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ĐẠI HỌC QUỐC GIA THÀNH PHỐ HỒ CHÍ MINH
ĐẠI HỌC CÔNG NGHỆ THÔNG TIN
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ĐẶC TẢ HÌNH THỨC
GVHD: Hầu Nguyễn Thành Nam
Ngày 20 tháng 05 năm 2013
Nhóm thực hiện:
• Trần Đức Yên 10520203
• Lê Tuấn Anh 10520211
• Tô Hồng Phong 10520411

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3.1 Predicate calculus
3.2 Quantiers and declarations
3.3 Substitution
3.4 Universal introduction and elimination
3.5 Existential introduction and elimination
3.6 Satisfaction and validity
CONTENTS PRESENTATION

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In this chapter we introduce another part of our logical language. The
language of propositions introduced in the previous chapter allows us to make state-
ments about specic objects, but it does not allow us to make statements such as
“Every cloud has a silver lining”. These are known as universal statements, since
they describe properties that must be satised by every object in some universe of
discourse
Example 3.1 The following are examples of universal statements:
 Each student must hand in course work
Example 3.2The following are examples of existential statements:
 I heard it from one of your friends

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We require a language that reveals the internal structure of our
propositional statements, a language that allows us to take them apart and
apply them to objects without proper names. The language we require is
the language of predicate calculus .
We could say that a predicate is a proposition with a gap for an
object of some kind.
For example, the statement “_ > 5” is a predicate.
 “0 > 5” A proposition that happens to be false.
 “x > 5” There is anx, which is a natural number, such that x > 5

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Example: Let Friends stand for the set of all your friends, and let x told y
mean that x has told y.
Example: Let Student stand for the set of all students, and let
Submit„ (x) …mean that x must hand in course work.

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As may be seen from the following equivalences :
There exists an x in a satisfying p, such that q
For all x in a satisfying p, q holds

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We can change the name of a bound variable without changing the
meaning of the quantied expression, as long as we avoid the names of
any other variables that appear.
This statement is false: there is no greatest
natural number
This statement is true; the meaning has
changed

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Example : The quantied predicate

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3.3 Substitution
We write p† [y/x] ‡ to denote the predicate that results from substituting
y for each free occurrence of x in predicate p; this new operator binds more
tightly than any other. The expression y need not be another variable; it can
be any expression whose possible values match those of x.
Example:

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3.3 Substitution
We write p† [t/x] [‡ † u/y] ‡ to denote the predicate p† [t/x] ‡ with the
expression u systematically substituted for free occurrences of the variable y
We write p[† t,u/x,y]‡ to denote the result of simultaneously substituting t for
x and u for y in predicate p. In general, this is different from the multiple
substitution p[† t/x][‡ † u/y]‡

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3.3 Substitution
If the major operator in an expression is not a quantier, then the effect
of substitution is easy to explain:
In every case, substitution distributes through the propositional
operators.

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If the major operator in an expression is not a quantier, then the effect
of substitution is easy to explain:
In every case, substitution distributes through the propositional
operators.

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Where x a means that x is a member of set a

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The constraint part of a universal quantication may be treated as
the an-tecedent of an implication.
From a conjunction, one may conclude either of the conjuncts; by
analogy, from a universally quantied predicate, one may conclude that
the predicate holds for any value in the range.

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The full form requires the equivalent of implication elimination, to
demon-strate that the expression chosen satises the constraint:
A special case of the last rule takes t as x:

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Now this subtree is finished, since we can use conjunction elimination
to con-nect top and bottom. The right-hand subtree is symmetric with the
left.

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The existential quantication true if and only if there
is some x in set a such that p and q are true. Of course, this object does not
have to be called x; it can be any expression t such that t has a value in a
and the following predicate is true:
That is, given that we are talking about t not x, both the constraint and the
quantied predicate should hold. To introduce an existential quantier, we must
show that a suitable expres-siont exists: we must provide an example
As before, the expression means that t is a member of set a

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If, in the course of a proof, we have established that and
x ≥ 0, then we may apply the special case of existential-introduction and
con-clude that
Elimination of the existential quantier is a more difficult affair. The pred-
icate states that there is some object x in a for whichs
is true. If x appears free in p then simply removing the quantier leaves us with an
un-justied statement about a free variable x. We cannot, in general, conclude p
from . To use the information contained in p, we must complete
anyreasoning that involves x before eliminating the quantier.

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Suppose that we assume only that and that p holds of x. If we
are then able to derive a predicate r that does not involve x, and we know
that there is some x in a for which p is true, then we may safely concluder.

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A predicate with free variables or “spaces” is neither true nor false; it
cannot be assigned a truth value until values are chosen for these variables
or the spaces are filled. Some predicates will become true whatever values
are chosen: these are said to be valid predicates.
Example : If n denotes a natural number, then the predicate
is valid: it will be true whichever value is chosen from the list 0, 1, 2, 3, ….
A predicate that is true for some, but not necessarily all, choices of values is
said to be satisable.

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Example : If n denotes a natural number, then the predicate
is satisable. There are natural numbers greater than or equal to 5.
A predicate that is false for all choices is said to be unsatisable. Valid,
satisfi-able, and unsatisable predicates are the analogues of tautologies,
contingen-cies, and contradictions in the language of propositions.

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