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Predicate calculus allows us to reason about properties of objects and relationships between objects.

**Predicate Calculus**

**Syntax**

Predicate
calculus allows us to reason about properties of objects and relationships
between objects.

In
propositional calculus, we could express the English statement “I like cheese”
by A. This enables us to create constructs such as ￢A, which means

“I do not
like cheese,” but it does not allow us to extract any information about the
cheese, or me, or other things that I like.

In
predicate calculus, we use predicates to express properties of objects. So the
sentence “I like cheese” might be expressed as L(me, cheese) where L is a
predicate that represents the idea of “liking.” Note that as well as expressing
a property of me, this statement also expresses a relationship between me and
cheese. This can be useful, as we will see, in describing environments for
robots and other agents.

For
example, a simple agent may be concerned with the
location of various blocks, and a statement
about the world might be T(A,B), which could mean: Block A is on top of Block
B.

It is also possible to make more general statements using the
predicate calculus.

For example, to express the idea that everyone likes cheese, we
might say ( x)(P(x)→L(x, C))

The symbol is read “for all,” so the statement above could be
read as “for every x it is true that if property P holds for x, then the
relationship L holds between x and C,” or in plainer English: “every x that is
a person likes cheese.” (Here we are interpreting P(x) as meaning “x is a
person” or, more precisely, “x has property P.”)

Note that we have used brackets rather carefully in the
statement above.

This statement can also be written with fewer brackets: x P(x)
→L(x, C), is called the universal quantifier.

The quantifier can be used to express the notion that some
values do have a certain property, but not necessarily all of them: (
x)(L(x,C))

This statement can be read “there exists an x such that x likes
cheese.”

This does not make any claims about the possible values of x, so
x could be a person, or a dog, or an item of furniture. When we use the
existential quantifier in this way, we are simply saying that there is at least
one value of x for which L(x,C) holds.

The following is true: ( x)(L(x,C))→(
x)(L(x,C)), but the following is not: ( x)(L(x,C))→( x)(L(x,C))

It is also possible to combine the
universal and existential quantifiers, such as in the following statement: ( x) ( y) (L(x,y)).

This statement can be read “for all x, there exists
a y such that L holds for x and y,” which we might interpret as “everyone likes
something.”

A useful
relationship exists between and . Consider the statement “not everyone likes
cheese.” We could write this as

￢( x)(P(x)→L(x,C))
-------------- (1)

As we
have already seen, A→B is equivalent to ￢A ∨ B. Using DeMorgan’s laws, we can
see that this is equivalent to ￢(A ∧ ￢B).
Hence, the statement (1)
above, can be rewritten

￢( x)￢(P(x) ∧ ￢L(x,C))
------------- (2)

This can
be read as “It is not true that for all x the following is not true: x is a
person and x does not like cheese.” If you examine this rather convoluted
sentence carefully, you will see that it is in fact the same as “there exists
an x such that x is a person and x does not like cheese.” Hence we can rewrite
it as ( x)(P(x) ∧ ￢L(x,C)) ------------- (3)

In making
this transition from statement (2) to statement (3), we have utilized the
following equivalence: x ￢( x)￢

In an
expression of the form ( x)(P(x, y)), the variable x is said to be bound,
whereas y is said to be free. This can be understood as meaning that the
variable y could be replaced by any other variable because it is free, and the
expression would still have the same meaning, whereas if the variable x were to
be replaced by some other variable in P(x,y), then the meaning of the
expression would be changed: ( x)(P(y,
z)) is not equivalent to ( x)(P(x, y)),

whereas
( x)(P(x, z)) is.

Note that a variable can occur both bound and free
in an expression, as in ( x)(P(x,y,z) → ( y)(Q(y,z)))

In this expression, x is bound throughout, and z is
free throughout; y is free in its first occurrence but is bound in (
y)(Q(y,z)). (Note that both occurrences of y are bound here.)

Making this kind of change is known as
substitution.

Substitution is allowed of any free variable for
another free variable.

**Functions**

In much the same way that functions
can be used in mathematics, we can express an object that relates to another
object in a specific way using functions.

For example, to represent the
statement “my mother likes cheese,” we might use L(m(me),cheese)

Here the function m(x) means the
mother of x. Functions can take more than

one argument, and in general a function with n arguments is
represented as f(x1, x2, x3, . . . , x_{n})

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