Propositional logic assumes that there are facts that either hold or do not in the world. Each fact can be in one of two states: true or false. Each fact requires a separate proposition. eg boyjohn, boyjack, girljill etc.
First-order predicate (first-order logic) calculus assumes that the world consists objects with properties that distinguish them from other objects. Among these objects, various relations hold or do not hold. Some of these relations are functions - relations in which there is only one value for a given input.
objects: people, houses,etc
relations: owns, is faster than, etc.
properties: colour, kind
functions: father of, one more than
First-order logic allows predicates to have arguments: boy(jack), boy(john),
girl(jill) etc. The arguments may be variables. First-order logic can also be
used to make statements about all the objects in the universe. This with the
implication connective from propositional logic enables one to represent general
laws or rules eg all men are mortal.
Second-order predicate calculus further allows predicates to be variables. Here we use only first-order predicate calculus.