# Monadic predicate calculus

In

Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments.

## Expressiveness

The absence of

Adding a single binary relation symbol to monadic logic, however, results in an undecidable logic.## Relationship with term logic

The need to go beyond monadic logic was not appreciated until the work on the logic of relations, by Augustus De Morgan and Charles Sanders Peirce in the nineteenth century, and by Frege in his 1879 *Begriffsschrift*. Prior to the work of these three, term logic (syllogistic 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 argument

- All dogs are mammals.
- No mammal is a bird.
- Thus, no dog is a bird.

can be notated in the language of monadic predicate calculus as

where , and denote the predicates^{[clarification needed]} of being, respectively, a dog, a mammal, and a bird.

Conversely, monadic predicate calculus is not significantly more expressive than term logic. Each formula in the monadic predicate calculus is equivalent to a formula in which quantifiers appear only in closed subformulas of the form

or

These formulas slightly generalize the 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

## Variants

The formal system described above is sometimes called the **pure** monadic predicate calculus, where "pure" signifies the absence of function symbols. Allowing monadic function symbols changes the logic only superficially^{[}*citation needed*]^{[clarification needed]}, whereas admitting even a single binary function symbol results in an undecidable logic.

Monadic second-order logic allows predicates of higher arity in formulas, but restricts second-order quantification to unary^{[clarification needed]} predicates, i.e. the only second-order variables allowed are subset variables.

## Footnotes

**^**Heinrich Behmann,*Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem*, in*Mathematische Annalen*(1922)**^**Löwenheim, L. (1915) "Über Möglichkeiten im Relativkalkül,"*Mathematische Annalen*76: 447-470. Translated as "On possibilities in the calculus of relatives" in Jean van Heijenoort, 1967.*A Source Book in Mathematical Logic*, 1879-1931. Harvard Univ. Press: 228-51.