Predicate vs Functions in First order logic?

Question

I have been so confused lately regarding difference between predicate and function in first order logic. My understanding so far is, Predicate is to show a comparison or showing a relation between two objects such as, President(Obama, America) Functions are to specify what a particular object is such as, Human(Obama) Now am I heading on right track to differentiate these two terms or I am completely wrong and need a brief explanation, I would like to have opinion from expert to clarify my knowledge(or approve my understanding). Thanks in advance Krio

Answer 1

Think of these in terms mathematical functions defined on sets.

For any set Α, a function F of arity n is defined as F : Aⁿ → A

For example: F(a₁, a₂, ... , a_n) = a_j where, ∀ i∈ {1,2, ..., n} a_i ∈ A and a_j ∈ A

Whereas a predicate P of arity n is defined as P : Aⁿ → {0,1}

For example: P(a₁, a₂, ... , a_n) = 0 where, ∀ i∈ {1,2, ..., n} a_i ∈ A

P(a₀, a₀, ... n times ... , a₀) = 1 where, a₀ ∈ A

Now, for your examples,

Predicate is to show a comparison or showing a relation between two objects such as, President(Obama, America) Functions are to specify what a particular object is such as, Human(Obama)

Let's consider the set A = {Obama, America}

A function President of arity 1 can be defined as President : A → A

to get something like, President(America) = Obama

A predicate President of arity 2 can be defined as President : A² → {0,1}

to get something like,

President(Obama, America) = 1

President(America, America) = 0 etc.

Another predicate Human of arity 1 can be defined as Human : A → {0,1}

to get something like,

Human(Obama) = 1

Human(America) = 0 etc.

A predicate is nothing but a boolen function.