What is the difference between a function and a relation in first-order logic?

Question

Consider this definition of first-order (or Herbrand) logic syntax. Here is the vocabulary:

Definition (Vocabulary): A vocabulary V consists of:

• A set of relation constants {r1, ..., rn}, each with an associated arity.
• A set of function constants {f1, ..., fm}, each with an associated arity.
• A non-empty set of object constants {c1, ..., ck}.
• A set of variables {x1,x2,...}.

Here is the definition for 'term':

Definition (Term): A term in V:

• A variable.
• An object constant.
• A function constant with arity n applied to n terms.
• Only expressions produced by the above rules are terms.

Here is the definition for 'sentence':

Definition (Sentence): A sentence in V:

• A relation constant with arity n applied to n terms.
• (¬ φ) where φ is a sentence.
• (φ ∨ ψ), where φ and ψ are sentences.
• (φ ∧ ψ), where φ and ψ are sentences.
• (φ ⇐ ψ), where φ and ψ are sentences.
• (φ ⇒ ψ), where φ and ψ are sentences.
• (φ ⇔ ψ), where φ and ψ are sentences.
• (∀x.φ), where φ is a sentence.
• (∃x.φ), where φ is a sentence.
• Only expressions produced by the above rules are sentences.

I noticed that a function is a term, but a relation is a sentence. At first I thought that was because of the quantifiers associated with a relation, but sentences using quantifiers are also defined. Also the definition of relation does not involve quantifiers.

This makes me wonder what is the difference between functions and relations in first-order logic that one would be considered a term and the other a sentence.

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Francois Bry, Mike Genesereth, Tim Hinrichs, Nat Love. Herbrand Logic. Retrieved on September 28, 2019 at https://www.cs.uic.edu/~hinrichs/herbrand/html/herbrandlogic.html

Answer 1

First of all, a relation by itself is not a sentence. A relation applied to one or more terms is a sentence. Similarly, a function by itself is not a term. A function applied to one or more terms is a term.

The difference between functions and relations is that functions yield terms when applied to terms, while relations yield truth values. For instance, 'the father of x' is a function. When applied to a term, say 'Bob', we get a new term: 'the father of Bob'. This is not a sentence because it doesn't have a truth value. On the other hand, 'x is the father of y' is a relation. When applied to 'Jim' and 'Bob' we get a sentence that is either true or false: 'Jim is the father of Bob'.

You can also think of relations as functions that return truth values instead of terms, although in logic they are not usually defined this way.