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Encyclopedia > Logical conjunction

In logic and mathematics, logical conjunction (usual symbol 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. AND Logic Gate File links The following pages link to this file: Logical conjunction Categories: User-created public domain images ... AND Logic Gate File links The following pages link to this file: Logical conjunction Categories: User-created public domain images ... A logic gate is an arrangement of controlled switches used to calculate operations using Boolean logic in digital circuits. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...

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. In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... Proposition is a term used in logic to describe the content of assertions. ...


The truth table of p AND q (also written as p ∧ q, p & q, or pcdotq) is as follows: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

Truth Table: Logical Conjunction
p q p ∧ q
F F F
F T F
T F F
T T T

Intuitively, the logical operator works the same as the common English word "and". The sentence "it's raining, and I'm inside" asserts that two things are simultaneously true: that it is raining outside, and that I am inside. Logically, this would be denoted by saying that A stands for "it's raining", B stands for "I'm inside", together A AND B.


For example, consider:

x > 13 AND x < 27.

If x is 36, then x > 13 is true, but x < 27 is false, so this sentence is false. But if x is 20, then both parts of the sentence are true, so the entire conjunction is also true.


The analogue of conjunction for a (possibly infinite) family of statements is universal quantification, which is part of predicate logic. Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... ...


Introduction and elimination rules

As a rule of inference. conjunction introduction is a classically valid, simple argument form. The argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction. This article discusses validity in logic, for the term in the social sciences see validity (psychometric). ... 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, along the lines of algebra; the letters represent logical variables. ...

A,
B.
Therefore, A and B.

or in logical operator notation: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

A,
B
vdash A and B

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

Everyone should vote.
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. Intuitively, it permits the inference from any conjunction of either element of that conjunction. This article discusses validity in logic, for the term in the social sciences see validity (psychometric). ... 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, along the lines of algebra; the letters represent logical variables. ...

A and B.
Therefore, A.

...or alternately,

A and B.
Therefore, B.

In logical operator notation: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

A and B
vdash A

...or alternately,

A and B
vdash B

Algebraic properties

One can also chain conjunctions, such as A AND B AND C, which is logically equivalent both to (A AND B) AND C and to A AND (B AND C). This statement is true if A, B, and C are simultaneously true. In fancier language, conjunction is associative. It's also commutative; A AND B is the same as B AND A. In logic, statements p and q are logically equivalent if they have the same logical content. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


Bitwise operation

Logical conjunction is often used for bitwise operations. Examples:

  • 0 and 0 = 0
  • 0 and 1 = 0
  • 1 and 0 = 0
  • 1 and 1 = 1
  • 1100 and 1010 = 1000

Note that in computer science, the AND operator can be used to set a bit to 0 by AND-ing the bit with 0 (A AND 0 = 0 for any (binary) value of A). This principle is called a "bit mask". For example, if you have a 4-byte-integer holding a color value, which could be described as 0xAABBGGRR (R-red; G-green; B-blue; A-alpha), you may want to select one of the colors. The bit mask for green would be 0x0000FF00. If you apply this bit mask to the 4-byte-integer, it only leaves the bits belonging to green intact (0x0000GG00).


Set-theoretic intersection

The intersection used in set theory is defined in terms of a logical conjunction: xAB if and only if (xA) ∧ (xB). Because of this, logical conjunction satisfies many of the same identities as set-theoretic intersection, such as associativity, commutativity, distributivity, and de Morgan's laws. 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, all elements of B that also belong to A), but no other elements. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... note that demorgans laws are also a big part in circut design. ...


Rhetorical considerations

The classical "trivium" divides the study of articulate argumentation into the disciplines of grammar, logic, and rhetoric. Grammar concerns those aspects of language that are internal to the language itself, in other words, that can be abstracted from ocnsiderations 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 langagues. Rhetoric treats those aspects of language and its use in reasoning that necessarily take the nature of the interpreter into consideration. ... Grammar is the study of rules governing the use of language. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Rhetoric from Greek ρήτωρ, rhêtôr, orator) is the art or technique of persuasion, usually through the use of language. ...


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.


At least on the face of it, English "and" has properties not captured by logical conjunction. Unlike logical conjunction, the use "and" in English is not always commutative. For example, "They got married and had a baby" is not normally interpreted to have the same connotations as "They had a baby and got married". Some lists conjoined by "and" do not allow an operation analogous to logical and-elimination. For example, "The American flag is red, white, and blue" does not support the inference that the whole American flag is white.


There are everyday examples when the English word "and" is used with the meaning of logical disjunction or OR. The following two sentences are often interpreted the same. "Damage caused by scratches or dents is chargeable." "Damage caused by scratches and dents is chargeable." OR logic gate. ...


Many examples of "or" in English seem to function as logical conjunctions. The sentence "It might work, or it might not" is equivalent to "It might work, and it might not". Arguably, the "or" in "You may have ice cream or you may have cake" functions conjunctively, since it quantifies truth universally, over both conjuncts---though obviously not also over an unstated third option, in which both ice cream and cake are eaten simultaneoously. Truth-functionality is what is at issue here, and this "or" guarantees the truth of both its conjuncts, even if in fact it forbids having both ice cream and cake. A "true" disjunction would function as a kind of existential quantifier, guaranteeing the truth of only one of the two elements it conjoined. In this sort of case an utterance of the cake sentence would initiate a kind of guessing game: one response might be "Is it ice cream?".


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, in natural language, "but" and "and" are semantically distinct. The former sentence suggests that the latter sentence is usually a contradiction. The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ...


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. In logic, Brazilian logic is a name given by Chris Mortensen, in his book Inconsistent Mathematics, to a system R# of relevance logic. ... Negation (i. ...


"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

Other operators

Exclusive disjunction (usual symbol XOR occasionally EOR) is a logical operator that results in true if one of the operands, but not both of them, is true. ... OR logic gate. ... XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ... In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...

Related topics

Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... Algebra of sets Ampheck Boole, George Boolean algebra Boolean domain Boolean function Boolean logic Boolean implicant Boolean prime ideal theorem Boolean-valued function Boolean-valued model Boolean satisfiability problem Booles syllogistic Canonical form (Boolean algebra) Characteristic function Compactness theorem Complete Boolean algebra De Morgan, Augustus De Morgans laws... A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ... In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the two-element boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is... Boolean logic is a system of syllogistic logic invented by 19th-century British mathematician George Boole, which attempts to incorporate the empty set, that is, a class of non-existent entities, such as round squares, without resorting to uncertain truth values. ... A boolean-valued function is a function of the type , where is an arbitrary set, where is a generic 2-element set, typically , and where the latter is frequently interpreted for logical applications as . ... It has been suggested that Predicate calculus be merged into this article or section. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In mathematical logic, propositional logic is the logic of mathematical objects called propositions. ... Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. ...

External links


  Results from FactBites:
 
Logical conjunction - Wikipedia, the free encyclopedia (1075 words)
In logic and mathematics, logical conjunction (usual symbol 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.
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.
The analogue of conjunction for a (possibly infinite) family of statements is universal quantification, which is part of predicate logic.
Logical conjunction - Gurupedia (431 words)
In logic and technical fields that use it, conjunction, or and, is a logical operator in logical calculi.
Conjunction as we've described it is a binary operator, meaning that it combines two terms into a single statement.
The analogue of conjunction for a (possibly infinite) familiy of statements is universal quantification, which is part of predicate logic.
  More results at FactBites »

 
 

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