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In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...
connective \lor can be used to join the two
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformul ...
s P and Q, rendering the complex formula P \lor Q . Common connectives include negation,
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
,
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...
, and implication. In standard systems of
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
, these connectives are interpreted as
truth function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly o ...
s, though they receive a variety of alternative interpretations in
nonclassical logic Non-classical logics (and sometimes alternative 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 done, including by way of ...
s. Their classical interpretations are similar to the meanings of natural language expressions such as
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
"not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a classical compositional semantics with a robust pragmatics. A logical connective is similar to, but not equivalent to, a syntax commonly used in programming languages called a
conditional operator The conditional operator is supported in many programming languages. This term usually refers to ?: as in C, C++, C#, and JavaScript. However, in Java, this term can also refer to && and , , . && and , , In some programming languages, e.g. Jav ...
.


Overview

In formal languages, truth functions are represented by unambiguous symbols. This allows logical statements to not be understood in an ambiguous way. These symbols are called ''logical connectives'', ''logical operators'', ''propositional operators'', or, in
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
, ''
truth-functional In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
connectives''. For the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives, see well-formed formula. Logical connectives can be used to link zero or more statements, so one can speak about '' -ary logical connectives''. The
boolean Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean. Related to this, "Boolean" may refer to: * Boolean data type, a form of data with only two possible values (usually "true" and "false" ...
constants ''True'' and ''False'' can be thought of as zero-ary operators. Negation is a 1-ary connective, and so on.


Common logical connectives


List of common logical connectives

Commonly used logical connectives include: * Negation (not): ¬ , N (prefix), ~ * Conjunction (and): ∧ , K (prefix), & , ∙ * Disjunction (or): ∨, A (prefix) * Material implication (if...then): → , C (prefix), ⇒ , ⊃ * Biconditional (if and only if): ↔ , E (prefix), ≡ , = Alternative names for biconditional are ''
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
'', ''
xnor The XNOR gate (sometimes XORN'T, ENOR, EXNOR or NXOR and pronounced as Exclusive NOR. Alternatively XAND, pronounced Exclusive AND) is a digital logic gate whose function is the logical complement of the Exclusive OR (XOR) gate. It is equivale ...
'', and ''bi-implication''. For example, the meaning of the statements ''it is raining'' (denoted by ''P'') and ''I am indoors'' (denoted by Q) is transformed, when the two are combined with logical connectives: * It is not raining (''P'') * It is raining and I am indoors (P \wedge Q) * It is raining or I am indoors (P \lor Q) * If it is raining, then I am indoors (P \rightarrow Q) * If I am indoors, then it is raining (Q \rightarrow P) * I am indoors if and only if it is raining (P \leftrightarrow Q) It is also common to consider the ''always true'' formula and the ''always false'' formula to be connective: * True formula (⊤, 1, V refix or T) * False formula (⊥, 0, O refix or F)


History of notations

* Negation: the symbol ¬ appeared in Heyting in 1929 Heyting (1929) ''Die formalen Regeln der intuitionistischen Logik''. (compare to
Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
's symbol ⫟ in his
Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept nota ...
); the symbol ~ appeared in
Russell Russell may refer to: People * Russell (given name) * Russell (surname) * Lady Russell (disambiguation) * Lord Russell (disambiguation) Places Australia * Russell, Australian Capital Territory * Russell Island, Queensland (disambiguation) ...
in 1908;
Russell Russell may refer to: People * Russell (given name) * Russell (surname) * Lady Russell (disambiguation) * Lord Russell (disambiguation) Places Australia * Russell, Australian Capital Territory * Russell Island, Queensland (disambiguation) ...
(1908) ''Mathematical logic as based on the theory of types'' (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort).
an alternative notation is to add a horizontal line on top of the formula, as in \overline; another alternative notation is to use a prime symbol as in P'. * Conjunction: the symbol ∧ appeared in Heyting in 1929 (compare to Peano's use of the set-theoretic notation of intersection ∩); the symbol & appeared at least in
Schönfinkel Schönfinkel ( yi, שײנפֿינק(ע)ל ''Sheynfinkel'', russian: Шейнфинкель ''Šejnfinkeľ''): * Moses (Ilyich) Schönfinkel, born ''Moisei (Moshe) Isai'evich Sheinfinkel'' (1889, Ekaterinoslav - 1942, Moscow) ** The Bernays–Schö ...
in 1924;
Schönfinkel Schönfinkel ( yi, שײנפֿינק(ע)ל ''Sheynfinkel'', russian: Шейнфинкель ''Šejnfinkeľ''): * Moses (Ilyich) Schönfinkel, born ''Moisei (Moshe) Isai'evich Sheinfinkel'' (1889, Ekaterinoslav - 1942, Moscow) ** The Bernays–Schö ...
(1924) '' Über die Bausteine der mathematischen Logik'', translated as ''On the building blocks of mathematical logic'' in From Frege to Gödel edited by van Heijenoort.
the symbol . comes from
Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Irel ...
's interpretation of logic as an elementary algebra. * Disjunction: the symbol ∨ appeared in
Russell Russell may refer to: People * Russell (given name) * Russell (surname) * Lady Russell (disambiguation) * Lord Russell (disambiguation) Places Australia * Russell, Australian Capital Territory * Russell Island, Queensland (disambiguation) ...
in 1908 (compare to Peano's use of the set-theoretic notation of
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
∪); the symbol + is also used, in spite of the ambiguity coming from the fact that the + of ordinary elementary algebra is an exclusive or when interpreted logically in a two-element
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
; punctually in the history a + together with a dot in the lower right corner has been used by Peirce, * Implication: the symbol → can be seen in Hilbert in 1917; ⊃ was used by Russell in 1908 (compare to Peano's inverted C notation); ⇒ was used in Vax. * Biconditional: the symbol ≡ was used at least by Russell in 1908; ↔ was used at least by Tarski in 1940; ⇔ was used in Vax; other symbols appeared punctually in the history, such as ⊃⊂ in
Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died o ...
, ~ in Schönfinkel or ⊂⊃ in Chazal. * True: the symbol 1 comes from
Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Irel ...
's interpretation of logic as an elementary algebra over the two-element Boolean algebra; other notations include \bigwedge (to be found in Peano). * False: the symbol 0 comes also from Boole's interpretation of logic as a ring; other notations include \bigvee (to be found in Peano). Some authors used letters for connectives at some time of the history: u. for conjunction (German's "und" for "and") and o. for disjunction (German's "oder" for "or") in earlier works by Hilbert (1904); N''p'' for negation, K''pq'' for conjunction, D''pq'' for alternative denial, A''pq'' for disjunction, X''pq'' for joint denial, C''pq'' for implication, E''pq'' for biconditional in Łukasiewicz (1929); cf. Polish notation.


Redundancy

Such a logical connective as
converse implication In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical propositi ...
"←" is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
), certain essentially different compound statements are logically equivalent. A less
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
example of a redundancy is the classical equivalence between and . Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugar for a compound having one negation and one disjunction. There are sixteen
Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth functio ...
s associating the input truth values and with four-digit
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...
outputs. These correspond to possible choices of binary logical connectives for
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
. Different implementations of classical logic can choose different
functionally complete In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression.. ("Complete set of logical connectives").. ( ...
subsets of connectives. One approach is to choose a ''minimal'' set, and define other connectives by some logical form, as in the example with the material conditional above. The following are the minimal functionally complete sets of operators in classical logic whose arities do not exceed 2: ;One element: , . ;Two elements: \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \. ;Three elements: \, \, \, \, \, \. Another approach is to use with equal rights connectives of a certain convenient and functionally complete, but ''not minimal'' set. This approach requires more propositional
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s, and each equivalence between logical forms must be either an
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
or provable as a theorem. The situation, however, is more complicated in intuitionistic logic. Of its five connectives, , only negation "¬" can be reduced to other connectives (see for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.


Natural language

The standard logical connectives of classical logic have rough equivalents in the grammars of natural languages. In
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
, as in many languages, such expressions are typically grammatical conjunctions. However, they can also take the form of
complementizer In linguistics (especially generative grammar), complementizer or complementiser (glossing abbreviation: ) is a functional category (part of speech) that includes those words that can be used to turn a clause into the subject or object of a s ...
s, verb suffixes, and
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
s. The denotations of natural language connectives is a major topic of research in formal semantics, a field that studies the logical structure of natural languages. The meanings of natural language connectives are not precisely identical to their nearest equivalents in classical logic. In particular, disjunction can receive an exclusive interpretation in many languages. Some researchers have taken this fact as evidence that natural language semantics is
nonclassical Nonclassical is a British independent record label and night club founded in 2004 by Gabriel Prokofiev, grandson of Sergei Prokofiev. History Nonclassical has released fourteen albums, each following a concept of recording new contemporary cl ...
. However, others maintain classical semantics by positing
pragmatic Pragmatism is a philosophical movement. Pragmatism or pragmatic may also refer to: *Pragmaticism, Charles Sanders Peirce's post-1905 branch of philosophy * Pragmatics, a subfield of linguistics and semiotics *'' Pragmatics'', an academic journal i ...
accounts of exclusivity which create the illusion of nonclassicality. In such accounts, exclusivity is typically treated as a scalar implicature. Related puzzles involving disjunction include
free choice inference Free choice is a phenomenon in natural language where a linguistic disjunction appears to receive a logical conjunctive interpretation when it interacts with a modal operator. For example, the following English sentences can be interpreted to mean ...
s, Hurford's Constraint, and the contribution of disjunction in
alternative question A question is an utterance which serves as a request for information. Questions are sometimes distinguished from interrogatives, which are the grammatical forms typically used to express them. Rhetorical questions, for instance, are interrogative ...
s. Other apparent discrepancies between natural language and classical logic include the
paradoxes of material implication The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formu ...
,
donkey anaphora Donkey sentences are sentences that contain a pronoun with clear meaning (it is bound semantically) but whose syntactical role in the sentence poses challenges to grammarians. Such sentences defy straightforward attempts to generate their formal ...
and the problem of
counterfactual conditionals Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactual ...
. These phenomena have been taken as motivation for identifying the denotations of natural language conditionals with logical operators including the strict conditional, the variably strict conditional, as well as various
dynamic Dynamics (from Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power") or dynamic may refer to: Physics and engineering * Dynamics (mechanics) ** Aerodynamics, the study of the motion of air ** Analytical dyn ...
operators. The following table shows the standard classically definable approximations for the English connectives.


Properties

Some logical connectives possess properties that may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are: ;
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
: Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed. ;
Commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name ...
:The operands of the connective may be swapped, preserving logical equivalence to the original expression. ; Distributivity: A connective denoted by · distributes over another connective denoted by +, if for all operands , , . ; Idempotence: Whenever the operands of the operation are the same, the compound is logically equivalent to the operand. ;
Absorption Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials *Absorption (skin), a route by which s ...
: A pair of connectives ∧, ∨ satisfies the absorption law if a\land(a\lor b)=a for all operands , . ; Monotonicity: If ''f''(''a''1, ..., ''a''''n'') ≤ ''f''(''b''1, ..., ''b''''n'') for all ''a''1, ..., ''a''''n'', ''b''1, ..., ''b''''n'' ∈ such that ''a''1 ≤ ''b''1, ''a''2 ≤ ''b''2, ..., ''a''''n'' ≤ ''b''''n''. E.g., ∨, ∧, ⊤, ⊥. ;
Affinity Affinity may refer to: Commerce, finance and law * Affinity (law), kinship by marriage * Affinity analysis, a market research and business management technique * Affinity Credit Union, a Saskatchewan-based credit union * Affinity Equity Par ...
: Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, ↔, \nleftrightarrow, ⊤, ⊥. ; Duality: To read the truth-value assignments for the operation from top to bottom on its
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
is the same as taking the complement of reading the table of the same or another connective from bottom to top. Without resorting to truth tables it may be formulated as . E.g., ¬. ; Truth-preserving: The compound all those arguments are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂ (see validity). ; Falsehood-preserving: The compound all those argument are
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
s is a contradiction itself. E.g., ∨, ∧, \nleftrightarrow, ⊥, ⊄, ⊅ (see validity). ; Involutivity (for unary connectives): . E.g. negation in classical logic. For classical and intuitionistic logic, the "=" symbol means that corresponding implications "...→..." and "...←..." for logical compounds can be both proved as theorems, and the "≤" symbol means that "...→..." for logical compounds is a consequence of corresponding "...→..." connectives for propositional variables. Some
many-valued logic Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "fals ...
s may have incompatible definitions of equivalence and order (entailment). Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law. In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic.


Order of precedence

As a way of reducing the number of necessary parentheses, one may introduce precedence rules: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, P \vee Q \wedge \rightarrow S is short for (P \vee (Q \wedge (\neg R))) \rightarrow S. Here is a table that shows a commonly used precedence of logical operators. However, not all compilers use the same order; for instance, an ordering in which disjunction is lower precedence than implication or bi-implication has also been used.. Sometimes precedence between conjunction and disjunction is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula.


Computer science

A truth-functional approach to logical operators is implemented as
logic gate A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate ...
s in digital circuits. Practically all digital circuits (the major exception is
DRAM Dynamic random-access memory (dynamic RAM or DRAM) is a type of random-access semiconductor memory that stores each bit of data in a memory cell, usually consisting of a tiny capacitor and a transistor, both typically based on metal-oxi ...
) are built up from NAND, NOR, NOT, and
transmission gate A transmission gate (TG) is an analog gate similar to a relay that can conduct in both directions or block by a control signal with almost any voltage potential. It is a CMOS-based switch, in which PMOS passes a strong 1 but poor 0, and NMOS passes ...
s; see more details in Truth function in computer science. Logical operators over bit vectors (corresponding to finite
Boolean algebras In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...
) are
bitwise operation In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic oper ...
s. But not every usage of a logical connective in computer programming has a Boolean semantic. For example, lazy evaluation is sometimes implemented for and , so these connectives are not commutative if either or both of the expressions , have side effects. Also, a
conditional Conditional (if then) may refer to: *Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B has occurred *Conditional proof, in logic: a proof that asserts a co ...
, which in some sense corresponds to the material conditional connective, is essentially non-Boolean because for if (P) then Q;, the consequent Q is not executed if the antecedent P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and constructivist views on the material conditional— rather than to classical logic's views.


Table and Hasse diagram

The 16 logical connectives can be partially ordered to produce the following
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents e ...
. The partial order is defined by declaring that x \leq y if and only if whenever x holds then so does y.


See also

*
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
*
Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth functio ...
*
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
*
Boolean-valued function A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = ), whose elements are ...
*
Four-valued logic In logic, a four-valued logic is any logic with four truth values. Several types of four-valued logic have been advanced. Belnap Nuel Belnap considered the challenge of question answering by computer in 1975. Noting human fallibility, he was conc ...
*
List of Boolean algebra topics This is a list of topics around Boolean algebra and propositional logic. Articles with a wide scope and introductions * Algebra of sets * Boolean algebra (structure) * Boolean algebra * Field of sets * Logical connective * Prop ...
* Logical constant *
Modal operator A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional in the following sen ...
* Propositional calculus *
Truth function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly o ...
*
Truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
* Truth values


References


Sources

* Bocheński, Józef Maria (1959), ''A Précis of Mathematical Logic'', translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland. * * * . *


External links

* *Lloyd Humberstone (2010),
Sentence Connectives in Formal Logic
, Stanford Encyclopedia of Philosophy (An
abstract algebraic logic In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 200 ...
approach to connectives.) *John MacFarlane (2005),
Logical constants
, Stanford Encyclopedia of Philosophy. {{DEFAULTSORT:Logical Connective Connective da:Logisk konnektiv