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 connective (meaning "or") can be used to join the two logical formulas and , producing the complex formula .
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 truth functions, 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.
Overview In formal languages, truth functions are denoted by fixed symbols, ensuring that well-formed statements have a single interpretation. 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 nullary operators. Negation is a unary connective, and so on.
Symbol, name Truthtable Venndiagram Zeroary connectives (constants) Truth/tautology 1
Falsity/contradiction 0
Unary connectives
0 1 Proposition 0 1
Negation 1 0
Binary connectives
0 0 1 1
0 1 0 1 Conjunction 0001 Alternative denial 1110 Disjunction 0111 Joint denial 1000 Exclusive or 0110 Biconditional 1001 Material conditional 1101 Material nonimplication 0010 Converse implication 1011 Converse nonimplication 0100 More information
List of common logical connectives Commonly used logical connectives include the following ones. Negation (not): , , (prefix) in which is the most modern and widely used, and is also common; Conjunction (and): , , (prefix) in which is the most modern and widely used; Disjunction (or): , (prefix) in which is the most modern and widely used; Implication (if...then): , , , (prefix) in which is the most modern and widely used, and is also common; Equivalence (if and only if): , , , , (prefix) in which is the most modern and widely used, and is commonly used where is also used.
For example, the meaning of the statements it is raining (denoted by ) and I am indoors (denoted by ) is transformed, when the two are combined with logical connectives:
It is not raining (); It is raining and I am indoors (); It is raining or I am indoors (); If it is raining, then I am indoors (); If I am indoors, then it is raining (); I am indoors if and only if it is raining ().
It is also common to consider the always true formula and the always false formula to be connective (in which case they are nullary). True formula: , , (prefix), or ; False formula: , , (prefix), or .
This table summarizes the terminology:
Connective In English Noun for parts Verb phrase Conjunction Both A and B conjunct A and B are conjoined Disjunction Either A or B, or both disjunct A and B are disjoined Negation It is not the case that A negatum/negand A is negated Conditional If A, then B antecedent, consequent B is implied by A Biconditional A if, and only if, B equivalents A and B are equivalent
History of notations Negation: the symbol appeared in Heyting in 1930Denis Roegel (2002), A brief survey of 20th century logical notations (see chart on page 2). (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 ; another alternative notation is to use a prime symbol as in . Conjunction: the symbol appeared in Heyting in 1930 (compare to Peano's use of the set-theoretic notation of intersection 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 comes from Boole'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.Peirce (1867) On an improvement in Boole's calculus of logic. Implication: the symbol appeared in Hilbert in 1918;; Reprinted as was used by Russell in 1908 (compare to Peano's Ɔ the inverted C); appeared in Bourbaki in 1954. Equivalence: the symbol in Frege in 1879; in Becker in 1933 (not the first time and for this see the following); appeared in Bourbaki in 1954; other symbols appeared punctually in the history, such as in Gentzen,Gentzen (1934) Untersuchungen über das logische Schließen. in Schönfinkel or in Chazal,Chazal (1996) : Éléments de logique formelle. True: the symbol comes from Boole's interpretation of logic as an elementary algebra over the two-element Boolean algebra; other notations include (abbreviation for the Latin word "verum") to be found in Peano in 1889. False: the symbol comes also from Boole's interpretation of logic as a ring; other notations include (rotated ) to be found in Peano in 1889.
Some authors used letters for connectives: for conjunction (German's "und" for "and") and for disjunction (German's "oder" for "or") in early works by Hilbert (1904); for negation, for conjunction, for alternative denial, for disjunction, for implication, for biconditional in Łukasiewicz in 1929.
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 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. (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 , . 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, as in many languages, such expressions are typically grammatical conjunctions. However, they can also take the form of complementizers, verb suffixes, 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 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 operators.
The following table shows the standard classically definable approximations for the English connectives.
English word Connective Symbol Logical gate not negation NOT and conjunction AND or disjunction OR if...then material implication IMPLY ...if converse implication either...or exclusive disjunction XOR if and only if biconditional XNOR not both alternative denial NAND neither...nor joint denial NOR but not material nonimplication NIMPLY not...but converse nonimplication
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. CommutativityThe 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 A pair of connectives ∧, ∨ satisfies the absorption law if for all operands , . Monotonicity If f(a1, ..., an) ≤ f(b1, ..., bn) for all a1, ..., an, b1, ..., bn ∈ {0,1} such that a1 ≤ b1, a2 ≤ b2, ..., an ≤ bn. E.g., ∨, ∧, ⊤, ⊥. Affinity Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, ↔, , ⊤, ⊥.