1.4 Predicates and Quantifiers
What is a predicate?
A predicate refers to a property that the subject of the statement can have. It must have a subject to go along with it.
What is a domain of discourse/ universe of discourse?
AKA as domain Many mathematical statements assert that a property is true for all values of a variable in a particular domain, which is called a domain of discourse.
What is quantification?
Another way that you can create a proposition from a propositional function.
What is a uniqueness quantifier?
Denoted by capital backwards e! or capital backwards e subscript 1. This notation states that "There exists a unique x such that P(x) is true." or " there is exactly one" and " there is one and only one"
Why is predicate logic necessary?
It's used to express the meaning of all statements that can't be expressed with propositional logic in mathematics and in natural languages. For example: Every computer connected to the university network is functioning properly. There are no rules of propositional logic that would allow us to conclude the truth of this statement.
What is De Morgan's Laws for Quantifiers?
NOT( backwards E)*P(x) is equal to (upside down A)x* NOT(P(x)) NOT(upside down A)x*P(x) is equal to (backwards E)*(NOT(P(x))
What are Prolog facts ?
Prolog facts define predicates by specifying the elements that satisfy these predicates.
What are Prolog rules?
Prolog rules are used to define new predicates using those already defined by Prolog facts.
What does it mean when a variable is not bound?
That particular variable is said to be free.
What is predicate calculus?
The area of logic that deals with predicates and quantifiers.
What is the scope of a quantifier?
The part of a logical expression to which a quantifier is applied.
What is an existential quantification/ existential quantifier ?
This quantifier has a higher precedence than all logical operators from propositional calculus. For example : "There exists an element x in the domain such that P(x) The backwards capital e notation is used for the existential quantification of P(x) When the statement there exists an element x in the domain such that P(x) is true. Then there is an x for which P(x) is true if it's false then P(x) is false for every x
What is a universal quantifier?
This quantifier has a higher precedence than all logical operators from propositional calculus. The notation (upside capital a) is called a universal quantifier. It's read as "for all xP(x) or "for every x P(x)" An element for which P(x) is false is called a counterexample of for every xP(x). When the statement for all x P(x) is true then P(x) is true for every x. if it's false then there is an x for which P(x) is false.
What does it mean when someone says this occurrence of the variable is bound?
When a quantifier is used on the variable x
What is a propositional function?
When there is a logic function the propositional function is the P(x). When a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value.
When are statements considered logically equivalent?
if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. The triple bar equal sign is used to indicate this.
