1.2 The predicate Calculus
In propositional calculus, each atomic symbol (P, 0, etc.) denotes a
proposition of some complexity. There is no way to access the components of
an individual assertion. Predicate calculus provides this ability. For example,
instead of letting a single propositional symbol, P denotes the entire sentence
"it rained on Tuesday," we can create a predicate weather that describes a
relationship between a date and the weather:
weather (tuesday, rain). Through inference rules we can manipulate
predicate calculus expressions, accessing their individual components and
inferring new sentences.
Predicate calculus also allows expressions to contain variables.
Variables let us create general assertions about classes of entities. For
example, we could state that for all values of X, where X is a day of the week,
the statement weather(X, rain) is true; i.e., it rains every day. As with
propositional calculus, we will first define the syntax of the language and then
discuss its semantics.
DEFINITION
PREDICATE CALCULUS SYMBOLS
The alphabet that makes up the symbols of the predicate calculus
consists of:
1. The set of letters, both upper- and lowercase, of the English alphabet.
2. The set of digits, 0, 1, ...,9.
3. The underscore, _.
Symbols in the predicate calculus begin with a tetter and are followed by any
sequence of these legal characters.
Legitimate characters in the alphabet of predicate calculus symbols include
a R 6 9 p_z
Examples of characters not in the alphabet include
# % @ / & ""
Legitimate predicate calculus symbols include
tom_and_jerry bill XXXX friends_of Ali George fire3
Examples of strings that are not legal symbols are
3jack "no blanks allowed" ab%cd ***71 duck!!!
Predicate calculus symbols may represent either variables, constants,
functions, or predicate. Constants name specific objects or properties in the
world. Constant symbols must begin with a lowercase letter. Thus george,
tree, tall, and blue are examples of well-formed constant symbols. The
constants true and false are reserved as truth symbols.
Variable symbols are used to designate general classes of objects or
properties in the world. Variables are represented by symbols beginning with
an uppercase letter. Thus George, BILL, and KAte are legal variables,
whereas geORGE and bill are not.
Predicate calculus also allows functions on objects in the world of
discourse. Function symbols (like constants) begin with a lowercase letter.
Functions denote a mapping of one or more elements in a set (called the
domain of the function) into a unique element of another set (the range of the
function). Elements of the domain and range are objects in the world of
discourse. In addition to common arithmetic functions such as addition and
multiplication, functions may define mappings between nonnumeric domains.