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In
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 prem ...
, 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 In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituenc ...
of propositional logic, the binary connective $\lor$ can be used to join the two atomic formulas $P$ 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 ...
, conjunction, and implication. In standard systems of classical logic, these connectives are interpreted as truth functions, 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 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

In
formal 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, '' 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. 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.

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 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 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 $\overline$; 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 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 Irel ...
's interpretation of logic as an elementary algebra. * 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 is an exclusive or 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 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 "←" 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), 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 functions associating the input truth values and with four-digit binary outputs. These correspond to possible choices of binary logical connectives for classical logic. 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 axioms, and each equivalence between logical forms must be either an axiom 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 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 constitu ...
s. However, they can also take the form of complementizers,
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 carr ...
es, and particles. 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 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 comput ...
is nonclassical. 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 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 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: 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: 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 $a\land\left(a\lor b\right)=a$ 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: 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 arg ...
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 contradictions 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 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, $P \vee Q \wedge \rightarrow S$ is short for $\left(P \vee \left(Q \wedge \left(\neg R\right)\right)\right) \rightarrow S$. 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 in digital circuits. Practically all digital circuits (the major exception is DRAM) are built up from NAND, NOR, NOT, and transmission gates; see more details in Truth function in computer science. Logical operators over bit vectors (corresponding to finite Boolean algebras) are bitwise operations. 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 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. The partial order is defined by declaring that $x \leq y$ if and only if whenever $x$ holds then so does $y.$

* Boolean domain * Boolean function * Boolean logic * Boolean-valued function * Four-valued logic * List of Boolean algebra topics * Logical constant * Modal operator *
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 b ...
* Truth function *
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 arg ...
* Truth values

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. * * * . *