Logical connective explained

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

and

, rendering the complex formula

P\lorQ

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]

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.

Symbol, name		Truth table						Venn diagram	- ! colspan=8	Zeroary connectives (constants)	-	⊤					1			-	⊥					0			- ! colspan=8	Unary connectives	-			p =	0	1		-		Proposition p		0		1			-	¬			1		0			- ! colspan=9	Binary connectives	- style="background-color:#ffff66; text-align:center;"	p =	0	0	1	1	- style="background-color:#ffff66; text-align:center;"	q =	0	1	0	1	-		Proposition p		0	0	1	1		-		Proposition q		0	1	0	1		-	∧			0	0	0	1		-	↑			1	1	1	0		-	∨			0	1	1	1		-	↓			1	0	0	0		-	→			1	1	0	1		-	\nleftrightarrow			0	1	1	0		-	↔			1	0	0	1		-	←			1	0	1	1		-

List of common logical connectives

Commonly used logical connectives include the following ones.^[2]

Negation (not):

\neg

\sim

(prefix) in which

\neg

is the most modern and widely used, and

\sim

is also common;

Conjunction (and):

\wedge

(prefix) in which

\wedge

is the most modern and widely used;

Disjunction (or):

\vee

(prefix) in which

\vee

is the most modern and widely used;

Implication (if...then):

\to

\supset

⇒

(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

(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 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 (

\negp

);

It is raining and I am indoors (

p\wedgeq

);

It is raining or I am indoors (

p\lorq

);

If it is raining, then I am indoors (

p → q

);

If I am indoors, then it is raining (

q → p

);

I am indoors if and only if it is raining (

p\leftrightarrowq

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:

\top

(prefix), or

;

False formula:

\bot

(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

\neg

appeared in Heyting in 1930^[3] ^[4] (compare to Frege's symbol ⫟ in his Begriffsschrift^[5]); the symbol

\sim

appeared in Russell in 1908;^[6] an alternative notation is to add a horizontal line on top of the formula, as in

\overline{p}

; another alternative notation is to use a prime symbol as in

Conjunction: the symbol

\wedge

appeared in Heyting in 1930^[3] (compare to Peano's use of the set-theoretic notation of intersection

\cap

^[7]); the symbol

appeared at least in Schönfinkel in 1924;^[8] the symbol

⋅

comes from Boole's interpretation of logic as an elementary algebra.

Disjunction: the symbol

\vee

appeared in Russell in 1908^[6] (compare to Peano's use of the set-theoretic notation of union

\cup

); 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.^[9]

Implication: the symbol

\to

appeared in Hilbert in 1918;^[10]

\supset

was used by Russell in 1908^[6] (compare to Peano's Ɔ the inverted C);

⇒

appeared in Bourbaki in 1954.^[11]

Equivalence: the symbol

\equiv

in Frege in 1879;^[12]

\leftrightarrow

in Becker in 1933 (not the first time and for this see the following);^[13]

\Leftrightarrow

appeared in Bourbaki in 1954;^[14] other symbols appeared punctually in the history, such as

\supset\subset

in Gentzen,^[15]

\sim

in Schönfinkel^[8] or

\subset\supset

in Chazal, ^[16]

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:

\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);^[17]

for negation,

Kpq

for conjunction,

Dpq

for alternative denial,

Apq

for disjunction,

Cpq

for implication,

Epq

for biconditional in Łukasiewicz in 1929.

Redundancy

Such a logical connective as converse implication "

\leftarrow

" 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

\negp\veeq

and

p\toq

. 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 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.^[18] 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:

\{\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, 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	Symbol	Logical gate
not	\neg	NOT
and	\wedge	AND
or	\vee	OR
if...then	\to	IMPLY
...if	\leftarrow
either...or	⊕	XOR
if and only if	\leftrightarrow	XNOR
not both	\uparrow	NAND
neither...nor	\downarrow	NOR
but not	\not\to	NIMPLY

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.

a\land(a\lorb)=a

for all operands, .

Monotonicity: If f(a₁, ..., a_n) ≤ f(b₁, ..., b_n) for all a₁, ..., a_n, b₁, ..., b_n ∈ such that a₁ ≤ b₁, a₂ ≤ b₂, ..., a_n ≤ b_n. E.g., ∨, ∧, ⊤, ⊥.

\nleftrightarrow

, ⊤, ⊥.

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., ¬.

\nleftrightarrow

, ⊥, ⊄, ⊅ (see validity).

Involutivity (for unary connectives): . 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 (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.

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,

P\veeQ\wedge{\negR} → S

is short for

(P\vee(Q\wedge(\negR))) → S

Here is a table that shows a commonly used precedence of logical operators.^[19] ^[20]

Operator	Precedence
\neg	1
\land	2
\lor	3
\to	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.^[21] 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

The 16 logical connectives can be partially ordered to produce the following Hasse diagram. The partial order is defined by declaring that

x\leqy

if and only if whenever

holds then so does

Applications

Logical connectives are used in computer science and in set theory.

Computer science

See main article: article and Logic gate. A truth-functional approach to logical operators is implemented as logic gates in digital circuits. Practically all digital circuits (the major exception is DRAM) are built up from NAND, NOR, NOT, and transmission gates; see more details in Truth function in computer science. Logical operators over bit vectors (corresponding to finite Boolean algebras) 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 (P) then Q;, the consequent Q is not executed if the antecedent P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and constructivist views on the material conditional— rather than to classical logic's views.

Set theory

See main article: article and Set theory. Logical connectives are used to define the fundamental operations of set theory,^[22] as follows:

Set theory operations and connectives
Set operation	Connective	Definition
Intersection	Conjunction	A\capB=\{x:x\inA\landx\inB\} ^[23] ^[24] ^[25]
Union		A\cupB=\{x:x\inA\lorx\inB\} ^[26]
Complement		\overline{A}=\{x:x\notinA\} ^[27] ^[28]
Subset	Implication	A\subseteqB\leftrightarrow(x\inA → x\inB) ^[29] ^[30]
Equality		A=B\leftrightarrow(\forallX)[A\inX\leftrightarrowB\inX] ^[31]

This definition of set equality is equivalent to the axiom of extensionality.

Sources

Bocheński, Józef Maria (1959), A Précis of Mathematical Logic, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
Book: Chao . C. . 数理逻辑：形式化方法的应用 . Mathematical Logic: Applications of the Formalization Method . 2023 . Preprint. . Beijing . 15–28 . Chinese.
Book: Enderton . Herbert . Herbert Enderton. A Mathematical Introduction to Logic . Academic Press . Boston, MA . 2nd . 978-0-12-238452-3 . 2001.
Book: Gamut, L.T.F. L. T. F. Gamut. Logic, Language and Meaning. University of Chicago Press. 1991. 1. 54 - 64. Chapter 2. 21372380.
Book: Rautenberg, W.. Wolfgang Rautenberg. 10.1007/978-1-4419-1221-3. A Concise Introduction to Mathematical Logic. . New York. 3rd. 978-1-4419-1220-6. 2010. .
Book: Humberstone, Lloyd. The Connectives. 2011. MIT Press. 978-0-262-01654-4.

External links

- Lloyd Humberstone (2010), "Sentence Connectives in Formal Logic", Stanford Encyclopedia of Philosophy (An abstract algebraic logic approach to connectives.)
John MacFarlane (2005), "Logical constants", Stanford Encyclopedia of Philosophy.

Notes and References

Web site: Cogwheel. What is the difference between logical and conditional /operator/. Stack Overflow. 9 April 2015.
Book: Chao . C. . 数理逻辑：形式化方法的应用 . Mathematical Logic: Applications of the Formalization Method . 2023 . Preprint. . Beijing . 15–28 . Chinese.
Heyting . A. . Die formalen Regeln der intuitionistischen Logik . Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse . 1930 . 42–56 . German.
Denis Roegel (2002), A brief survey of 20th century logical notations (see chart on page 2).
Book: Frege . G. . Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens . 1879 . Verlag von Louis Nebert . Halle a/S. . 10.
[Bertrand Russell|Russell]
[Giuseppe Peano|Peano]
[Moses Schönfinkel|Schönfinkel]
[Charles Sanders Peirce|Peirce]
Book: Hilbert . D. . Bernays . P. . Prinzipien der Mathematik . 1918 . Lecture notes at Universität Göttingen, Winter Semester, 1917-1918 . none.

Reprinted as Encyclopedia: Prinzipien der Mathematik . Hilbert . D. . David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933 . 2013 . Ewald . W. . Sieg . W. . Springer . Heidelberg, New York, Dordrecht and London . 59–221.
Book: Bourbaki . N. . Théorie des ensembles . 1954 . Hermann & Cie, Éditeurs . Paris . 14.
Book: Frege . G. . Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens . 1879 . Verlag von Louis Nebert . Halle a/S. . 15 . German.
Book: Becker . A. . Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I . 1933 . Junker und Dünnhaupt Verlag . Berlin . 4 . German.
Book: Bourbaki . N. . Théorie des ensembles . 1954 . Hermann & Cie, Éditeurs . Paris . 32 . French.
[Gerhard Gentzen|Gentzen]
Chazal (1996) : Éléments de logique formelle.
Encyclopedia: Hilbert . D. . Über die Grundlagen der Logik und der Arithmetik . Verhandlungen des Dritten Internationalen Mathematiker Kongresses in Heidelberg vom 8. bis 13. August 1904 . Krazer . K. . 1904 . 1905 . 174–185.
[Józef Maria Bocheński|Bocheński]
Book: Discrete Mathematics Using a Computer. John. O'Donnell. Cordelia. Hall. Rex. Page. Springer. 2007. 9781846285981. 120. .
Book: Allen, Colin . Logic primer . Hand . Michael . 2022 . The MIT Press . 978-0-262-54364-4 . 3rd . Cambridge, Massachusetts.
Book: Jackson, Daniel. Software Abstractions: Logic, Language, and Analysis . MIT Press. 2012. 9780262017152. 263. .
Book: Pinter, Charles C. . A book of set theory . 2014 . Dover Publications, Inc . 978-0-486-49708-2 . Mineola, New York . 26-29.
Web site: Set operations . 2024-06-11 . www.siue.edu.
Web site: 1.5 Logic and Sets . 2024-06-11 . www.whitman.edu.
Web site: Theory Set . 2024-06-11 . mirror.clarkson.edu.
Web site: Set Inclusion and Relations . 2024-06-11 . autry.sites.grinnell.edu.
Web site: Complement and Set Difference . 2024-06-11 . web.mnstate.edu.
Web site: Cooper . A. . Set Operations and Subsets – Foundations of Mathematics . 2024-06-11 . en-US.
Web site: Basic concepts . 2024-06-11 . www.siue.edu.
Web site: Cooper . A. . Set Operations and Subsets – Foundations of Mathematics . 2024-06-11 . en-US.
Web site: Cooper . A. . Set Operations and Subsets – Foundations of Mathematics . 2024-06-11 . en-US.

Logical connective explained

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

Sources

External links

Notes and References