Predicate (mathematical logic)

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In mathematical logic, the predicate is generally understood as a Boolean function P : X -> {true, false}, is called a predicate on X >. However, predicates have many different uses and interpretations in mathematics and logic, and the exact definition, meaning and usage will vary from theory to theory. So, for example, when the theory defines the concept of a relation, the predicate is only a characteristic function or otherwise known as a function of a relation indicator. However, not all theories have a relationship, or are based on set theory, so one should be careful with the precise definition and semantic interpretation of a predicate.

Video Predicate (mathematical logic)

A simplified summary

Informally, a predicate is a statement that may be true or false depending on its variable values. This can be considered an operator or function that returns the correct or wrong value. For example, predicates are sometimes used to denote assigned membership: when speaking of a set, it is sometimes inconvenient or impossible to describe a set by listing all its elements. Thus, the predicate P (x) will be true or false, depending on whether x is included in a set.

Predictions are also commonly used to talk about object properties, by defining the set of all objects that have multiple identical properties. So, for example, when P is a predicate on X , one may sometimes say P >. Similarly, the notation P ( x ) is used to denote the sentence or statement P about the variable object x. Groups defined by P ( x ) are written as { x | P ( x )}, and is a collection of objects whose P is correct.

For example, { x | x is a natural value less than 4} is a set of {1,2,3}.

If t is an element of the set { x | P ( x )}, then the P statement ( t ) is true

Here, P ( x ) is termed predicates and x of the propositions . Sometimes, P ( x ) is also called proposeional function (template in a role), since each placeholder option x produces the argument.

The simple predicate form is a Boolean expression, in which case the input to the expression itself is a Boolean value, combined using a Boolean operation. Similarly, the Boolean expression with the predicate input itself is a more complex predicate.

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Formal definition

The exact semantic interpretation of atomic formulas and atomic sentences will vary from theory to theory.

• In propositional logic, the atomic formula is called a propositional variable. In a sense, this is a zero (ie 0-arity) predicate.
• In the first-order logic, the atomic formula consists of a predicate symbol applied to a number of corresponding terms.
• In set theory, predicate is understood as a characteristic function or set an indicator function, yes functioning from element set to truth value. The set-builder notation makes use of predicates to define sets.
• In autoepistemic logic, which rejects the excluded law in the middle, the predicate may be true, false, or simply unknown ; yes. a given set of facts may not be enough to determine the truth or falseness of a predicate.
• In fuzzy logic, predicate is a characteristic function of the probability distribution. That is, a strictly correct/false assessment of the predicate is replaced by the quantity interpreted as the degree of truth.

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See also

• Independent variable and dependent variable
• Functor logic predicate
• Truthbearer
• Multigrade predicate
• Predicate blur
• Classify topos
• binary relations

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References

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External links

• Introduction to predicate

Source of the article : Wikipedia