Boolean algebra in the context of Disjunction

Non-classical logics (and sometimes alternative logics or non-Aristotelian logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is commonly the case, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.

Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computerized proof methods may not use classical logic in the reasoning process.

Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra.

Binary data occurs in many different technical and scientific fields, where it can be called by different names including bit (binary digit) in computer science, truth value in mathematical logic and related domains and binary variable in statistics.

A relay is an electrically operated switch. It has a set of input terminals for one or more control signals, and a set of operating contact terminals. The switch may have any number of contacts in multiple contact forms, such as make contacts, break contacts, or combinations thereof.

Relays are used to control a circuit by an independent low-power signal and to control several circuits by one signal. They were first used in long-distance telegraph circuits as signal repeaters that transmit a refreshed copy of the incoming signal onto another circuit. Relays were used extensively in telephone exchanges and early computers to perform logical operations.

George Boole (/buːl/ BOOL; 2 November 1815 – 8 December 1864) was an English autodidact, mathematician, philosopher and logician who served as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854), which contains Boolean algebra. Boolean logic, essential to computer programming, is credited with helping to lay the foundations for the Information Age.

Boole was the son of a shoemaker. He received a primary school education and learned Latin and modern languages through various means. At 16, he began teaching to support his family. He established his own school at 19 and later ran a boarding school in Lincoln. Boole was an active member of local societies and collaborated with fellow mathematicians. In 1849, he was appointed the first professor of mathematics at Queen's College, Cork (now University College Cork) in Ireland, where he met his future wife, Mary Everest. He continued his involvement in social causes and maintained connections with Lincoln. In 1864, Boole died due to fever-induced pleural effusion after developing pneumonia.

Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.

The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by mathematician Lotfi Zadeh. Basic fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. The works of Zadeh and Joseph Goguen in the 1960's and 1970's went further by considering issues such as linguistic variables and lattices.

Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, computer scientist, cryptographer and inventor known as the "father of information theory" and the man who laid the foundations of the Information Age.

Shannon was the first to describe the use of Boolean algebra—essential to all digital electronic circuits—and helped found the field of artificial intelligence. Roboticist Rodney Brooks declared Shannon the 20th century engineer who contributed the most to 21st century technologies, and mathematician Solomon W. Golomb described his intellectual achievement as "one of the greatest of the twentieth century".

In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.

Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality$${\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}$$is always true in elementary algebra.For example, in elementary arithmetic, one has$${\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}$$Therefore, one would say that multiplication distributes over addition.

This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ${\displaystyle \,\land \,}$) and the logical or (denoted ${\displaystyle \,\lor \,}$) distributes over the other.

In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1,1}). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory.

A Boolean function takes the form ${\displaystyle f:\{0,1\}^{k}\to \{0,1\}}$, where ${\displaystyle \{0,1\}}$ is known as the Boolean domain and ${\displaystyle k}$ is a non-negative integer called the arity of the function. In the case where ${\displaystyle k=0}$, the function is a constant element of ${\displaystyle \{0,1\}}$. A Boolean function with multiple outputs, ${\displaystyle f:\{0,1\}^{k}\to \{0,1\}^{m}}$ with ${\displaystyle m>1}$ is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography).

In propositional logic and Boolean algebra, there is a duality between conjunction and disjunction, also called the duality principle. It is the most widely known example of duality in logic. The duality consists in these metalogical theorems:

• In classical propositional logic, the connectives for conjunction and disjunction can be defined in terms of each other, and consequently, only one of them needs to be taken as primitive.
• If ${\displaystyle \varphi ^{D}}$ is used as notation to designate the result of replacing every instance of conjunction with disjunction, and every instance of disjunction with conjunction (e.g. ${\displaystyle p\land q}$ with ${\displaystyle q\lor p}$, or vice-versa), in a given formula ${\displaystyle \varphi }$, and if ${\displaystyle {\overline {\varphi }}}$ is used as notation for replacing every sentence-letter in ${\displaystyle \varphi }$ with its negation (e.g., ${\displaystyle p}$ with ${\displaystyle \neg p}$), and if the symbol ${\displaystyle \models }$ is used for semantic consequence and ⟚ for semantical equivalence between logical formulas, then it is demonstrable that ${\displaystyle \varphi ^{D}}$ ⟚ ${\displaystyle \neg {\overline {\varphi }}}$, and also that ${\displaystyle \varphi \models \psi }$ if, and only if, ${\displaystyle \psi ^{D}\models \varphi ^{D}}$, and furthermore that if ${\displaystyle \varphi }$ ⟚ ${\displaystyle \psi }$ then ${\displaystyle \varphi ^{D}}$ ⟚ ${\displaystyle \psi ^{D}}$. (In this context, ${\displaystyle {\overline {\varphi }}^{D}}$ is called the dual of a formula ${\displaystyle \varphi }$.)

In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted true and false) which is intended to represent the two truth values of logic and Boolean algebra. It is named after George Boole, who first defined an algebraic system of logic in the mid-19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions by changing control flow depending on whether a programmer-specified Boolean condition evaluates to true or false. It is a special case of a more general logical data type—logic does not always need to be Boolean (see probabilistic logic).

Logical equality is a logical operator that compares two truth values, or more generally, two formulas, such that it gives the value True if both arguments have the same truth value, and False if they are different. In the case where formulas have free variables, we say two formulas are equal when their truth values are equal for all possible resolutions of free variables. It corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus.

It is customary practice in various applications, if not always technically precise, to indicate the operation of logical equality on the logical operands x and y by any of the following forms: