Logical conjunction
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AND.jpg
Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true.
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Definition
In logic and technical fields that use it, conjunction, or and, is a logical operator in logical calculi, and a rule of inference in deductive systems. The result of conjoining two propositions is also called their conjunction. The conjunction is true if both of the combined propositions are true; else it is false.
For two inputs A and B, the truth table of the operator is as follows.
| A | B | A and B |
|---|---|---|
| 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's raining outside, and that I'm 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.
As a rule of inference Conjunction is a validityity, simple argument form:
- A,
- B.
- Therefore, A and B.
or in logical operator notation:
- <math> A, <math>
- <math> B <math>
- <math> \vdash A \and B <math>
The argument form has two premises. The first premise is the left conjunct, the second premise the right conjunct. From these two premises it can be logically concluded that A and B, must be true as well.
Here is an example of an argument that fits the form conjunction:
- Everyone should vote.
- Democracy is the best system of government.
- Therefore, everyone should vote and Democracy is the best system of government.
Associativity and commutativity
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.
Symbol
The symbol "<math>\land<math>" is typically used for conjunction, and "P <math>\land<math> Q" is read "P and Q". The word "AND" in all-caps is also used, and the ordinary English word "and" is probably most common of all.
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.
"But"
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.
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 doesn't shine when it rains. 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.
See also
- Boolean algebra
- Logical disjunction
- Exclusive disjunction
- Sheffer stroke (or NAND)de:Konjunktion (Logik)
eo:Konjunkcio (logiko) es:Puerta AND it:AND ja:論理積 pl:AND pt:E (lgica) fr:Conjonction_logique th:การเชื่อมเชิงตรรกศาสตร์ uk:Кон'юнкція