Monadic predicate calculus

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

${\displaystyle [(\mathrm{\forall }xD(x)\Rightarrow M(x))\wedge \mathrm{\neg }(\mathrm{\exists }yM(y)\wedge B(y))]\Rightarrow \mathrm{\neg }(\mathrm{\exists }zD(z)\wedge B(z))}$

where ${\displaystyle D}$, ${\displaystyle M}$ and ${\displaystyle B}$ denote the predicates 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

${\displaystyle \mathrm{\forall }x{P}_1(x)\vee \cdots \vee P_n(x)\vee \mathrm{\neg }{P}_1^{\prime }(x)\vee \cdots \vee \mathrm{\neg }{P}_m^{\prime }(x)}$

or

${\displaystyle \mathrm{\exists }x\mathrm{\neg }{P}_1(x)\wedge \cdots \wedge \mathrm{\neg }{P}_n(x)\wedge P_1^{\prime }(x)\wedge \cdots \wedge P_m^{\prime }(x),}$

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)", ${\displaystyle \mathrm{\forall }xM(x)\Rightarrow (H(x)\vee C(x))}$. 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.