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In logic and/or mathematics, logical conjunction or and is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. Logic is the study of the principles of valid demonstration and Inference. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence

Contents

Definition

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 both of its operands are true. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence

Truth table

The truth table of p AND q (also written as p ∧ q or p & q (logic), p && q (some programming languages), or p\cdotq (electronics)). A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional

p q ∧
T T T
T F F
F T F
F F F

Venn diagram

The Venn diagram of "A and B" (the red area is true)

A and B

The analogue of conjunction for a (possibly infinite) family of statements is universal quantification, which is part of predicate logic. Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every In Mathematical logic, predicate logic is the generic term for symbolic Formal systems like First-order logic, Second-order logic, Many-sorted

Introduction and elimination rules

As a rule of inference, conjunction introduction is a classically valid, simple argument form. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of In Logic, the argument form or test form of an Argument results from replacing the different words or sentences that make up the argument with letters The argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction.

A,
B.
Therefore, A and B.

or in logical operator notation:

A,
B
\vdash A \and B

Here is an example of an argument that fits the form conjunction introduction:

Everyone should vote. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence Conjunction introduction is the Inference that if p is true and q is true then the conjunction p and q is true
Democracy is the best system of government.
Therefore, everyone should vote and democracy is the best system of government.

Conjunction elimination is another classically valid, simple argument form. In Mathematical logic, simplification (equivalent to conjunction elimination) is a valid Argument and Rule of inference which makes The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of In Logic, the argument form or test form of an Argument results from replacing the different words or sentences that make up the argument with letters Intuitively, it permits the inference from any conjunction of either element of that conjunction.

A and B.
Therefore, A.

. . . or alternately,

A and B.
Therefore, B.

In logical operator notation:

A \and B
\vdash A

. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence . . or alternately,

A \and B
\vdash B

Properties

The following properties apply to conjunction:

(a \land (b \land c)) \equiv ((a \land b) \land (a \land c))
(a \rightarrow b) \rightarrow ((a \land c) \rightarrow (b \land c))

If using binary values for true (1) and false (0), then logical conjunction works exactly like normal arithmetic multiplication. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1.

Applications in computer programming

AND logic gate
AND logic gate

In high-level computer programming, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "AND" or the ampersand symbol "&". A logic gate performs a logical operation on one or more logic inputs and produces a single logic output Many languages also provide short-circuit control structures corresponding to logical conjunction. Short-circuit evaluation or minimal evaluation denotes the semantics of some Boolean operators in some Programming languages in which the second argument

Logical conjunction is often used for bitwise operations, where 0 corresponds to false and 1 to true:

The operation can also be applied to two binary words viewed as bitstrings of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. A word is a unit of Language that carries meaning and consists of one or more Morphemes which are linked more or less tightly together and has a Phonetic A bitstring is a sequence of Bits Anything on a Discrete computer can be represented by a bitstring For example:

This can be used to select part of a bitstring using a bit mask. In Computer science, a mask is data that is used for Bitwise operations. For example, 10011101 AND 00001000  =  00001000 extracts the fifth bit of an 8-bit bitstring.

In computer networking, bit masks are used to derive the network address of a subnet within an existing network from a given IP address, by ANDing the IP address and the subnet mask. Computer networking is the Engineering Discipline concerned with communication between Computer systems or devices Networking routers In Computer networks based on the Internet Protocol Suite, a subnetwork, or subnet, is a portion of the network's computers and network devices that have An Internet Protocol ( IP) address is a numerical identification ( Logical address) that is assigned to devices participating in a Computer network In Computer networks based on the Internet Protocol Suite, a subnetwork, or subnet, is a portion of the network's computers and network devices that have

Logical conjunction "AND" is also used in SQL operations to form database queries. A Computer Database is a structured collection of records or data that is stored in a computer system

Set-theoretic intersection

The intersection used in set theory is defined in terms of a logical conjunction: x ∈ A ∩ B if and only if (x ∈ A) ∧ (x ∈ B). In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Because of this, set-theoretic intersection shares several properties with logical conjunction, such as associativity, commutativity, and idempotence. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

Rhetorical considerations

The classical "trivium" divides the study of articulate argumentation into the disciplines of grammar, logic, and rhetoric. In medieval universities, the trivium comprised the three subjects taught first Grammar, Logic, and Rhetoric. Grammar is the field of Linguistics that covers the Rules governing the use of any given natural language. Logic is the study of the principles of valid demonstration and Inference. Rhetoric has had many definitions no simple definition can do it justice Grammar concerns those aspects of language that are internal to the language itself, in other words, that can be abstracted from considerations of the object world and the language user. Logic deals with the properties of language and reasoning that are independent of particular manners of interpretation and invariant over conceivable languages. Rhetoric treats those aspects of language and its use in reasoning that necessarily take the nature of the interpreter into consideration.

Natural languages are evolved for many purposes beyond their use in logical argumentation, and so any study of logic in a natural language context must sort out those aspects of natural language that are pertinent to its use in logic and those that are not.

English "and" has properties not captured by logical conjunction, because "and" can sometimes imply order. For example, "They got married and had a child" in common discourse means that the marriage came before the child. Then again the word "and" in common usage can imply a partition of a thing into parts, as "The American flag is red, white, and blue. " Here it is not meant that the flag is at once red, white, and blue, but rather that it has a part of each color.

A minor issue of logic and language is the role of the word "but". Logically, the sentence "it's raining, but the sun is shining" is equivalent to "it's raining, and the sun is shining", so logically, "but" is equivalent to "and". However, "but" and "and" are semantically distinct in natural language. Speakers use "but", a conjunction of contradiction, to mark their surprise or reservation vis-a-vis a circumstance that goes against a trend.

One way to resolve this problem of correspondence between symbolic logic and natural language is to observe that the first sentence (using "but"), implies the existence of a hidden but mistaken assumption, namely that the sun does not shine when it rains. We might say that, given probability p that it rains and the sun shines, and probability 1 − p that it rains and the sun does not shine, or that it does not rain at all, we would say "but" in place of "and" when p was low enough to warrant our incredulity.

That implication captures the semantic difference of "and" and "but" without disturbing their logical equivalence. On the other hand, in Brazilian logic, the logical equivalence is broken between A BUT NOT B (where "BUT NOT" is a single operator) and A AND (NOT B), which is a weaker statement. A paraconsistent logic is a Logical system that attempts to deal with Contradictions in a discriminating way In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition

"But" is also sometimes disjunctive (It never rains but it pours); sometimes minutive (Canada has had but three shots on goal); sometimes contrastive (He was not God, but merely an exalted man); sometimes a spatial preposition (He's waiting but the house); and sometimes interjective (My, but that's a lovely boat). These uses await semantic assimilation with conjunctive "but".

Like "and", "but" is sometimes non-commutative: "He got here, but he got here late" is not equivalent to "He got here late, but he got here". This example shows also that unlike "and", "but" can be felicitously used to conjoin sentences that entail each other; compare "He got here late, and he got here".

See also

External links

An and-inverter graph (AIG is a directed acyclic graph that represents a structural implementation of the logical functionality of a circuit or network. If two conditions are combined by and, they must both be true for the compound condition to be true as well In Computer programming, a bitwise operation operates on one or two Bit patterns or binary numerals at the level of their individual Bits On most Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s This is a list of topics around Boolean algebra and propositional logic. In the theory of Relational databases, a Boolean conjunctive query is a Conjunctive query without distinguished predicates i In Mathematics and Abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and In Mathematics, a (finitary Boolean function is a function of the form f: B k &rarr B, where B  = {0 1} A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f: X → B, where X is an arbitrary set Conjunction introduction is the Inference that if p is true and q is true then the conjunction p and q is true In Mathematical logic, simplification (equivalent to conjunction elimination) is a valid Argument and Rule of inference which makes First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science A logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values Peano-Russell notation was Bertrand Russell 's application of Peano 's logical notation to the logical notions of Frege and was used in the writing of This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic"
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