• Also known as Propositional Functions & Open Sentences
  • Predicate is a function(Predicate Symbol) that accepts a valid input to its Predicate Variable and return either true or false
  • Example: let , the predicate symbol stands for is a student at NUS and be the predicate variable. Together, we get which stands for is a student at NUS

Dynamic Truth Value

Depends on the values we substituted to the Predicate Variable

Not a Mathematical Statement

Only becomes a Mathematical Statement when specific values are substituted to the Predicate Variable, and we call it Closed Predicate. Otherwise Open Predicate

Predicate is not a Predicate Variable

Can’t be used as a Predicate Variable that is substituted into Predicate Symbol like . Because Predicate is meant to return either true or false with value substituted

Predicate Variable

  • Holds value that determines if Predicate is true or false

Not a Mathematical Statement

It doesn’t hold the value of true or false! Only Mathematical Statement can be true or false

Predicate Symbol

Domain of Predicate Variable

  • The set of all values that may be substituted in place of the Predicate Variable
  • Also known as Domain of Discourse, Universe of Discourse, Universal Set & Universe

Truth Set


  • Instead of manually assign a specific value to Predicate Variable to obtain a Mathematical Statement
  • Quantifiers is another way to specify how many of a particular type of values that can make the Predicate a closed predicate
  • There are 2 types, represented with either or

Order of Quantifiers

Unless the Quantifiers are of the same type. Otherwise, the meaning is different

Given is x loves y

  1. in english: For all people x, there is a person y such that x loves y.

  2. in english: There is a person y such that all people x, x loves y.

  • The first one means for everyone (you, me, he), there is someone we love
  • The second one means there is someone who is loved by everyone (you, me, he)


Predicate with more than one Quantifiers

Implicitly Quantified

The Quantifiers are assumed without specified explicitly