In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic (that is, they take only one argument), and there are no function letters. This means that all atomic formulae have the form P(x), where P is a predicate letter and x is a variable. Logic, from Classical Greek λόγος logos (the word), is the study of patterns found in reasoning. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... In linguistics and logic, a predicate is an expression that can be true of something. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ...

The absence of predicates with more than one argument severely restricts what can be expressed in the monadic predicate calculus. In fact, and in contrast to full predicate calculus, it is decidable whether a given monadic predicate calculus formula is logically valid. According to the modern viewpoint, the decidability of monadic predicate calculus exposes it as inadequate for general mathematical reasoning. It is, however, interesting that the need to go beyond monadic logic was only realized quite late in the history of mathematics. In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion. ... A logical system is decidable iff there exists an algorithm such that for every well-formed formula in that system there exists a maximum finite number N of steps such that the algorithm is capable of deciding in less than or equal to N algorithmic steps whether the formula is...

Prior to the work of Frege in the late 19th century, syllogistic term logic was widely considered to be the universal foundation for all deductive reasoning. Inferences in term logic can all be represented in the monadic predicate calculus; for example the syllogism Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen, IPA: ) was a German mathematician who became a logician and philosopher. ... Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ... A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ...

All dogs are mammals

No mammal is a herbivore

Thus, no dog is a herbivore

can be notated in the language of monadic predicate calculus as

where D, M and H denote the predicates of being, respectively, a dog, a mammal, and a herbivore.

Conversely, monadic predicate calculus is arguably not significantly more expressive than term logic. One easily proves that every formula in the monadic predicate calculus is equivalent to a formula in which quantifiers do not nest and appear only in closed subformulae of the shape In logic, statements p and q are logically equivalent if they have the same logical content. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

or

which are each other's negations and slightly generalize 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. Thus, if propositional logic is taken for granted, every formula in monadic predicate calculus expresses something that can also be spoken about in traditional logic. On the other hand, a modern view of the problem of multiple generality in traditional logic is exactly that quantifiers cannot usefully nest if one has no multiary predicates with which to relate the bound variables. A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... The problem of multiple generality names a failure in Aristotelian logic to describe certain intuitively valid inferences. ...

Variants

The concept described here is sometimes called the pure monadic predicate calculus, where "pure" signifies the absence of function letters. Allowing monadic function letters changes the logic only superficially, whereas even a single binary function letter would allow an easy encoding of full predicate calculus.

Monadic predicate calculus is also called monadic first-order logic. Monadic second-order logic keeps the requirement that all predicates are unary, but allows for quantification over predicates. In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ...