logical connectives

In logic, 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, 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, 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 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.

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

Overview

In formal languages, 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.

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 ($P \wedge Q$)
* 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'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 $\overline$; 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'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'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 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