better tracking tools. Many DS users felt that Avid had not been giving DS the attention that it deserved. On July 7, 2013, Avid sent out an email marking the end of life of the DS product. "To Our Avid DS customers, We are writing to inform you that Avid will be realigning our business strategy to focus on a core suite of products to best leverage our developmental and creative resources. As part of this transition, we will be ceasing future development of Avid DS with a final sale date of September 30th, 2013" == Hardware == Up until version 10.5, DS was sold as a turn-key system; the software was not available without purchasing CPU, I/O and storage hardware from Avid. Beginning with 10.5, customers were able to configure their own systems using widely available components, based on recommended system requirements. In turn-key systems, there were many hardware refreshes over time. StudioZ single stream: Intergraph TDZ-425 with 30 minutes of uncompressed SCSI storage. CPUs at the time were Pentium II/300 MHz. StudioZ dual stream: Intergraph TDZ-2000 GT1 with one hour of fibre channel storage. CPUs on first systems were Pentium II/400 MHz, but last shipping systems had Pentium III/1 GHz. DS was one of the first applications to show that real-time effects could be processed with just the CPUs of the system, not requiring special video cards with real-time effect hardware. Equinox: Developed by Avid, it was one of the first uncompressed HD video cards available. Systems were available on CPUs from Pentium III/1 GHz to Pentium 4/2.8 GHz. Storage was typically SCSI, but fibre channel was also supported. Nitris DNA: Developed by Avid, the Nitris hardware was probably the largest hardware update to the system since it was released. 10-bit HD and SD support was standard. Real-time down and cross convert. This was the only hardware for DS that had on-board effect processing. This allowed a system at the time to play back dual-stream uncompressed HD effects in real-time at 16-bit precision. This was also the first hardware from Avid to support the DNxHD codec. Starting with Pentium 4, Intel Core Xeons were supported. SCSI storage was primarily used. AJA Video Systems: First available as a 4:4:4 option to be used in conjunction with Nitris hardware. Final-generation DS systems used the AJA Video Systems Kona 3 (Xena 2K) card as the only I/O for the system. The last systems shipped with two Intel Core Xeon 6-core processors. SAS is the recommended storage for these systems. == History ==

VueScan

VueScan is a computer program for image scanning, especially of photographs, including negatives. It supports optical character recognition (OCR) of text documents. The software can be downloaded and used free of charge, but adds a watermark on scans until a license is purchased. == Purpose == VueScan is intended to work with a large number of image scanners, excluding specialised professional scanners such as drum scanners, on many computer operating systems (OS), even if drivers for the scanner are not available for the OS. These scanners are supplied with device drivers and software to operate them, included in their price. A 2014 review considered that the reasons to purchase VueScan are to allow older scanners not supported by drivers for newer operating systems to be used in more up-to-date systems and for better scanning and processing of photographs (prints; also slides and negatives when supported by scanners) than is afforded by manufacturers' software. The review did not report any advantages to VueScan's processing of documents over other software. The reviewer considered VueScan comparable to SilverFast, a similar program, with support for some specific scanners better in one or the other. Vuescan supports more scanners, with a single purchase giving access to the full range of both film and flatbed scanners, and costs less. The VueScan program can be used with its own drivers or with drivers supplied by the scanner manufacturer, if supported by the operating system. VueScan drivers can also be used without the VueScan program by application software that supports scanning directly, such as Adobe Photoshop, again enabling the use of scanners without current manufacturers' drivers. In 2019 when Apple released macOS Catalina, they removed support for running 32-bit programs, including 32-bit drivers for scanning equipment. In response, Hamrick released VueScan 9.7, effectively saving thousands of scanners from being rendered obsolete. == Overview == VueScan enables the user to modify and fine-tune the scanning parameters. The program uses its own independent method to interface with scanner hardware, and can support many older scanners under computer operating systems for which drivers are not available, allowing old scanners to be used with newer platforms that do not otherwise support them. VueScan supports an increasing number of scanners and digital cameras; 2,400 on Windows, 2,100 on Mac OS X and 1,900 on Linux in 2018. VueScan is supplied as one downloadable file for each operating system, which supports the full range of scanners. Without the purchase of a license, the program runs in fully functional demonstration mode, identical to Professional mode, except that watermarks are superimposed on saved and printed images. Purchase of a license removes the watermark. A standard license allows updates for one year; a professional license allows unlimited updates and provides some additional features. VueScan supports optical character recognition (OCR), with English included, and 32 additional language packages available on its website. In September 2011, VueScan co-developer Ed Hamrick said that he was selling US$3 million per year of VueScan licenses.

Tales from the Loop (role-playing game)

Tales from the Loop (Swedish: Ur Varselklotet), subtitled "Roleplaying in the '80s That Never Was", is an alternative history science fiction tabletop role-playing game published in 2017 by Free League Publishing, the international arm of Swedish game and book publisher Fria Ligan AB, and Modiphius Entertainment. The game, based on the art of Simon Stålenhag, envisions an alternative world where a group of bored and ignored preteens and teens solve mysteries caused by new technology near their hometown. == Description == === Setting === Tales from the Loop is set in an alternative history world taken from the artwork of Simon Stålenhag. According to this alternative timeline, back in the 1940s, research began on particle accelerators. In the 1960s, two massive underground particle accelerators were built in Sweden and Colorado with the promise of a harvest of technological marvels that would change everyone's lives. Tales from the Loop is set twenty years later, in the late 1980s, and the better life has not materialized. Although the particle accelerators have created robots and large skyships, the detritus of failed experiments and the ruins of abandoned high tech company buildings litter the landscape. Generally the life of the average family has not changed for the better. A campaign can either be set in the Mälaren Islands, west of the Swedish capital of Stockholm, or in a city in the Southwest United States that resembles Boulder City, Nevada. There is also a step-by-step guide for the gamemaster to use their own hometown. === Character generation === Player characters are preteens and young teenagers age 10–15 who live in a society where they are bored and largely left to themselves. Players can choose archetypes for their characters including Bookworm, Jock, Troublemaker, Popular Kid and Weirdo. Unlike most role-playing games, characters in Tales from the Loop cannot be killed, although in an ongoing campaign or due to an in-game effect, they are removed from the game if they reach the age of sixteen. === Game system === The game uses the Year Zero Engine first developed by Tomas Härenstam for the post-apocalyptic role-playing game Mutant: Year Zero. (Härenstam served as the editor and project manager for Tales from the Loop.) Problems are resolved by rolling a pool of six-sided dice, with any 6 rolled marking success. Attributes and skills (Sneak, Force, Move, Build, Tinker, Calculate, Contact, Charm, Lead, Investigate, Comprehend, and Empathize) may allow the player to add more dice to the dice pool, increasing the chances of success. However, if a character has earned a condition such as Scared or Injured, dice are removed from the dice pool. === Gameplay === The game principles are that life for the characters is dull and boring, but the area around the town is full of wonderful, mysterious things. An adventure is set up as a Mystery, and in order to successfully resolve the Mystery, characters must overcome a series of Troubles, which can range from having to be home by a certain time to dealing with a bully to disarming or otherwise overcoming a booby-trap on a door that must be opened. Each Mystery is played as a series of scenes, much like a TV drama. Although the gamemaster leads the players into the Mystery, each scene is set collaboratively with the players before action continues. As critic Jukka Kauppinen noted, "The players and the gamemaster take turns verbally staging a new scene — where we are, what it's like there — and only then what we do." === Campaign === The book presents a chronologically-linked set of four Mysteries called "The Four Seasons of Mad Science" that take place over a calendar year: "Summer Break and Killer Birds": The Kids hears pigeons having a conversation and investigate "Grown-Up Attraction": Adults start disappearing without any sign of struggle. "Creatures from the Cretaceous": The search for a missing dog leads to the discovery of creatures that don't belong in our time "I, Wagner": The Kids discover a body in a stream, and are drawn into a Mystery with robots and humans that may affect them closely. == Publication history == In 2017, Swedish artist Simon Stålenhag was raising money on Kickstarter to publish a book of his art titled Tales from the Loop. One of the stretch goals offered was the creation of a role-playing game. A second Kickstarter campaign to publish the role-playing game was initiated by Fria Ligan AB, who surpassed their crowdfunding goal and raised a total of 3,745,896 kr from 5,600 backers. The role-playing game Tales from the Loop was subsequently published as a 184-page hardcover book in 2017 by Free League Publishing, the international arm of Swedish game and book publisher Fria Ligan AB, and Modiphius Entertainment. Cover art and interior art were by Stålenhag, and cartography was by Christian Granath. A stand-alone expansion, Things from the Flood (Swedish: Flodskörden), based on Stålenhag's art book of the same name, was created by Nils Hintze, Rickard Antroia, and Tomas Härenstam. The 216-page hardcover book was published in 2019 with cover art by Stålenhag, interior art by Stålenhag and Reine Rosenberg, and cartography by Christian Granath. In 2020, the setting of the role-playing game was transferred to the TV series Tales from the Loop developed by Nathanial Halpern and Simon Stålenhag. The series tells eight stories of children's encounters with strange technology. == Reception == Shut Up & Sit Down praised Tales from the Loop for its comfortable, contemporary setting, simple rules that make the game easy to run, and the alternation between sci-fi and the kids' lives, but criticized the Type system for characters, noting "a suggested 'Pride' for the Weirdo involved being homosexual –– the only mention of queerness in the entire game. Those of us who identify as GLBTQ bristled at that: why was only the Weirdo queer, with queerness as a (possibly secret) Pride? Why not more fully address being a GLBTQ kid in the 1980s?" The review concluded, "For new RPG players, Tales is a decent game that you'll enjoy and that will make your heart burst. But you need an experienced GM who’s able to either alter the book’s mysteries or create their own, and who can put in work when poor dice rolls hold the players back." Rob Weiland of Geek & Sundry named Tales from the Loop 2017's best RPG release and praised Stålenhag's art, the collaborative nature between the GM and players, and the simplicity of running the game. Weiland concluded, "It has a simple system that is easy to explain but holds up under several plays. It has a setting that’s immediately evocative but also leaves plenty of room for GMs to build out their own world. It offers players a chance to experience the rush of memory, the pain of childhood and the wonder of movies." In a review of Tales from the Loop in Black Gate, Andrew Zimmerman Jones said, "Though not based directly on an established franchise, it draws richly from elements of popular culture that will make it resonate with many players. The focus on narrative play also means it’s a good game for people who aren’t necessarily big into learning a ton of new rules." Jukka Kauppinen, writing for the Finnish games magazine Skrolli, called the game, "downright delicious in its diversity. The science fiction world created by the Swedish artist Simon Stälenhag is, after all, both delightful vintage and tickling novelty." Kauppinen concluded, "This mutual storytelling and interaction makes this game more of a campfire circle than a traditional role-playing game. At the same time, its setting in the real world, tinged with science fiction and even horror, creates a delicious and unique adventure environment." In his 2023 book Monsters, Aliens, and Holes in the Ground, RPG historian Stu Horvath noted that the game system "pushes the players to constantly reevaluate their characters' relationships with the everyday world, for better or worse. It won't be long before navigating entanglements with parents, teachers, siblings and bullies proves just as risky to the characters, and central to the players' experience, as trying to find out what happened with the time portal or dealing with a rampaging robot." Horvath concluded, "The appeal of Tales from the Loop is Stålenhag's deep shadows and purple dusks. They hide the dangers and mysteries that often act [as] an escape hatch, a way to avoid prosaic problems." == Awards == At the 2017 Golden Geek Awards, Tales of the Loop won "RPG of the Year", and was a finalist for " Best RPG Artwork/Presentation" At the 2017 ENnie Awards, Tales from the Loops won five Gold Medals: Product of the Year Best Writing Best Setting Best Game Best Art, Interior

Split Up (expert system)

Split Up is an intelligent decision support system, which makes predictions about the distribution of marital property following divorce in Australia. It is designed to assist judges, registrars of the Family Court of Australia, mediators and lawyers. Split Up operates as a hybrid system, combining rule – based reasoning with neural network theory. Rule based reasoning operates within strict parameters, in the form: IF < condition(s) > then . Neural networks, by contrast, are considered to be better suited to generate decisions in uncertain domains, since they can be taught to weigh the factors considered by judicial decision makers from case data. Yet, they do not provide an explanation for the conclusions they reach. Split_up, with a view to overcome this flaw, uses argument structures proposed by Toulmin as the basis for representations from which explanations can be generated. == Application == In Australian family law, a judge in determining the distribution of property will: identify the assets of the marriage included in the common pool establish what percentage of the common pool each party will receive determine a final property order in line with the decisions made in 1. and 2. Split_Up implements step 1 and 2 : the common pool determination and the prediction of a percentage split. === The common pool determination === Since the determination of marital property is rule based, it is implemented using directed graphs. However, the percentage split between the parties is discretionary in that a judge has a wide discretion to look at each party's contributions to the marriage under section 79(4) of the Family Law Act 1975. Broadly, the contributions can be taken as financial or non-financial. The party who can demonstrate a larger contribution to the marital relationship will receive a larger proportion of the assets. The court may further look at each party's financial resources and future needs under section 75(2)of the Family Law Act 1975. These needs can include factors such as the inability to gain employment, the continued care of a child under 18 years of age or medical expenses. This means that different judges may and will reach different conclusions based on the same facts, since each judge assigns different relevant weights to each factor. Split_up determines the percentage split by using a combination of rule- based reasoning and neural networks. === The percentage split determination === In order to determine how judges weigh the different factors, 103 written judgements of commonplace cases were used to establish a database comprising 94 relevant factors for percentage split determination. The factors relevant for a percentage split determination are: Past contributions of a husband relative to those of a wife The husband's future needs relative to those of the wife The wealth of the marriage The factors relevant for a determination of past contributions are The relative direct and indirect contributions of both parties The length of the marriage The relative contributions of both parties to the homemaking role The hierarchy provides a structure that is used to decompose the task of predicting an outcome into 35 subtasks. Outputs of tasks further down the hierarchy are used as inputs into sub-tasks higher up the hierarchy. Each sub-task is treated as a separate and smaller data mining exercise. Twenty one solid arcs represent inferences performed with the use of rule sets. For example, the level of wealth of a marriage is determined by a rule, which uses the common pool value. By contrast, the fourteen dashed arcs establish inferences performed with the use of neural networks. These receive their name from the fact that they resemble a nervous system in the brain. They consist of many self – adjusting processing elements cooperating in a densely interconnected network. Each processing element generates a single output that is transmitted to the other processing element. The output signal of a processing element depends on the input to the processing element, i.e. each input is gated by a weighting factor that determines the amount of influence that the input will have on the output. The strength of the weighting factors is adjusted autonomously by the processing element as the data is processed. In Split_Up, the neural network is a statistical technique for learning the weights of each of the relevant attributes used in a percentage split determination of marital property. Hence the inputs to the neural network are contributions, future needs and wealth, and the output the percentage split predicted. On each arc there is a statistical weight. Using back propagation the neural network learns the necessary pattern to recognize the prediction. It is trained by repeatedly exposing it to examples of the problem and learning the significance (weights) of the input nodes. The neural network used by Split_up is said to generalise well if the output of the network is correct (or nearly correct) for examples not seen during training, which classifies it as an intelligent system. === Toulmin Argument Structure === Since the manner in which these weights are learned is primarily statistical, domain knowledge of legal rules and principles is not modelled directly. However, explanations for a legal conclusion in a domain as discretionary as the determining the distribution of property following divorce, are at least as important as the conclusion reached. Hence the creators of Split_Up used Toulmin Argument structures, to provide independent explanations of the conclusions reached. These operate on the basis that every argument makes an assertion based on some data. The assertion of the argument stands as the claim of the argument. Since knowing the data and the claim, does not necessarily mean that the claim follows from the data, a mechanism is required to justify the claim in the light of the data. The justification is known as the warrant. The backing of an argument supports the validity of the warrant. In the legal domain, this is typically a reference to a statute or a precedent. Here, a neural network (or rules), produce a conclusion from the data of an argument and the data, warrant and backing are reproduced to generate an explanation. It is noteworthy, though, that an argument's warrant is reproduced as an explanation regardless of the claim values used. This lack of claim - sensitivity must be overcome by the different users, i.e., the judge, the representatives for the wife and the representatives for the husband, each of whom is encouraged to use the system to prepare their cases, but not to rely exclusively on its outcome.

Pulsar (social listening platform)

Pulsar is a software platform for social media monitoring, audience intelligence and social listening that allows organizations to monitor and analyze online conversations across social media, news, and other digital sources. The platform combines social media listening, media monitoring, trend analysis, and audience segmentation to help users understand public discussions and audience behavior in real time. The platform is a social listening platform, which aggregates data from networks such as X, Facebook, Instagram, and forums) and applies artificial intelligence for text and sentiment analysis. Pulsar is offered as a cloud-based Software as a Service (SaaS) tool and insights consultancy. It has been part of Pulsar Group (formerly Access Intelligence), a publicly listed group of communications software products, since 2019. As well as commercial uses, the platform has been used in peer-reviewed academic research analysing online discourse. The platform is listed on the UK government's G-Cloud 14 Digital Marketplace for the provision of social listening and audience intelligence services. == History == Pulsar originated in the early 2010s as a project within Face, a London-based innovation and market research consultancy. The platform's first product, Pulsar TRAC, launched in 2013 as a social media analytics tool. Pulsar TRAC was designed to measure the reach of conversations, mapping brand audiences, and tracking how content spreads through networks. The development was led by Dr Francesco D'Orazio, who created the Pulsar brand and led the development of the platform while serving as VP of Product and Innovation at Face. Face itself had been acquired by the Cello Group Plc (a UK-based advisory firm) in 2012, and Pulsar became part of Cello's portfolio of research and data tools. In January 2017, Cello Group made a significant investment to scale Pulsar and announced the merger of Face's qualitative research business into Pulsar, unifying both under the Pulsar brand for global expansion. In 2018, Pulsar opened an office in Los Angeles to better serve its growing U.S. client base in media, healthcare, and entertainment sectors and Francesco D'Orazio was appointed CEO. The company focused on developing new products amid a wave of consolidation in the social listening industry. In October 2019, Pulsar was acquired by Access Intelligence Plc (now Pulsar Group), an AIM-listed communications software company. The group, which also owns PR and media tools Isentia, Vuelio and ResponseSource, integrated Pulsar to their end-to-end marketing and communications insights offering. Pulsar established a new office in Sydney, Australia in 2022 as part of this global expansion, adding to its existing offices in London and Los Angeles. In 2023, Pulsar Group (then Access Intelligence) was recognised as one of Europe's fastest growing companies by the Financial Times. In May 2024, Access Intelligence PLC changed its name to Pulsar Group PLC. The company has since continued to develop its platform. In March 2025 it introduced new tool Narratives AI, described as a "search engine for public opinion" and the first of its kind for analyzing public narratives and their evolutions in both social media and the news. In October 2025, Pulsar launched Insight Agents, a set of AI agents embedded into the platform advertised to "proactively anticipate user needs or issues, carry out routine tasks, uncover anomalies in your datasets, and prompt responses at scale, 24/7." == Products == Pulsar's architecture integrates four main products into a single interface. The core product suite is often broken into three main components: Pulsar TRAC (for social listening and audience analysis), Pulsar TRENDS (for trend discovery and analysis), and Pulsar CORE (for owned-channel and web analytics). Pulsar's fourth product is Narratives AI. === Pulsar TRAC === Pulsar TRAC is a social listening and audience intelligence platform that allows users to configure searches that track public conversations and measure audience behaviour. Pulsar TRAC is focused on conversation insights and audience segmentations - the platform is reported to collect and analyse data from a wide range of sources, including major social networks, forums, news and review sites, and ecommerce platforms, with real-time visualisations and AI-supported analytics used to find patterns and communities of interest. Pulsar TRAC can be incorporated into workflows with other audience tools, such as an integration with Audiense that connects TRAC's conversation insights to external audience-segmentation datasets. === Pulsar CORE === Pulsar CORE centres on the analysis of owned-channel data, such as brand social media profiles, website interaction and other in-house digital assets, to generate audience and content insights. CORE can monitor published content, evaluate competitors, and extract demographic and behavioural segmentation from owned channels. === Narratives AI === Narratives AI is a tool within the Pulsar audience intelligence platform that uses artificial intelligence to detect, cluster and analyse narratives forming across social and news media. It was launched in March 2025 as a standalone search interface that processes real-time and historical data to find cultural trends, behaviours and beliefs. It uses clustering algorithms and visualisation to show how conversations form and spread online, and their relative importance within wider discourse. == Notable features == === Insight Agents === Pulsar's Insight Agents are AI-powered agents within the Pulsar platform designed to automate and augment common tasks in media, social, audience and narrative intelligence. Branded as TeamMates, these agents are grouped into four functional types: Sentinels for real-time monitoring, anomaly detection and alerting Oracles for forecasting and scenario planning Custodians for governance, compliance and policy enforcement Analysts for research, reporting and recommendations Each agent is trained on Pulsar's multi-source data and domain-specific workflows. In February 2026, Pulsar introduced 'Crisis Oracle,' an AI-driven system designed to quantify narrative momentum and predict reputational risk. == Academic research == Pulsar has been used as a data collection and analysis tool in peer-reviewed academic research across public health, infodemiology, veterinary science, and policy research. Published uses include a World Health Organization report on infodemic management, a Journal of Medical Internet Research study on headache and migraine discourse across Japan, Germany, and France, a Frontiers in Big Data study of Long COVID narratives, and Frontiers in Veterinary Science studies on canine chronic kidney disease and oral medication administration in dogs.

TAPPS2

TAPPS2 (Technische Alternative Planungs- und Programmier-System) is a tool used for developing the program logic for the universal, heating and solar thermal controllers by Austrian manufacturer Technische Alternative. Its primary usecase is defining the exact reaction of the controller to a certain event. Other than its predecessor, TAPPS, which could only be used to program controllers of type UVR1611, TAPPS2 is mainly used to program the UVR16x2 and RSM610 controllers, as well as several extension modules. == Development == Development in TAPPS2 is done on a vector-based drawing surface using components that can be placed via drag and drop. The components, which can be separated into inputs, functions and outputs are then being connected according to their individual features. Available components vary according to the current solar thermal control unit.

BL (logic)

In mathematical logic, basic fuzzy logic (or shortly BL), the logic of the continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic MTL of all left-continuous t-norms. == Syntax == === Language === The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives: Implication → {\displaystyle \rightarrow } (binary) Strong conjunction ⊗ {\displaystyle \otimes } (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation ⊗ {\displaystyle \otimes } follows the tradition of substructural logics. Bottom ⊥ {\displaystyle \bot } (nullary — a propositional constant); 0 {\displaystyle 0} or 0 ¯ {\displaystyle {\overline {0}}} are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL). The following are the most common defined logical connectives: Weak conjunction ∧ {\displaystyle \wedge } (binary), also called lattice conjunction (as it is always realized by the lattice operation of meet in algebraic semantics). Unlike MTL and weaker substructural logics, weak conjunction is definable in BL as A ∧ B ≡ A ⊗ ( A → B ) {\displaystyle A\wedge B\equiv A\otimes (A\rightarrow B)} Negation ¬ {\displaystyle \neg } (unary), defined as ¬ A ≡ A → ⊥ {\displaystyle \neg A\equiv A\rightarrow \bot } Equivalence ↔ {\displaystyle \leftrightarrow } (binary), defined as A ↔ B ≡ ( A → B ) ∧ ( B → A ) {\displaystyle A\leftrightarrow B\equiv (A\rightarrow B)\wedge (B\rightarrow A)} As in MTL, the definition is equivalent to ( A → B ) ⊗ ( B → A ) . {\displaystyle (A\rightarrow B)\otimes (B\rightarrow A).} (Weak) disjunction ∨ {\displaystyle \vee } (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics), defined as A ∨ B ≡ ( ( A → B ) → B ) ∧ ( ( B → A ) → A ) {\displaystyle A\vee B\equiv ((A\rightarrow B)\rightarrow B)\wedge ((B\rightarrow A)\rightarrow A)} Top ⊤ {\displaystyle \top } (nullary), also called one and denoted by 1 {\displaystyle 1} or 1 ¯ {\displaystyle {\overline {1}}} (as the constants top and zero of substructural logics coincide in MTL), defined as ⊤ ≡ ⊥ → ⊥ {\displaystyle \top \equiv \bot \rightarrow \bot } Well-formed formulae of BL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence: Unary connectives (bind most closely) Binary connectives other than implication and equivalence Implication and equivalence (bind most loosely) === Axioms === A Hilbert-style deduction system for BL has been introduced by Petr Hájek (1998). Its single derivation rule is modus ponens: from A {\displaystyle A} and A → B {\displaystyle A\rightarrow B} derive B . {\displaystyle B.} The following are its axiom schemata: ( B L 1 ) : ( A → B ) → ( ( B → C ) → ( A → C ) ) ( B L 2 ) : A ⊗ B → A ( B L 3 ) : A ⊗ B → B ⊗ A ( B L 4 ) : A ⊗ ( A → B ) → B ⊗ ( B → A ) ( B L 5 a ) : ( A → ( B → C ) ) → ( A ⊗ B → C ) ( B L 5 b ) : ( A ⊗ B → C ) → ( A → ( B → C ) ) ( B L 6 ) : ( ( A → B ) → C ) → ( ( ( B → A ) → C ) → C ) ( B L 7 ) : ⊥ → A {\displaystyle {\begin{array}{ll}{\rm {(BL1)}}\colon &(A\rightarrow B)\rightarrow ((B\rightarrow C)\rightarrow (A\rightarrow C))\\{\rm {(BL2)}}\colon &A\otimes B\rightarrow A\\{\rm {(BL3)}}\colon &A\otimes B\rightarrow B\otimes A\\{\rm {(BL4)}}\colon &A\otimes (A\rightarrow B)\rightarrow B\otimes (B\rightarrow A)\\{\rm {(BL5a)}}\colon &(A\rightarrow (B\rightarrow C))\rightarrow (A\otimes B\rightarrow C)\\{\rm {(BL5b)}}\colon &(A\otimes B\rightarrow C)\rightarrow (A\rightarrow (B\rightarrow C))\\{\rm {(BL6)}}\colon &((A\rightarrow B)\rightarrow C)\rightarrow (((B\rightarrow A)\rightarrow C)\rightarrow C)\\{\rm {(BL7)}}\colon &\bot \rightarrow A\end{array}}} The axioms (BL2) and (BL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012). == Semantics == Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for BL, with three main classes of algebras with respect to which the logic is complete: General semantics, formed of all BL-algebras — that is, all algebras for which the logic is sound Linear semantics, formed of all linear BL-algebras — that is, all BL-algebras whose lattice order is linear Standard semantics, formed of all standard BL-algebras — that is, all BL-algebras whose lattice reduct is the real unit interval [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous t-norm.