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"And" (∧) is a logical operator, a connector in logical calculi for denoting conjunction.
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 ∧ B.
"And" is a binary operator, meaning that it takes two terms and asserts that both are simultaneously true. However, it's common usage to chain conjunctions, such as A ∧ B ∧ C. Properly written, using parenthesis for proper grouping, this would be written as ( ( A ∧ B ) ∧ C ), but the intent is easily understood: the statement is true if A and B and C are simultaneously true.
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, as demonstrated by the preceding sentences, "but" and "and" are semantically distinct. The former sentences 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.