Logical connective

 ( Connective ) 

In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is an operator that combines or modifies one or more logical variables or formulas, similarly to how arithmetic connectives like $+$ and $-$ combine or negate arithmetic expressions. For instance in the syntax of propositional logic, the Binary relation connective $\backslash lor$ (meaning "or") can be used to join the two logical formulas $P$ and $Q$, producing the complex formula $P\; \backslash lor\; Q$.

Unlike in algebra, there are many symbols in use for each logical connective. The table "Logical connectives" shows examples.

Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as , 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 language "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.

Logical connectives can be used to link zero or more statements, so one can speak about arity logical connectives. The Boolean algebra constants True and False can be thought of as zero-ary operators. Negation is a unary connective, and so on.

• negation: $\backslash neg$, $\backslash sim$, $N$ (prefix) in which $\backslash neg$ is the most modern and widely used, and $\backslash sim$ is also common;
• Conjunction (and): $\backslash wedge$, , $K$ (prefix) in which $\backslash wedge$ is the most modern and widely used;
• Disjunction (or): $\backslash vee$, $A$ (prefix) in which $\backslash vee$ is the most modern and widely used;
• Implication (if...then): $\backslash to$, $\backslash supset$, $\backslash Rightarrow$, $C$ (prefix) in which $\backslash to$ is the most modern and widely used, and $\backslash supset$ is also common;
• Equivalence (if and only if): $\backslash leftrightarrow$, $\backslash subset\backslash !\backslash !\backslash !\backslash supset$, $\backslash Leftrightarrow$, $\backslash equiv$, $E$ (prefix) in which $\backslash leftrightarrow$ is the most modern and widely used, and $\backslash subset\backslash !\backslash !\backslash !\backslash supset$ is commonly used where $\backslash supset$ is also used.

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 ($\backslash neg\; p$);
• 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 (in which case they are nullary).

• Truth formula: $\backslash top$, $1$, $V$ (prefix), or $\backslash mathrm\{T\}$;
• False formula: $\backslash bot$, $0$, $O$ (prefix), or $\backslash mathrm\{F\}$.

This table summarizes the terminology:

• Negation: the symbol $\backslash neg$ appeared in Arend Heyting in 1930Denis Roegel (2002),

• Conjunction: the symbol $\backslash wedge$ appeared in Heyting in 1930 (compare to Giuseppe Peano's use of the set-theoretic notation of intersection $\backslash cap$Giuseppe Peano (1889) Arithmetices principia, nova methodo exposita.); 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 $\backslash cdot$ comes from George Boole's interpretation of logic as an elementary algebra.
• Disjunction: the symbol $\backslash vee$ appeared in Bertrand Russell in 1908 (compare to Giuseppe Peano's use of the set-theoretic notation of union $\backslash cup$); 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 Boolean ring; punctually in the history a $+$ together with a dot in the lower right corner has been used by Peirce.Peirce (1867) ''On an improvement in Boole's calculus of logic.
• Implication: the symbol $\backslash to$ appeared in David Hilbert in 1918;; Reprinted as $\backslash supset$ was used by Russell in 1908 (compare to Peano's Ɔ the inverted C); $\backslash Rightarrow$ appeared in Nicolas Bourbaki in 1954.
• Equivalence: the symbol $\backslash equiv$ in Gottlob Frege in 1879; $\backslash leftrightarrow$ in Becker in 1933 (not the first time and for this see the following); $\backslash Leftrightarrow$ appeared in Nicolas Bourbaki in 1954; other symbols appeared punctually in the history, such as $\backslash supset\backslash subset$ in Gerhard Gentzen,Gerhard Gentzen (1934) Untersuchungen über das logische Schließen. $\backslash sim$ in Schönfinkel or $\backslash subset\backslash supset$ in Chazal,Chazal (1996) : Éléments de logique formelle.
• True: the symbol $1$ comes from George Boole's interpretation of logic as an elementary algebra over the two-element Boolean algebra; other notations include $\backslash mathrm\{V\}$ (abbreviation for the Latin word "verum") to be found in Peano in 1889.
• False: the symbol $0$ comes also from Boole's interpretation of logic as a ring; other notations include $\backslash Lambda$ (rotated $\backslash mathrm\{V\}$) to be found in Peano in 1889.

Some authors used letters for connectives: $\backslash operatorname\{u.\}$ for conjunction (German's "und" for "and") and $\backslash operatorname\{o.\}$ for disjunction (German's "oder" for "or") in early works by Hilbert (1904); $Np$ for negation, $Kpq$ for conjunction, $Dpq$ for alternative denial, $Apq$ for disjunction, $Cpq$ for implication, $Epq$ for biconditional in Łukasiewicz in 1929.

There are sixteen associating the input $p$ and $q$ with four-digit binary outputs.Bocheński (1959), A Précis of Mathematical Logic, passim. These correspond to possible choices of binary logical connectives for classical logic. 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

$\backslash \{\backslash uparrow\backslash \}$, $\backslash \{\backslash downarrow\backslash \}$.

Two elements

$\backslash \{\backslash vee,\; \backslash neg\backslash \}$, $\backslash \{\backslash wedge,\; \backslash neg\backslash \}$, $\backslash \{\backslash to,\; \backslash neg\backslash \}$, $\backslash \{\backslash gets,\; \backslash neg\backslash \}$, $\backslash \{\backslash to,\; \backslash bot\backslash \}$, $\backslash \{\backslash gets,\; \backslash bot\backslash \}$, $\backslash \{\backslash to,\; \backslash nleftrightarrow\backslash \}$, $\backslash \{\backslash gets,\; \backslash nleftrightarrow\backslash \}$, $\backslash \{\backslash to,\; \backslash nrightarrow\backslash \}$, $\backslash \{\backslash to,\; \backslash nleftarrow\backslash \}$, $\backslash \{\backslash gets,\; \backslash nrightarrow\backslash \}$, $\backslash \{\backslash gets,\; \backslash nleftarrow\backslash \}$, $\backslash \{\backslash nrightarrow,\; \backslash neg\backslash \}$, $\backslash \{\backslash nleftarrow,\; \backslash neg\backslash \}$, $\backslash \{\backslash nrightarrow,\; \backslash top\backslash \}$, $\backslash \{\backslash nleftarrow,\; \backslash top\backslash \}$, $\backslash \{\backslash nrightarrow,\; \backslash leftrightarrow\backslash \}$, $\backslash \{\backslash nleftarrow,\; \backslash leftrightarrow\backslash \}$.

Three elements

$\backslash \{\backslash lor,\; \backslash leftrightarrow,\; \backslash bot\backslash \}$, $\backslash \{\backslash lor,\; \backslash leftrightarrow,\; \backslash nleftrightarrow\backslash \}$, $\backslash \{\backslash lor,\; \backslash nleftrightarrow,\; \backslash top\backslash \}$, $\backslash \{\backslash land,\; \backslash leftrightarrow,\; \backslash bot\backslash \}$, $\backslash \{\backslash land,\; \backslash leftrightarrow,\; \backslash nleftrightarrow\backslash \}$, $\backslash \{\backslash land,\; \backslash nleftrightarrow,\; \backslash top\backslash \}$.

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 , 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.

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. However, others maintain classical semantics by positing pragmatics 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 operators.

The following table shows the standard classically definable approximations for the English connectives.

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 Law

A pair of connectives ∧, ∨ satisfies the absorption law if $a\backslash land(a\backslash lor\; b)=a$ for all operands , .

Monotonicity

If f( a₁, ..., a _n) ≤ f( b₁, ..., b _n) for all a₁, ..., a _n, b₁, ..., b _n ∈ {0,1} such that a₁ ≤ b₁, a₂ ≤ b₂, ..., 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 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.

Here is a table that shows a commonly used precedence of logical operators.

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, 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.

• Boolean domain
• Boolean function
• Boolean logic
• Boolean-valued function
• Catuṣkoṭi
• Dialetheism
• Four-valued logic
• List of Boolean algebra topics
• Logical conjunction
• Logical constant
• Modal operator
• Propositional calculus
• Term logic
• Tetralemma
• Truth function
• Truth table

• Bocheński, Józef Maria (1959), A Précis of Mathematical Logic, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
• (2025). 9780122384523, Academic Press. ISBN 9780122384523
• Rautenberg, W. (2025). 9781441912206, Springer Science+Business Media. ISBN 9781441912206
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• Lloyd Humberstone (2025). 9780262016544, MIT Press. ISBN 9780262016544

• Lloyd Humberstone (2010), " Sentence Connectives in Formal Logic", Stanford Encyclopedia of Philosophy (An abstract algebraic logic approach to connectives.)
• John MacFarlane (2005), " Logical constants", Stanford Encyclopedia of Philosophy.