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Logical connective

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Symbol connecting formulas in logic

For other logical symbols, seeList of logic symbols.

Logical connectives

NOT	$\neg A,-A,{\overline {A}},{\sim }A$
AND	$A\land B,A\cdot B,AB,A\mathop {\&} B,A\mathop {\&\&} B$
NAND	$A\mathrel {\overline {\land }} B,A\uparrow B,A\mid B,{\overline {A\cdot B}}$
OR	$A\lor B,A+B,A\mid B,A\parallel B$
NOR	$A\mathrel {\overline {\lor }} B,A\downarrow B,{\overline {A+B}}$
XNOR	$A\odot B,{\overline {A\mathrel {\overline {\lor }} B}}$
└equivalent	$A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B$
XOR	$A\mathrel {\underline {\lor }} B,A\oplus B$
└ nonequivalent	$A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B$
implies	$A\Rightarrow B,A\supset B,A\rightarrow B$
nonimplication (NIMPLY)	$A\not \Rightarrow B,A\not \supset B,A\nrightarrow B$
converse	$A\Leftarrow B,A\subset B,A\leftarrow B$
converse nonimplication	$A\not \Leftarrow B,A\not \subset B,A\nleftarrow B$

Related concepts

Applications

Overview

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Informal languages, truth functions are denoted by fixed symbols, ensuring that well-formed statements have a single interpretation. These symbols are calledlogical connectives,logical operators,propositional operators, or, inclassical 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, seewell-formed formula.

Logical connectives can be used to link zero or more statements, so one can speak aboutn-ary logical connectives. Theboolean constantsTrue andFalse can be thought of as nullary operators. Negation is a unary connective, and so on.

Symbol, name		Truth table
Zeroary connectives (constants)
$\top$	Truth/tautology	1
$\bot$	Falsity/contradiction	0
Unary connectives
$p {\displaystyle p}$ =		0		1
	Proposition $p {\displaystyle p}$	0		1
$\neg$	Negation	1		0
Binary connectives
$p {\displaystyle p}$ =		0	0	1	1
$q {\displaystyle q}$ =		0	1	0	1
$\land$	Conjunction	0	0	0	1
$\uparrow$	Alternative denial	1	1	1	0
$\vee$	Disjunction	0	1	1	1
$\downarrow$	Joint denial	1	0	0	0
$\nleftrightarrow$	Exclusive or	0	1	1	0
$\leftrightarrow$	Biconditional	1	0	0	1
$\rightarrow$	Material conditional	1	1	0	1
$\nrightarrow$	Material nonimplication	0	0	1	0
$\leftarrow$	Converse implication	1	0	1	1
$\nleftarrow$	Converse nonimplication	0	1	0	0
More information

List of common logical connectives

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Commonly used logical connectives include the following ones.^[1]

Negation (not): $\neg$ , $\sim$ , $N {\displaystyle N}$ (prefix) in which $\neg$ is the most modern and widely used, and $\sim$ is also common;
Conjunction (and): $\wedge$ , $\&$ , $K {\displaystyle K}$ (prefix) in which $\wedge$ is the most modern and widely used;
Disjunction (or): $\vee$ , $A {\displaystyle A}$ (prefix) in which $\vee$ is the most modern and widely used;
Implication (if...then): $\to$ , $\supset$ , $\Rightarrow$ , $C {\displaystyle C}$ (prefix) in which $\to$ is the most modern and widely used, and $\supset$ is also common;
Equivalence (if and only if): $\leftrightarrow$ , $\subset \!\!\!\supset$ , $\Leftrightarrow$ , $\equiv$ , $E {\displaystyle E}$ (prefix) in which $\leftrightarrow$ is the most modern and widely used, and $\subset \!\!\!\supset$ is commonly used where $\supset$ is also used.

For example, the meaning of the statementsit is raining (denoted by $p {\displaystyle p}$ ) andI am indoors (denoted by $q {\displaystyle q}$ ) is transformed, when the two are combined with logical connectives:

It isnot raining ( $\neg p$ );
It is rainingand I am indoors ( $p\wedge q$ );
It is rainingor 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 indoorsif and only if it is raining ( $p\leftrightarrow q$ ).

It is also common to consider thealways true formula and thealways false formula to be connective (in which case they arenullary).

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

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

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Negation: the symbol $\neg$ appeared inHeyting in 1930^[2]^[3] (compare toFrege's symbol ⫟ in hisBegriffsschrift^[4]); the symbol $\sim$ appeared inRussell in 1908;^[5] an alternative notation is to add a horizontal line on top of the formula, as in ${\overline {p}}$ ; another alternative notation is to use aprime symbol as in $p^{'} {\displaystyle p'}$ .
Conjunction: the symbol $\wedge$ appeared in Heyting in 1930^[2] (compare toPeano's use of the set-theoretic notation ofintersection $\cap$ ^[6]); the symbol $\&$ appeared at least inSchönfinkel in 1924;^[7] the symbol $\cdot$ comes fromBoole's interpretation of logic as anelementary algebra.
Disjunction: the symbol $\vee$ appeared inRussell in 1908^[5] (compare toPeano's use of the set-theoretic notation ofunion $\cup$ ); the symbol $+$ is also used, in spite of the ambiguity coming from the fact that the $+$ of ordinaryelementary algebra is anexclusive or when interpreted logically in a two-elementring; punctually in the history a $+$ together with a dot in the lower right corner has been used byPeirce.^[8]
Implication: the symbol $\to$ appeared inHilbert in 1918;^[9]^: 76 $\supset$ was used by Russell in 1908^[5] (compare to Peano's Ɔ the inverted C); $\Rightarrow$ appeared inBourbaki in 1954.^[10]
Equivalence: the symbol $\equiv$ inFrege in 1879;^[11] $\leftrightarrow$ in Becker in 1933 (not the first time and for this see the following);^[12] $\Leftrightarrow$ appeared inBourbaki in 1954;^[13] other symbols appeared punctually in the history, such as $\supset \subset$ inGentzen,^[14] $\sim$ in Schönfinkel^[7] or $\subset \supset$ in Chazal,^[15]
True: the symbol $1 {\displaystyle 1}$ comes fromBoole's interpretation of logic as anelementary algebra over thetwo-element Boolean algebra; other notations include $\mathrm {V}$ (abbreviation for the Latin word "verum") to be found in Peano in 1889.
False: the symbol $0 {\displaystyle 0}$ comes also from Boole's interpretation of logic as a ring; other notations include $\Lambda$ (rotated $\mathrm {V}$ ) to be found in Peano in 1889.

Some authors used letters for connectives: $\operatorname {u.}$ for conjunction (German's "und" for "and") and $\operatorname {o.}$ for disjunction (German's "oder" for "or") in early works by Hilbert (1904);^[16] $N p {\displaystyle Np}$ for negation, $K p q {\displaystyle Kpq}$ for conjunction, $D p q {\displaystyle Dpq}$ for alternative denial, $A p q {\displaystyle Apq}$ for disjunction, $C p q {\displaystyle Cpq}$ for implication, $E p q {\displaystyle Epq}$ for biconditional inŁukasiewicz in 1929.

Redundancy

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Such a logical connective asconverse implication " $\leftarrow$ " is actually the same asmaterial conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, inclassical logic), certain essentially different compound statements arelogically equivalent. A lesstrivial example of a redundancy is the classical equivalence between $\neg p\vee q$ and $p\to q$ . Therefore, a classical-based logical system does not need the conditional operator " $\to$ " if " $\neg$ " (not) and " $\vee$ " (or) are already in use, or may use the " $\to$ " only as asyntactic sugar for a compound having one negation and one disjunction.

There are sixteenBoolean functions associating the inputtruth values $p {\displaystyle p}$ and $q {\displaystyle q}$ with four-digitbinary outputs.^[17] These correspond to possible choices of binary logical connectives forclassical logic. Different implementations of classical logic can choose differentfunctionally complete subsets of connectives.

One approach is to choose aminimal set, and define other connectives by some logical form, as in the example with the material conditional above. The following are theminimal functionally complete sets of operators in classical logic whose arities do not exceed 2:

One element: $\{\uparrow \}$ , $\{\downarrow \}$ .
Two elements: $\{\vee ,\neg \}$ , $\{\wedge ,\neg \}$ , $\{\to ,\neg \}$ , $\{\gets ,\neg \}$ , $\{\to ,\bot \}$ , $\{\gets ,\bot \}$ , $\{\to ,\nleftrightarrow \}$ , $\{\gets ,\nleftrightarrow \}$ , $\{\to ,\nrightarrow \}$ , $\{\to ,\nleftarrow \}$ , $\{\gets ,\nrightarrow \}$ , $\{\gets ,\nleftarrow \}$ , $\{\nrightarrow ,\neg \}$ , $\{\nleftarrow ,\neg \}$ , $\{\nrightarrow ,\top \}$ , $\{\nleftarrow ,\top \}$ , $\{\nrightarrow ,\leftrightarrow \}$ , $\{\nleftarrow ,\leftrightarrow \}$ .
Three elements: $\{\lor ,\leftrightarrow ,\bot \}$ , $\{\lor ,\leftrightarrow ,\nleftrightarrow \}$ , $\{\lor ,\nleftrightarrow ,\top \}$ , $\{\land ,\leftrightarrow ,\bot \}$ , $\{\land ,\leftrightarrow ,\nleftrightarrow \}$ , $\{\land ,\nleftrightarrow ,\top \}$ .

Another approach is to use with equal rights connectives of a certain convenient and functionally complete, butnot minimal set. This approach requires more propositionalaxioms, and each equivalence between logical forms must be either anaxiom or provable as a theorem.

The situation, however, is more complicated inintuitionistic logic. Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (seeFalse (logic) § False, negation and contradiction for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.

Natural language

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The standard logical connectives of classical logic have rough equivalents in the grammars of natural languages. InEnglish, as in many languages, such expressions are typicallygrammatical conjunctions. However, they can also take the form ofcomplementizers,verb suffixes, andparticles. Thedenotations of natural language connectives is a major topic of research informal 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 anexclusive interpretation in many languages. Some researchers have taken this fact as evidence that natural languagesemantics isnonclassical. However, others maintain classical semantics by positingpragmatic accounts of exclusivity which create the illusion of nonclassicality. In such accounts, exclusivity is typically treated as ascalar implicature. Related puzzles involving disjunction includefree choice inferences,Hurford's Constraint, and the contribution of disjunction inalternative questions.

Other apparent discrepancies between natural language and classical logic include theparadoxes of material implication,donkey anaphora and the problem ofcounterfactual conditionals. These phenomena have been taken as motivation for identifying the denotations of natural language conditionals with logical operators including thestrict conditional, thevariably strict conditional, as well as variousdynamic operators.

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

English word	Connective	Symbol	Logical gate
not	negation	$\neg$	NOT
and	conjunction	$\land$	AND
or	disjunction	$\vee$	OR
if...then	material implication	$\rightarrow$	IMPLY
...if	converse implication	$\leftarrow$
either...or	exclusive disjunction	$\nleftrightarrow$	XOR
if and only if	biconditional	$\leftrightarrow$	XNOR
not both	alternative denial	$\uparrow$	NAND
neither...nor	joint denial	$\downarrow$	NOR
but not	material nonimplication	$\nrightarrow$	NIMPLY
not...but	converse nonimplication	$\nleftarrow$

Properties

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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.
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 +, ifa · (b +c) = (a ·b) + (a ·c) for all operandsa,b,c.
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 $a\land (a\lor b)=a$ for all operandsa,b.
Monotonicity: Iff(a₁, ...,a_n) ≤f(b₁, ...,b_n) for alla₁, ...,a_n,b₁, ...,b_n ∈ {0,1} such thata₁ ≤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., ¬, ↔, $\nleftrightarrow$ , ⊤, ⊥.
Duality: To read the truth-value assignments for the operation from top to bottom on itstruth 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 asg̃(¬a₁, ..., ¬a_n) = ¬g(a₁, ...,a_n). E.g., ¬.
Truth-preserving: The compound all those arguments are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂ (seevalidity).
Falsehood-preserving: The compound all those argument arecontradictions is a contradiction itself. E.g., ∨, ∧, $\nleftrightarrow$ , ⊥, ⊄, ⊅ (seevalidity).
Involutivity (for unary connectives): f(f(a)) =a. 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. Somemany-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.

This sectionneeds expansion. You can help byadding missing information.(March 2012)

Order of precedence

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As a way of reducing the number of necessary parentheses, one may introduceprecedence rules: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, $P\vee Q\land {\neg R}\rightarrow S$ is short for $(P\vee (Q\land (\neg R)))\rightarrow S$ .

Here is a table that shows a commonly used precedence of logical operators.^[18]^[19]

Operator	Precedence
$\neg$	1
$\land$	2
$\vee$	3
$\rightarrow$	4
$\leftrightarrow$	5

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.^[20] 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.

Table and Hasse diagram

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The 16 logical connectives can bepartially ordered to produce the followingHasse diagram. The partial order is defined by declaring that $x\leq y$ if and only if whenever $x {\displaystyle x}$ holds then so does $y . {\displaystyle y.}$

Applications

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Logical connectives are used incomputer science and inset theory.

Computer science

[edit]

Main article:Logic gate

A truth-functional approach to logical operators is implemented aslogic gates indigital circuits. Practically all digital circuits (the major exception isDRAM) are built up fromNAND,NOR,NOT, andtransmission gates; see more details inTruth function in computer science. Logical operators overbit vectors (corresponding to finiteBoolean algebras) arebitwise operations.

But not every usage of a logical connective incomputer programming has a Boolean semantic. For example,lazy evaluation is sometimes implemented forP ∧ Q andP ∨ Q, so these connectives are not commutative if either or both of the expressionsP,Q haveside effects. Also, aconditional, which in some sense corresponds to thematerial conditional connective, is essentially non-Boolean because forif (P) then Q;, the consequent Q is not executed if theantecedent P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer tointuitionist andconstructivist views on the material conditional— rather than to classical logic's views.

Set theory

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Main articles:Set theory andAxiomatic set theory

Logical connectives are used to define the fundamental operations ofset theory,^[21] as follows:

Set theory operations and connectives
Set operation	Connective	Definition
Intersection	Conjunction	$A\cap B=\{x:x\in A\land x\in B\}$ ^[22]^[23]^[24]
Union	Disjunction	$A\cup B=\{x:x\in A\lor x\in B\}$ ^[25]^[22]^[23]
Complement	Negation	${\overline {A}}=\{x:x\notin A\}$ ^[26]^[23]^[27]
Subset	Implication	$A\subseteq B\leftrightarrow (x\in A\rightarrow x\in B)$ ^[28]^[23]^[29]
Equality	Biconditional	$A=B\leftrightarrow (\forall X)[A\in X\leftrightarrow B\in X]$ ^[28]^[23]^[30]

This definition of set equality is equivalent to theaxiom of extensionality.

References

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^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.
^^a ^bHeyting, A. (1930). "Die formalen Regeln der intuitionistischen Logik".Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (in German):42–56.
^Denis Roegel (2002),A brief survey of 20th century logical notations (see chart on page 2).
^Frege, G. (1879).Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a/S.: Verlag von Louis Nebert. p. 10.
^^a ^b ^cRussell (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).
^Peano (1889)Arithmetices principia, nova methodo exposita.
^^a ^bSchönfinkel (1924) Über die Bausteine der mathematischen Logik, translated asOn the building blocks of mathematical logic in From Frege to Gödel edited by van Heijenoort.
^Peirce (1867)On an improvement in Boole's calculus of logic.
^Hilbert, D. (1918). Bernays, P. (ed.).Prinzipien der Mathematik. Lecture notes at Universität Göttingen, Winter Semester, 1917-1918; Reprinted asHilbert, D. (2013). "Prinzipien der Mathematik". In Ewald, W.; Sieg, W. (eds.).David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933. Heidelberg, New York, Dordrecht and London: Springer. pp. 59–221.
^Bourbaki, N. (1954).Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14.
^Frege, G. (1879).Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (in German). Halle a/S.: Verlag von Louis Nebert. p. 15.
^Becker, A. (1933).Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I (in German). Berlin: Junker und Dünnhaupt Verlag. p. 4.
^Bourbaki, N. (1954).Théorie des ensembles (in French). Paris: Hermann & Cie, Éditeurs. p. 32.
^Gentzen (1934)Untersuchungen über das logische Schließen.
^Chazal (1996) : Éléments de logique formelle.
^Hilbert, D. (1905) [1904]. "Über die Grundlagen der Logik und der Arithmetik". In Krazer, K. (ed.).Verhandlungen des Dritten Internationalen Mathematiker Kongresses in Heidelberg vom 8. bis 13. August 1904. pp. 174–185.
^Bocheński (1959),A Précis of Mathematical Logic, passim.
^O'Donnell, John; Hall, Cordelia; Page, Rex (2007).Discrete Mathematics Using a Computer. Springer. p. 120.ISBN 9781846285981..
^Allen, Colin; Hand, Michael (2022).Logic primer (3rd ed.). Cambridge, Massachusetts: The MIT Press.ISBN 978-0-262-54364-4.
^Jackson, Daniel (2012).Software Abstractions: Logic, Language, and Analysis. MIT Press. p. 263.ISBN 9780262017152..
^Pinter, Charles C. (2014).A book of set theory. Mineola, New York: Dover Publications, Inc. pp. 26–29.ISBN 978-0-486-49708-2.
^^a ^b"Set operations".www.siue.edu. Retrieved2024-06-11.
^^a ^b ^c ^d ^e"1.5 Logic and Sets".www.whitman.edu. Retrieved2024-06-11.
^"Theory Set".mirror.clarkson.edu. Retrieved2024-06-11.
^"Set Inclusion and Relations".autry.sites.grinnell.edu. Retrieved2024-06-11.
^"Complement and Set Difference".web.mnstate.edu. Retrieved2024-06-11.
^Cooper, A."Set Operations and Subsets – Foundations of Mathematics". Retrieved2024-06-11.
^^a ^b"Basic concepts".www.siue.edu. Retrieved2024-06-11.
^Cooper, A."Set Operations and Subsets – Foundations of Mathematics". Retrieved2024-06-11.
^Cooper, A."Set Operations and Subsets – Foundations of Mathematics". Retrieved2024-06-11.

Sources

[edit]

Bocheński, Józef Maria (1959),A Précis of Mathematical Logic, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
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.
Enderton, Herbert (2001).A Mathematical Introduction to Logic (2nd ed.). Boston, MA: Academic Press.ISBN 978-0-12-238452-3.
Gamut, L.T.F (1991). "Chapter 2".Logic, Language and Meaning. Vol. 1. University of Chicago Press. pp. 54–64.OCLC 21372380.
Rautenberg, W. (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..
Humberstone, Lloyd (2011).The Connectives. MIT Press.ISBN 978-0-262-01654-4.

External links

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Wikimedia Commons has media related toLogical connectives.

"Propositional connective".Encyclopedia of Mathematics.EMS Press. 2001 [1994].
Lloyd Humberstone (2010), "Sentence Connectives in Formal Logic",Stanford Encyclopedia of Philosophy (anabstract algebraic logic approach to connectives)
John MacFarlane (2005), "Logical constants",Stanford Encyclopedia of Philosophy.

v t e Commonlogical connectives
Tautology/True $\top$
Alternative denial (NAND gate) ${\overline {\wedge }}$ Converse implication $\Leftarrow$ Implication (IMPLY gate) $\Rightarrow$ Disjunction (OR gate) $\lor$
Negation (NOT gate) $\neg$ Exclusive or (XOR gate) $\oplus$ Biconditional (XNOR gate) $\odot$ Statement (Digital buffer)
Joint denial (NOR gate) ${\overline {\vee }}$ Nonimplication (NIMPLY gate) $\nRightarrow$ Converse nonimplication $\nLeftarrow$ Conjunction (AND gate) $\land$
Contradiction/False $\bot$
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Movatterモバイル変換

Overview

List of common logical connectives

History of notations

Redundancy

Natural language

Properties

Order of precedence

Table and Hasse diagram

Applications

Computer science

Set theory

See also

References

Sources

External links