Logical connective

In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives 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 $P\lor Q$ .

For other logical symbols, see List of logic symbols.

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 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.^[1]

: $\neg$ , $\sim$ , $N$ (prefix) in which $\neg$ is the most modern and widely used, and $\sim$ is used by many people too;

Negation (not)

: $\wedge$ , $\&$ , $K$ (prefix) in which $\wedge$ is the most modern and widely used;

Conjunction (and)

: $\vee$ , $A$ (prefix) in which $\vee$ is the most modern and widely used;

Disjunction (or)

: $\to$ , $\supset$ , $\Rightarrow$ , $C$ (prefix) in which $\to$ is the most modern and widely used, and $\supset$ is used by many people too;

Implication (if...then)

: $\leftrightarrow$ , $\subset \!\!\!\supset$ , $\Leftrightarrow$ , $\equiv$ , $E$ (prefix) in which $\leftrightarrow$ is the most modern and widely used, and $\subset \!\!\!\supset$ may be also a good choice compared to $\supset$ denoting implication just like $\leftrightarrow$ to $\to$ .

Equivalence (if and only if)

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Bocheński, Józef Maria

Chao, C. (2023). 数理逻辑：形式化方法的应用 [Mathematical Logic: Applications of the Formalization Method] (in Chinese). Beijing: Preprint. pp. 15–28.

(2001), A Mathematical Introduction to Logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3

Enderton, Herbert

(1991), "Chapter 2", Logic, Language and Meaning, vol. 1, University of Chicago Press, pp. 54–64, OCLC 21372380

Gamut, L.T.F

(2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.

Rautenberg, W.

Humberstone, Lloyd (2011). The Connectives. MIT Press. 978-0-262-01654-4.

ISBN

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Propositional connective"

Lloyd Humberstone (2010), "", Stanford Encyclopedia of Philosophy (An abstract algebraic logic approach to connectives.)

Sentence Connectives in Formal Logic

John MacFarlane (2005), "", Stanford Encyclopedia of Philosophy.