## Foundations of Mathematics## Mathematical and Formal Logic |

## How is predicate calculus interpreted? |

Predicate calculus may be a general system of logic, but it accurately expresses a large variety of assertions and provides many types of reasoning. It is definitely more flexible than Aristotle’s syllogisms and more useful (in many cases) than propositional calculus.

Predicate calculus makes heavy use of symbolic notation, using lower-case letters *a, b, c, …,x, y, z,…* to denote the subject (in predicate calculus, often referred to as “individuals”); upper-case letters *M, N, P, Q, R,…* to denote predicates. The simplest of assertions are formed by moving the predicate with the subject.

For example, using the “all” quantifier means that when you have an arbitrary variable, you must prove something true about that variable, and then prove that it does not matter what variable you chose, it will always be true. Thus, from propositional calculus, the sentence, “All humans are mortal,” becomes in predicate calculus, “All things *x* are such that, if *x* is a human, then *x* is a mortal.” This sentence may also be written symbolically under predicate calculus. (To compare, the sentence “*x* is a human” is not a statement in propositional calculus [see above] because it involves an unknown entity *x*; therefore, a truth value cannot be assigned without knowing what *x* represents.)