Predicates and their arguments
A predicate is an expression that says something about one or more objects. 'Is mortal,' 'is tall,' and 'is prime' are one-place predicates: they apply to a single object. 'Loves,' 'is taller than,' and 'divides' are two-place predicates: they apply to a pair of objects. 'Is between,' 'gives to,' and 'introduces to' are three-place predicates. The arity of a predicate is the number of objects it relates, and it matters because a predicate with n arguments cannot be used as one with m ≠ n.
In predicate logic, we write predicates with uppercase letters, and their arguments inside parentheses or as subscripts. 'Socrates is mortal' becomes Ms (or M(s)). 'Alice loves Bob' becomes Lab (or L(a,b)). The first position is filled by the first argument, the second by the second, and so on. Order matters for relational predicates: Lab and Lba mean different things, just as 'Alice loves Bob' and 'Bob loves Alice' mean different things in English.
Arity errors are among the most common mistakes in formalization. If you declare L as a two-place predicate meaning 'loves,' then writing Lx is ill-formed — it provides only one argument where two are required. Conversely, if M is a one-place predicate meaning 'is mortal,' then Mab is ill-formed because it provides two arguments to a predicate that takes one. Checking arity is like checking that a function receives the right number of inputs: get the count wrong and the expression is meaningless.
In scientific contexts, relational predicates are ubiquitous. 'Enzyme e catalyzes reaction r' uses a two-place predicate C(e, r). 'City c is between cities a and b along route r' uses a four-place predicate B(c, a, b, r). The greater the arity, the more information the predicate encodes about how its arguments relate. Practice identifying the arity of predicates in everyday and technical language, because correct arity assignment is the foundation of accurate formalization.
- Identify the arity of every predicate you use.
- Respect the order of arguments in relational predicates.
- Use uppercase letters for predicates and lowercase for constants and variables.
- Double-check that every atomic formula supplies exactly as many arguments as the predicate's arity requires.
Takeaway: Predicates have arities, and relational predicates care about argument order. Getting either wrong changes the meaning of the formula.