The Handy Math Answer Book

matter what variable you chose because it will always be true. Thus, from propositional

calculus the sentence, “All humans are mortal,” becomes, in predicate calculus, “All

things xare such that, if xis a human, then xis a mortal.” This sentence may also be

written symbolically under predicate calculus. (To compare, the sentence “xis 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 xrepresents.)

What is the atomic formulaof predicate calculus?

The atomic formula of predicate calculus is when a predicate and special case of a subject

(individual) are written together. For example, if Mis the predicate “to be a human,” and b

is the subject (individual) “Socrates,” then Mbmeans the assertion “Socrates is a human.”

This atomic formula is phrased “bis the argument of M.” Thus, M,as the predicate, may be

applied to any subject, and that subject is then an argument of M. But if cis the subject

“Vermont,” then Mcis a false assertion, because Vermont is not a human. Some predicates

require more than one argument; thus, you can have formulas such as Mxy.

What are logical operatorsin predicate calculus?

Predicate calculus commonly uses seven special symbols—called logical operators—

to express a formula (in this case, a formula is a meaningful expression built up from

atomic formulas by repeated application of the logical operators). The following table

lists the symbols and their meanings. (Note: Many of these symbols are also used as

logical operators in truth tables; see above):

Logical Operators in Predicate Calculus

Symbol Name Usage Meaning*

& conjunction ... & ... “both ... and ...”

 disjunction ... ... “either ... or ... (or both)”

~ negation ~... “it is not the case that ...”

 implication ... ... “if ... then ...”

bi-implication ... ... “... if and only if ...”

 universal quantifier x... “for all x, ...”

 existential quantifier x... “there exists xsuch that ...”

*Where xis any variable

What is an algorithm?
The word “algorithm” is a distortion of Muhammad ibn Musa al-Khuwarizmi’s name
(783–c. 850; also seen as al-Khowarizmi and al-Khwarizmi), the Persian mathemati-
cian who wrote about algebraic methods (for more about al-Khuwarizmi, see “History
112 of Mathematics”). In general, an algorithm is a specific set of instructions that, if fol-