Semantics
Worlds
- There are infinite set of objects in the world and these are also called as individuals. The objects can be concrete (example, Block Q), abstract (example, 7, π), fictional (example, Beauty), set of integers etc., which we want to say something about it by giving it a name.
- There are infinite set of functions that can be applied to individuals. These functions can be of any arity.
- Example: A function which maps a person into his or her parents.
- The individuals can participate in any number of relations. Every relation will have arity. A relation which has arity as 1 is called a property.
Interpretations
- In predicate calculus; interpretation of an expression refers to mapping of object constants into object that are in the world, n-ary function constants into n-ary functions, n-ary relation constants into n-array relation.
- The assignments which are made are called the denotations of their corresponding predicate-calculus-expressions.
- The object assignment are made to a set of objects, this set of object is called domain of interpretation.
- The values of an atom may be true or false depending on given interpretation.
- Example: Consider discs world problem. In this world the entities are P, Q, R and floor.
- The relations that are present in this world are ON an Clear. The relations On and Clear can be defined by η tuples of the objects which participate in these relations.
- Let we have the discs as shown in Fig. 2.4.1.
Fig. A Configuration of Discs
- From the above Figure the relations of On are <P, Q>, <Q, P>, <R, Floor> and relations of Clear are <P> it is a singleton set.
- The discs world problem can be described in predicate calculus. Let us use the object constants P, Q, R and F and binary relational constant On and unary relational constant Clear.
- The interpretation of the predicate calculus expression is shown in the Table
- Table A Mapping between Predicate Calculus and the World
Table: A Mapping between Predicate Calculus and the World
From the above Table, we can find the values of some predicate-calculus wffs,
On(P, Q) is True because <P, Q> is in relation On.
Clear(R) is False because <R> is not in the relation Clear.
On(Q, P) is False because <Q, P>is not in the relation On.
On(P, Q) ∧ On(Q, R) ∧ Clear(P) is true because both <P, Q> and <Q, R> are present in relation On and <P> is present in Clear relation.
