Information about Logical And
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!
q (electronics)).
The analogue of conjunction for a (possibly infinite) family of statements is universal quantification, which is part of predicate logic.
or in logical operator notation:
Here is an example of an argument that fits the form conjunction introduction:
Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.
...or alternately,
In logical operator notation:
...or alternately,
Boolean "and" is also used in SQL operations. Some database systems are case-sensitive and require "AND".
In computer science, the AND operator can be used to select part of a bitstring using a bit mask. For example,
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.
"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".
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.Truth table
The truth table of p AND q (also written as p ∧ q (logic), p && q (computer science), or pq (electronics)).
| p | q | & |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Venn diagram
The Venn diagram of "A and B")
A and B
The analogue of conjunction for a (possibly infinite) family of statements is universal quantification, which is part of predicate logic.
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.- A,
- B.
- Therefore, A and B.
or in logical operator notation:
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.
- A and B.
- Therefore, A.
...or alternately,
- A and B.
- Therefore, B.
In logical operator notation:
...or alternately,
Properties
The following properties apply to conjunction:
- ::
- ::
- truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of conjunction.
- falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false' as a result of conjunction.
Applications in computer science
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
Use in programming
In high-level computer programming, the logical conjunction "and" is commonly represented by the infix operator "and" or the ampersand (&). Many languages also provide short-circuit control structures corresponding to logical conjunction.Boolean "and" is also used in SQL operations. Some database systems are case-sensitive and require "AND".
In computer science, the AND operator can be used to select part of a bitstring using a bit mask. For example,
10011101 AND 00001000 = 00001000 examines the fifth bit of a bitstring.
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). 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.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 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.
"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
- And-inverter graph
- AND gate
- Binary and
- Bitwise AND
- Boolean algebra (logic)
- Boolean algebra topics
- Boolean conjunctive query
- Boolean domain
- Boolean function
- Boolean-valued function
- Conjunction introduction
- Conjunction elimination
- First-order logic
- Grammatical conjunction
- Logical disjunction
- Logical graph
- Logical value
- Operation
- Peano-Russell notation
- Propositional calculus
- Zeroth order logic
External links
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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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 classical logic, the only possible truth values are true and false.
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In classical logic, the only possible truth values are true and false.
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proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
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A truth table is a mathematical table used in logic — specifically in connection with Boolean algebra, boolean functions, and propositional calculus — to compute the functional values of logical expressions on each of their functional arguments, that is, on each
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Venn diagrams are illustrations used in the branch of mathematics known as set theory. They show all of the possible mathematical or logical relationships between sets (groups of things).
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The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.
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In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing.
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predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified.
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validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false).
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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.
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Conjunction introduction is the inference that, if p is true, and q is true, then the conjunction p and q is true.
For example, if it's true that it's raining, and it's true that I'm inside, then it's true that it's raining, and I'm inside.
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For example, if it's true that it's raining, and it's true that I'm inside, then it's true that it's raining, and I'm inside.
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In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true.
For instance, if it's true that it's raining, and I'm inside, then one may assert either term of the conjunction alone: it's raining, or I'm inside.
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For instance, if it's true that it's raining, and I'm inside, then one may assert either term of the conjunction alone: it's raining, or I'm inside.
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validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false).
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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.
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associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:
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- 4 • (2 + 3) = (4 • 2) + (4 • 3).
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Idempotence IPA: /ˌaɪdɨmˈpoʊtənts/ describes the property of operations in mathematics and computer science that yield the same result after the operation is applied multiple times.
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monotonic function (or monotone function) is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
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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 classical logic, the only possible truth values are true and false.
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In classical logic, the only possible truth values are true and false.
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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 classical logic, the only possible truth values are true and false.
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In classical logic, the only possible truth values are true and false.
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binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.
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Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:
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Short-circuit evaluation or minimal evaluation denotes the semantics of some boolean operators in some programming languages in which the second argument is only executed or evaluated if the first argument does not suffice to determine the value of the expression: when the
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SQL
Paradigm: multi-paradigm
Appeared in: 1974
Designed by: Donald D. Chamberlin and Raymond F. Boyce
Developer: IBM
Latest release: SQL:2003/ 2003
Typing discipline: static, strong
Major implementations: Many
SQL
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Paradigm: multi-paradigm
Appeared in: 1974
Designed by: Donald D. Chamberlin and Raymond F. Boyce
Developer: IBM
Latest release: SQL:2003/ 2003
Typing discipline: static, strong
Major implementations: Many
SQL
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