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Concept
Logical conjunction
Summary
In logic, mathematics and linguistics, and () is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as or or (prefix) or or in which is the most modern and widely used.
The and of a set of operands is true if and only if all of its operands are true, i.e., is true if and only if is true and is true.
An operand of a conjunction is a conjunct.
Beyond logic, the term "conjunction" also refers to similar concepts in other fields:
In natural language, the denotation of expressions such as English "and";
In programming languages, the short-circuit and control structure;
In set theory, intersection.
In lattice theory, logical conjunction (greatest lower bound).
And is usually denoted by an infix operator: in mathematics and logic, it is denoted by (Unicode ), or ; in electronics, ; and in programming languages, &, &&, or and. In Jan Łukasiewicz's prefix notation for logic, the operator is , for Polish koniunkcja.
Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if (also known as iff) both of its operands are true.
The conjunctive identity is true, which is to say that AND-ing an expression with true will never change the value of the expression. In keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result true.
The truth table of :
In systems where logical conjunction is not a primitive, it may be defined as
or
As a rule of inference, conjunction introduction is a classically valid, simple argument form. The argument form has two premises, and . Intuitively, it permits the inference of their conjunction.
Therefore, A and B.
or in logical operator notation:
Here is an example of an argument that fits the form conjunction introduction:
Bob likes apples.
Bob likes oranges.
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Logical connective
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 can be used to join the two atomic formulas and , rendering the complex formula . Common connectives include negation, disjunction, conjunction, implication, and equivalence.
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬.
Logical disjunction
In logic, disjunction, also known as logical disjunction or logical or or logical addition or inclusive disjunction , is a logical connective typically notated as and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula , assuming that abbreviates "it is sunny" and abbreviates "it is warm". In classical logic, disjunction is given a truth functional semantics according to which a formula is true unless both and are false.