Chapter 01: Logic concepts

Predicates and quantifiers

A predicate is an expression of one or more variables determined on some specific domain. A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable.

Definition

A predicate is statement that contains variables. A predicate may be true or false depending on the values of these variables.

Universal quantifier

means that the predicate is true for all possible values of .

A universal quantification is a type of quantifier[1], a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

Example

  1. , i.e., "the square of any number is not negative".

  2. , i.e., "all squares are rectangles."

Existential quantifier

An existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

means that there exists an is true. Sometimes, we will use also . It means that there exists a unique where is true.

Example

  1. is true.

  2. is true, but is false.

  3. is true. The order of the quantifiers is very important, this statement is false.

A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation.

Example

  1. Statement: and negation:

  2. Statement: and negation:

  3. Statement: and negation:

  1. quantifier

    A quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula

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