Logical connective


In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.
The most common logical connectives are binary connectives which join two sentences which can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective.
Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic. Semantics of a logical connective is often, but not always, presented as a truth function.
A logical connective is similar to but not equivalent to a conditional operator.

In language

Natural language

In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a grammatically compound sentence. Some but not all such grammatical conjunctions are truth functional. For example, consider the following sentences:
The words and and so are grammatical conjunctions joining the sentences and to form the compound sentences and. The and in is a logical connective, since the truth of is completely determined by and : it would make no sense to affirm and but deny. However, so in is not a logical connective, since it would be quite reasonable to affirm and but deny : perhaps, after all, Jill went up the hill to fetch a pail of water, not because Jack had gone up the hill at all.
Various English words and word pairs express logical connectives, and some of them are synonymous. Examples are:
WordConnectiveSymbolLogical gate
andconjunction"∧"AND
and thenconjunction"∧"AND
and then withinconjunction"∧"AND
ordisjunction"∨"OR
either...orexclusive disjunction"⊕"XOR
either, but not bothexclusive disjunction"⊕"XOR
impliesmaterial implication"→"IMPLY
is implied byconverse implication"←"
if...thenmaterial implication"→"IMPLY
...ifconverse implication"←"
if and only ifbiconditional"↔"XNOR
just in casebiconditional"↔"XNOR
butconjunction"∧"AND
howeverconjunction"∧"AND
not bothalternative denial"↑"NAND
neither...norjoint denial"↓"NOR
not, not thatnegation"¬"NOT
it is false thatnegation"¬"NOT
it is not the case thatnegation"¬"NOT
althoughconjunction"∧"AND
even thoughconjunction"∧"AND
thereforematerial implication"→"IMPLY
somaterial implication"→"IMPLY
that is to saybiconditional"↔"XNOR
furthermoreconjunction"∧"AND
but notmaterial nonimplication"↛"NIMPLY
not...butconverse nonimplication"↚"
without...there is nomaterial implication"→"IMPLY
no... withoutconverse implication"←"

Formal languages

In formal languages, truth functions are represented by unambiguous symbols. These symbols are called logical connectives, logical operators, propositional operators, or, in classical logic, truth-functional connectives. See well-formed formula for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives.
Logical connectives can be used to link more than two statements, so one can speak about -ary logical connective.

Common logical connectives

List of common logical connectives

Commonly used logical connectives include
Alternative names for biconditional are iff, xnor, and bi-implication.
For example, the meaning of the statements it is raining and I am indoors is transformed when the two are combined with logical connectives. For statement P = It is raining and Q = I am indoors:
It is also common to consider the always true formula and the always false formula to be connective:
Some authors used letters for connectives at some time of the history: u. for conjunction and o. for disjunction in earlier works by Hilbert ; Np for negation, Kpq for conjunction, Dpq for alternative denial, Apq for disjunction, Xpq for joint denial, Cpq for implication, Epq for biconditional in Łukasiewicz ; 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 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 "¬" and "∨" 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 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. Neither conjunction, disjunction, nor material conditional has an equivalent form constructed of the other four logical connectives.

Properties

Some logical connectives possess properties which 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: A pair of connectives ∧, ∨ satisfies the absorption law if for all operands,.
; Monotonicity: If f ≤ f for all a1,..., an, b1,..., bn ∈ 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., ¬, ↔, , ⊤, ⊥.
; 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 argument are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂.
; Falsehood-preserving: The compound all those argument are contradictions is a contradiction itself. E.g., ∨, ∧,, ⊥, ⊄, ⊅.
; Involutivity :. 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.
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, is short for.
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 are built up from NAND, NOR, NOT, and transmission gates; see more details in Truth function in computer science. Logical operators over bit vectors are bitwise operations.
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 then Q; the consequent Q is not executed if the antecedent P is false. This is closer to intuitionist and constructivist views on the material conditional, rather than to classical logic's ones.