Definition:Conjunction
Definition
Conjunction is a binary connective written symbolically as $p \land q$ whose behaviour is as follows:
- $p \land q$
is defined as:
- $p$ is true and $q$ is true.
This is called the conjunction of $p$ and $q$.
$p \land q$ is voiced:
- $p$ and $q$.
General Definition
Let $p_1, p_2, \ldots, p_n$ be statements.
The conjunction of $p_1, p_2, \ldots, p_n$ is defined as:
- $\ds \bigwedge_{i \mathop = 1}^n \ p_i = \begin {cases}
p_1 & : n = 1 \\ & \\ \ds \paren {\bigwedge_{i \mathop = 1}^{n - 1} p_i} \land p_n & : n > 1 \end {cases}$
That is:
- $\ds \bigwedge_{i \mathop = 1}^n \ p_i = p_1 \land p_2 \land \cdots \land p_{n - 1} \land p_n$
In terms of the set $P = \set {p_1, \ldots, p_n}$, this can also be rendered:
- $\ds \bigwedge P$
and is referred to as the conjunction of $P$.
Truth Function
The conjunction connective defines the truth function $f^\land$ as follows:
\(\ds \map {f^\land} {\F, \F}\) | \(=\) | \(\ds \F\) | ||||||||||||
\(\ds \map {f^\land} {\F, \T}\) | \(=\) | \(\ds \F\) | ||||||||||||
\(\ds \map {f^\land} {\T, \F}\) | \(=\) | \(\ds \F\) | ||||||||||||
\(\ds \map {f^\land} {\T, \T}\) | \(=\) | \(\ds \T\) |
Truth Table
The characteristic truth table of the logical conjunction operator $p \land q$ is as follows:
- $\begin{array}{|cc||c|} \hline
p & q & p \land q \\ \hline \F & \F & \F \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$
Boolean Interpretation
The truth value of $\mathbf A \land \mathbf B$ under a boolean interpretation $v$ is given by:
- $\map v {\mathbf A \land \mathbf B} = \begin{cases}
\T & : \map v {\mathbf A} = \map v {\mathbf B} = \T \\ \F & : \text{otherwise} \end{cases}$
Conjunct
The statements $p$ and $q$ are known as the conjuncts.
Semantics of the Conjunction
The conjunction is used to symbolise any statement in natural language such that two substatements are held to be true simultaneously.
Thus it is also used to symbolise the concept of but as well as and.
Thus $p \land q$ can be also interpreted as:
- $p$ and $q$
- $p$ but $q$
- $p$, yet $q$
- $p$, although $q$
- $p$; still, $q$
- $p$; however, $q$
- $p$; on the other hand $q$
- $p$; moreover $q$
- $p$; furthermore, $q$
- $p$; nevertheless, $q$
- Not only $p$ but also $q$
- Despite $p$, $q$
Warning
Beware of the usage of and in natural language which has the following form:
- He fell out of bed and broke his leg
which does not mean the same as:
- He broke his leg and fell out of bed.
This use of and actually means and then, as it is implicit that the two occurrences are neither simultaneous nor independent, but that the second occurrence happens as a result of the first.
Notational Variants
Various symbols are encountered that denote the concept of logical conjunction:
Symbol | Origin | Known as |
---|---|---|
$p \land q$ | wedge | |
$p \mathop {\mathsf {AND} } q$ | ||
$p \mathop . q$ | 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica | dot |
$p \mathop \And q$ | Ampersand | |
$p \mathop {\And \And} q$ | Used in various computer programming languages | |
$\operatorname K p q$ | Łukasiewicz's Polish notation |
Also known as
The conjunction is also known as the logical product.
The conjuncts are thence known as the factors of the logical product.
Some sources refer to the conjunction merely as and.
Treatments which consider logical connectives as functions may refer to this operator as the conjunctive function.
Examples
Roses and Violets
- Roses are red and violets are blue
is a conjunction whose conjuncts are:
- Roses are red
- Violets are blue.
London and Manchester
- London is a city and Manchester is a city
is a conjunction whose conjuncts are:
- London is a city
- Manchester is a city.
Also see
- Results about conjunction can be found here.
Sources
- 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.7$: Sentential Calculus
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.3$: Basic Truth-Tables of the Propositional Calculus
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 2$: The Axiom of Specification
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 1$
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $3$ Conjunction and Disjunction
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Connectives
- 1972: Patrick Suppes: Axiomatic Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.1$: Simple and Compound Statements
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 2$: Logical Constants $(1)$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
- 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): and
- 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.1$: Boolean operators
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set?
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.1$: Introduction
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): and
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conjunction
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.1$: Declarative sentences
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): and
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conjunction
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 6$ Significance of the results
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjunction