Logical conjunction
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 example, consider:
- x > 13 AND x < 27.
The symbol "" is typically used for conjunction, and "P 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.
Input1 | Input2 | Output |
---|---|---|
FALSE | FALSE | FALSE |
FALSE | TRUE | FALSE |
TRUE | FALSE | FALSE |
TRUE | TRUE | TRUE |
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.
Conjunction as we've described it is a binary operator, meaning that it combines two terms into a single statement. However, 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.
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 valid, simple argument form:
- A,
- B.
- Therefore, A and B.
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.
"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:
- Logical disjunction
- Sheffer stroke (or NAND)