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In mathematical logic, **predicate logic** is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic, or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified. Two common quantifiers are the existential ∃ ("there exists") and universal ∀ ("for all") quantifiers. The variables could be elements in the universe under discussion, or perhaps relations or functions over that universe. For instance, an existential quantifier over a function symbol would be interpreted as modifier "there is a function". The foundations of predicate logic were developed independently by Gottlob Frege and Charles Sanders Peirce.

In informal usage, the term "predicate logic" occasionally refers to first-order logic. Some authors consider the **predicate calculus** to be an axiomatized form of predicate logic, and the predicate logic to be derived from an informal, more intuitive development.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Predicate_logic

In mathematics, a **predicate** is commonly understood to be a Boolean-valued function *P*: *X*→ {true, false}, called the predicate on *X*. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

Informally, a **predicate** is a statement that may be true or false depending on the values of its variables. It can be thought of as an operator or function that returns a value that is either true or false. For example, predicates are sometimes used to indicate set membership: when talking about sets, it is sometimes inconvenient or impossible to describe a set by listing all of its elements. Thus, a predicate *P(x)* will be true or false, depending on whether *x* belongs to a set.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Predicate_(mathematical_logic)

**First-order logic** is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as **first-order predicate calculus**, the **lower predicate calculus**, **quantification theory**, and **predicate logic**. First-order logic uses quantified variables over (non-logical) objects. This distinguishes it from propositional logic, which does not use quantifiers.

A theory about some topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things. Sometimes "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic.

The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/First-order_logic

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