The difference between Predicate calculus and Universal quantifier
When used as nouns, predicate calculus means the branch of logic that deals with quantified statements such as "there exists an x such that..." or "for any x, it is the case that...", where x is a member of the domain of discourse, whereas universal quantifier means the operator, represented by the symbol ∀, used in predicate calculus to indicate that a predicate is true for all members of a specified set.
check bellow for the other definitions of Predicate calculus and Universal quantifier
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Predicate calculus as a noun (logic):
The branch of logic that deals with quantified statements such as "there exists an x such that..." or "for any x, it is the case that...", where x is a member of the domain of discourse.
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Universal quantifier as a noun (logic):
The operator, represented by the symbol ∀, used in predicate calculus to indicate that a predicate is true for all members of a specified set.
Examples:
"Some verbal equivalents are ''"for each"'' or ''"for every"''."