Question

In standard mathematics binary relations are treated as an expansion on the concept of the function, and as such we get into a difficult situation where 'xFy' could be the application of f on to x and y or a sentence xFy that states that x and y are related by F.

In FOL it is not clear to me how we avoid this issue, if feels inadequately formal to have such ambiguity. I assume this is solved in the interpretation, from what I've read it seems a relation can be treated distinctly to a standard function.

Answer 1

A function is a mathematical concept that describes a relationship between two sets of values. It can be expressed in different forms, such as equations or verbal descriptions. In the context of mathematical logic, binary relations and functions are treated as distinct concepts to avoid ambiguity.

Step-by-step explanation:

In mathematics, a function describes a relationship between two sets of values, where each input value from one set corresponds to exactly one output value from the other set. A function can be expressed in various forms, such as equations or verbal descriptions.

For example, in the statement 'Professor = Adam Smith', the function 'Professor' represents the relationship between the input 'Adam Smith' and the output 'Professor'. Similarly, in the statement 'Friends = Bob + Shawn + Margaret', the function 'Friends' describes the relationship between the inputs 'Bob', 'Shawn', 'Margaret' and the output 'Friends'.

In the context of mathematical logic and formal languages, binary relations and functions are distinct concepts. Binary relations represent a relationship between two elements, while functions are a special kind of binary relation where each element in the first set corresponds to exactly one element in the second set. This distinction helps avoid ambiguity in the interpretation of formal statements.