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 ...

. Different implementations of classical logic can choose different functionally complete 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: $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $.
;Three elements: $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $, $\backslash $.
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

_{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: Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, ↔, $\backslash nleftrightarrow$, ⊤, ⊥.
; Duality: To read the truth-value assignments for the operation from top to bottom on its truth table 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

Sentence Connectives in Formal Logic

,

Logical constants

,

logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

, 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
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

, the binary connective $\backslash lor$ can be used to join the two atomic formulas $P$ and $Q$, rendering the complex formula $P\; \backslash lor\; Q$.
Common connectives include negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fal ...

, 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 astronomy, a conjunction occu ...

, 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 logics. Their classical interpretations are similar to the meanings of natural language expressions such as English "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
In linguistics and related fields, pragmatics is the study of how context contributes to meaning. The field of study evaluates how human language is utilized in social interactions, as well as the relationship between the interpreter and the int ...

.
A logical connective is similar to, but not equivalent to, a syntax commonly used in programming languages called a conditional operator.
Overview

Informal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of sym ...

s, 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 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
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...

.
Logical connectives can be used to link zero or more statements, so one can speak about '' -ary logical connectives''. The boolean 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'', '' xnor'', 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 ($P\; \backslash wedge\; Q$) * It is raining or I am indoors ($P\; \backslash lor\; Q$) * If it is raining, then I am indoors ($P\; \backslash rightarrow\; Q$) * If I am indoors, then it is raining ($Q\; \backslash rightarrow\; P$) * I am indoors if and only if it is raining ($P\; \backslash leftrightarrow\; Q$) 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's symbol ⫟ in his Begriffsschrift); the symbol ~ appeared in Russell in 1908; Russell (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 $\backslash overline$; 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 ofintersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

∩); the symbol & appeared at least in Schönfinkel in 1924; Schönfinkel (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 Ire ...

's interpretation of logic as an elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entai ...

.
* Disjunction: the symbol ∨ appeared in Russell in 1908 (compare to Peano's use of the set-theoretic notation of union ∪); the symbol + is also used, in spite of the ambiguity coming from the fact that the + of ordinary elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entai ...

is an exclusive or
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...

when interpreted logically in a two-element ring; 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, ~ 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 Ire ...

's interpretation of logic as an elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entai ...

over the two-element Boolean algebra; other notations include $\backslash bigwedge$ (to be found in Peano).
* False: the symbol 0 comes also from Boole's interpretation of logic as a ring; other notations include $\backslash bigvee$ (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 asconverse 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 proposit ...

"←" is actually the same as material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...

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 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
In computer science, syntactic sugar is syntax within a programming language that is designed to make things easier to read or to express. It makes the language "sweeter" for human use: things can be expressed more clearly, more concisely, or in an ...

for a compound having one negation and one disjunction.
There are sixteen Boolean functions associating the input truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some prog ...

s and with four-digit binary outputs. These correspond to possible choices of binary logical connectives for 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 ...

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 ...

or provable as a theorem.
The situation, however, is more complicated in intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...

. 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, as in many languages, such expressions are typicallygrammatical conjunction
In grammar, a conjunction ( abbreviated or ) is a part of speech that connects words, phrases, or clauses that are called the conjuncts of the conjunctions. That definition may overlap with that of other parts of speech and so what constitutes ...

s. 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, verb
A verb () is a word (part of speech) that in syntax generally conveys an action (''bring'', ''read'', ''walk'', ''run'', ''learn''), an occurrence (''happen'', ''become''), or a state of being (''be'', ''exist'', ''stand''). In the usual descr ...

suffix
In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns, adjectives, and verb endings, which form the conjugation of verbs. Suffixes can carry ...

es, and particles. The denotation
In linguistics and philosophy, the denotation of an expression is its literal meaning. For instance, the English word "warm" denotes the property of being warm. Denotation is contrasted with other aspects of meaning including connotation. For inst ...

s 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
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comp ...

is nonclassical. However, others maintain classical semantics by positing pragmatic 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 inferences, Hurford's Constraint, and the contribution of disjunction in alternative questions.
Other apparent discrepancies between natural language and classical logic include the paradoxes of material implication, donkey anaphora and the problem of counterfactual conditionals. 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 dy ...

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:The operands of the connective may be swapped, preserving logical equivalence to the original expression.
; Distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic ...

: A connective denoted by · distributes over another connective denoted by +, if for all operands , , .
; Idempotence
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

: Whenever the operands of the operation are the same, the compound is logically equivalent to the operand.
; Absorption: A pair of connectives ∧, ∨ satisfies the absorption law if $a\backslash land(a\backslash lor\; b)=a$ for all operands , .
; Monotonicity
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

: If ''f''(''a''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., ∨, ∧, $\backslash nleftrightarrow$, ⊥, ⊄, ⊅ (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 logics 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, $P\; \backslash vee\; Q\; \backslash wedge\; \backslash rightarrow\; S$ is short for $(P\; \backslash vee\; (Q\; \backslash wedge\; (\backslash neg\; R)))\; \backslash rightarrow\; S$. 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 gates indigital circuit In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematical ...

s. Practically all digital circuits (the major exception is DRAM) 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) are bitwise operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a Binary numeral system, 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-l ...

s.
But not every usage of a logical connective in computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...

has a Boolean semantic. For example, lazy evaluation
In programming language theory, lazy evaluation, or call-by-need, is an evaluation strategy which delays the evaluation of an expression until its value is needed ( non-strict evaluation) and which also avoids repeated evaluations (sharing).
The ...

is sometimes implemented for and , so these connectives are not commutative if either or both of the expressions , have side effect
In medicine, a side effect is an effect, whether therapeutic or adverse, that is secondary to the one intended; although the term is predominantly employed to describe adverse effects, it can also apply to beneficial, but unintended, consequence ...

s. Also, a conditional, which in some sense corresponds to the material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...

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. The partial order is defined by declaring that $x\; \backslash leq\; y$ 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
* Boolean logic
* 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
* List of Boolean algebra topics
* 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 se ...

* Propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

* 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
* Truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some prog ...

s
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
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

(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
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

.
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