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As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications. First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of generalized quantifiers.
Example requires a quantifier over predicates, which cannot be implemented in single-sorted first-order logic: Zj → ∃X(Xj∧Xp). Quantification over properties Santa Claus has all the attributes of a sadist. Example requires quantifiers over predicates, which cannot be implemented in single-sorted first-order logic: ∀X(∀x(Sx → Xx) → ...
This language uses the same operators as tuple calculus, the logical connectives ∧ (and), ∨ (or) and ¬ (not). The existential quantifier (∃) and the universal quantifier (∀) can be used to bind the variables. Its computational expressiveness is equivalent to that of relational algebra. [2]
The relational calculus is similar to the relational algebra, which is also part of the relational model: While the relational calculus is meant as a declarative language that prescribes no execution order on the subexpressions of a relational calculus expression, the relational algebra is meant as an imperative language: the sub-expressions of ...
The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains).
A model with this condition is called a full model, and these are the same as models in which the range of the second-order quantifiers is the powerset of the model's first-order part. [3] Thus once the domain of the first-order variables is established, the meaning of the remaining quantifiers is fixed.
Set-builder notation makes use of predicates to define sets. In autoepistemic logic, which rejects the law of excluded middle, predicates may be true, false, or simply unknown. In particular, a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.
Since the calculus is a query language for relational databases we first have to define a relational database. The basic relational building block is the domain (somewhat similar, but not equal to, a data type). A tuple is a finite sequence of attributes, which are ordered pairs of domains and values. A relation is a set of (compatible) tuples ...