# Thread: Is a Name or Variable alone a WFF?

1. ## Is a Name or Variable alone a WFF?

So the formation rules of predicate logic sit pretty well with me...but one thing that we don't seem to have covered explicitly is precisely whether a name or variable by itself is a wff. It doesn't seem to me intuitively that say "Superman" by itself should be a well-formed formula, but unfortunately I can't explain why this would be the case :P

I would have thought that for a name or variable to be part of a wff, it would have to be preceded by a predicate of some sort. So "Superhero(Superman)" is a wff, and "Superman" is not.

Am I right? If not, why? Thanks in advance!

2. A constant (a superman in your example) and a variable are terms and terms alone are not atomic sentences. An atomic sentence can have a form like "Predicate(term)" [superhero(superman) in your example] or a term = term. Atomic sentences alone can be WFFs in first order logic (see below syntax).

I assume you can read a BNF form. Belows are the syntax of first order logic with equality. A "Sentence" below denotes a WFF in FOL.

$\displaystyle \text{Sentence} \rightarrow \text{AtomicSentence}$
$\displaystyle | \text{(Sentence Connective Sentence)}$
$\displaystyle | \text{Quantifier Variable, ... Sentence}$
$\displaystyle | \neg Sentence$

$\displaystyle \text{AtomicSentence} \rightarrow \text{Predicate(Term,...)}$ | $\displaystyle \text{Term=Term}$

$\displaystyle \text{Term} \rightarrow \text{Function(Term,..)} | \text{Constant} | \text{Variable}$

$\displaystyle \text{Quantifier} \rightarrow \forall | \exists$
$\displaystyle \text{Connective} \rightarrow \wedge | \lor | \Rightarrow | \Leftrightarrow$
$\displaystyle \text{Constant} \rightarrow \text{Superman} |...$ //your example in Q
$\displaystyle \text{Predicate} \rightarrow \text{Superhero} |...$ //your exampe

3. Much thanks for the concise reply!