Adding a single binary predicate letter to monadic logic would result in a system with the expressive power of the full predicate calculus. Therefore the absence of polyadic predicates severely restricts what can be expressed in the monadic predicate calculus. That calculus is so weak that, unlike the full predicate calculus, it is decidable whether a given formula of that calculus is logically valid (true for all nonempty domains). Because the monadic predicate calculus is decidable, it is ipso facto inadequate for general mathematical reasoning, if only because the tiny fragment of mathematics called Peano arithmetic is known to be undecidable.
Notwithstanding the above deficiencies, the need to go beyond monadic logic was not appreciated until the work on the logic of relations, by Augustus DeMorgan and Charles Peirce in the 19th century, and by Frege in his little-read 1879 Begriffsschrifft. Prior to the work of these three men, syllogistic term logic was widely considered adequate for formal deductive reasoning.
Inferences in term logic can all be represented in the monadic predicate calculus. For example the syllogism
Conversely, monadic predicate calculus is not significantly more expressive than term logic. It is easily proved that every formula in the monadic predicate calculus is equivalent to a formula in which quantifiers appear only in closed subformulae of the form
Each of these formulas is the negation of the other, and the quantifiers do not nest. These formulas also generalize slightly the form of basic judgements considered in term logic. For example, this form allows statements such as "Every mammal is either a herbivore or a carnivore (or both)", . Reasoning about such statements can, however, still be handled within the framework of term logic, although not by the 19 classical Aristotelian syllogisms alone.
Taking propositional logic as given, every formula in the monadic predicate calculus expresses something that can likewise be formulated in term logic. On the other hand, a modern view of the problem of multiple generality in traditional logic concludes that quantifiers cannot nest usefully if there are no polyadic predicates to relate the bound variables.
Monadic predicate calculus is also called monadic first-order logic. Monadic second-order logic keeps the requirement that all predicates be unary, but allows for quantification over predicates as well as variables.