logical

Dictionary: log·i·cal (lŏj'ĭ-kəl) pronunciation

adj.

Of, relating to, in accordance with, or of the nature of logic.
Based on earlier or otherwise known statements, events, or conditions; reasonable: Rain was a logical expectation, given the time of year.
Reasoning or capable of reasoning in a clear and consistent manner.

logicality log'i·cal'i·ty (-kăl'ĭ-tē) or log'i·cal·ness n.
logically log'i·cal·ly adv.

SYNONYMS logical, analytic, ratiocinative, rational. These adjectives mean capable of or reflecting the capability for correct and valid reasoning: a logical mind; an analytic thinker; the ratiocinative process; a rational being.
ANTONYM illogical

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Computer Desktop Encyclopedia: logical

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(1) A reasonable solution to a problem.

(2) A higher level view of an object; for example, the user's view versus the computer's view. See logical vs. physical.

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Thesaurus: logical

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adjective

Consistent with reason and intellect: consequent, intelligent, rational, reasonable. See reason/unreason.
Able to reason validly: analytic, analytical, ratiocinative, rational. See reason/unreason.

Antonyms: logical

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adj

Definition: probable, reasonable
Antonyms: illogical, improbable, irrational, unlikely, unreasonable

Hacker Slang: logical

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[from the technical term logical device, wherein a physical device is referred to by an arbitrary ‘logical’ name] Having the role of. If a person (say, Les Earnest at SAIL) who had long held a certain post left and were replaced, the replacement would for a while be known as the logical Les Earnest. (This does not imply any judgment on the replacement.) Compare virtual.

At Stanford, ‘logical’ compass directions denote a coordinate system relative to El Camino Real, in which ‘logical north’ is always toward San Francisco and ‘logical south’ is always toward San Jose--in spite of the fact that El Camino Real runs physical north/south near San Francisco, physical east/west near San Jose, and along a curve everywhere in between. (The best rule of thumb here is that, by definition, El Camino Real always runs logical north-south.)

In giving directions, one might say: “To get to Rincon Tarasco restaurant, get onto El Camino Bignum going logical north.” Using the word ‘logical’ helps to prevent the recipient from worrying about that the fact that the sun is setting almost directly in front of him. The concept is reinforced by North American highways which are almost, but not quite, consistently labeled with logical rather than physical directions. A similar situation exists at MIT: Route 128 (famous for the electronics industry that grew up along it) wraps roughly 3 quarters around Boston at a radius of 10 miles, terminating near the coastline at each end. It would be most precise to describe the two directions along this highway as ‘clockwise’ and ‘counterclockwise’, but the road signs all say “north” and “south”, respectively. A hacker might describe these directions as logical north and logical south, to indicate that they are conventional directions not corresponding to the usual denotation for those words.

Word Tutor: logical

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IN BRIEF: Capable of or characterized by clear reasoning.

I see mysteries and complications wherever I look, and I have never met a steadily logical person. — Martha Gellhorn

Wikipedia: Logical conjunction

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In logic and mathematics, logical conjunction or and is a two-place logical connective that has the value true if both of its operands are true, otherwise a value of false.

1 Notation
2 Definition
- 2.1 Truth table
- 2.2 Venn diagram
3 Introduction and elimination rules
4 Properties
5 Applications in computer programming
6 Set-theoretic intersection
7 Natural language
8 See also
9 External links

Notation

And is usually expressed with an infix operator. In mathematics and logic, it is usually ∧; in electronics $\cdot$ ; and in programming languages, & or and. Some programming languages have a related control structure, the short-circuit and, written &&, and then, etc.

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:

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)

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.

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.

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:

$A \and B$

$\vdash A$

...or alternately,

$A \and B$

$\vdash B$

Properties

The following properties apply to conjunction:

associativity: $a \land (b \land c) \equiv (a \land b) \land c$

commutativity: $a \land b \equiv b \land a$

distributivity: $(a \land (b \lor c)) \equiv ((a \land b) \lor (a \land c))$

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

idempotency: $a \land a \equiv a$

monotonicity: $(a \rightarrow b) \rightarrow ((c \land a) \rightarrow (c \land b))$

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

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.

If using binary values for true (1) and false (0), then logical conjunction works exactly like normal arithmetic multiplication.

Applications in computer programming

AND logic gate

In high-level computer programming and digital electronics, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "AND", an algebraic multiplication, or the ampersand symbol "&". Many languages also provide short-circuit control structures corresponding to logical conjunction.

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

0 AND 0 = 0,
0 AND 1 = 0,
1 AND 0 = 0,
1 AND 1 = 1.

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. For example:

11000110 AND 10100011 = 10000010.

This can be used to select part of a bitstring using a bit mask. 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.

Logical conjunction "AND" is also used in SQL operations to form database queries.

The Curry-Howard correspondence relates logical conjunction to product types.

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, set-theoretic intersection shares several properties with logical conjunction, such as associativity, commutativity, and idempotence.

Natural language

The logical conjunction and in logic is related to, but not the same as, the grammatical conjunction and in natural languages.

English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order. For example, "They got married and had a child" in common discourse means that the marriage came before the child. The word "and" can also 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.

External links

Wolfram Mathematics Conjunction

v • d • e Logical connectives

Tautology ( $\top$ )

NAND ( $\uparrow$ ) · Converse implication ( $\leftarrow$ ) · Implication ( $\rightarrow$ ) · OR ( $\or$ )

Negation ( $\neg$ ) · XOR ( $\oplus$ ) · Biconditional ( $\leftrightarrow$ ) · Proposition

NOR ( $\downarrow$ ) · Nonimplication ( $\nrightarrow$ ) · Converse nonimplication ( $\nleftarrow$ ) · AND ( $\and$ )

Contradiction ( $\bot$ )

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Translations: Logical

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Dansk (Danish)
adj. - logisk

Nederlands (Dutch)
logisch, rationeel, analytisch beredeneerd, vanzelfsprekend, systematisch uitgewerkt, samenhangend, weloverdacht

Français (French)
adj. - logique

Deutsch (German)
adj. - logisch

Ελληνική (Greek)
adj. - λογικός

Italiano (Italian)
logico

Português (Portuguese)
adj. - lógico, razoável, natural

Русский (Russian)
логический, последовательный, разумный

Español (Spanish)
adj. - lógico, dialéctico, consecuente, natural

Svenska (Swedish)
adj. - logisk, följdriktig

中文（简体）(Chinese (Simplified))
合乎逻辑的, 合理的

中文（繁體）(Chinese (Traditional))
adj. - 合乎邏輯的, 合理的

한국어 (Korean)
adj. - 논리적인 , 논리적으로 생각하는, 조리에 닿는

日本語 (Japanese)
adj. - 論理的な, 当然の, 論理学の

العربيه (Arabic)
‏(صفه) منطقي, منطقي التفكير‏

עברית (Hebrew)
adj. - ‮הגיוני, שכלי, לוגי, סביר, עקבי‬

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Best of the Web: logical

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Some good "logical" pages on the web:

Math
mathworld.wolfram.com

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