CPSC 125 Ch 1 sec 3

Formal Logic

Mathematical Structures
for Computer Science
Chapter 1

Copyright © 2006 W.H. Freeman & Co. MSCS Slides Formal Logic

Variables and Statements
●  Variables in Logic

●  A variable is a symbol that stands for an individual in a collection
or set. For example, the variable x may stand for one of the days.
We may let x = Monday or x = Tuesday, etc.

●  Incomplete Statements

●  A sentence containing a variable is called an incomplete statement.
An incomplete statement is about the individuals in a deﬁnite
domain or set. When we replace the variable by the name of an
individual in the set we obtain a statement about that individual.

■  Example of an incomplete statement : “x has 30 days.”

●  Here, x can be any month and substituting that, we will get a
complete statement.

Section 1.3 Quantifiers, Predicates and Validity 1

Quantifiers and Predicates
●  Quantifiers:

●  Quantifiers are phrases that refer to given quantities, such as
“for some” or “for all” or “for every,” indicating how many
objects have a certain property.

●  Two kinds of quantifiers: Universal and Existential

●  Universal Quantifier: represented by ∀

●  The symbol is translated as and means “for all”, “given any”,
“for each,” or “for every,” and is known as the universal
quantifier.

●  Existential Quantifier: represented by ∃

●  The symbol is translated as and means variously “for some,”
“there exists,” “there is a,” or “for at least one”.


Quantifiers and Predicates
●  Predicate

●  It is the verbal statement that describes the property of a
variable. Usually represented by the letter P, the notation P(x) is
used to represent some unspecified property or predicate that x
may have

■  e.g. P(x) = x has 30 days.

■  P(April) = April has 30 days.

●  Combining the quantifier and the predicate, we get a complete
statement of the form ∀(x)P(x) or ∃(x)P(x).

●  The collection of objects that satisfy the property P(x) is called
the domain of interpretation.

●  Truth value of expressions formed using quantifiers and
predicates

■  What is the truth value of ∀(x)P(x) where x is all the months and

P(x) = x has less than 32 days.

■  Undoubtedly, the above is true since no month has 32 days.


Truth value of the following expressions
●  Truth of expression ∀(x)P(x)

1.  P(x) is the property that x is yellow, and the domain of
interpretation is the collection of all flowers:

not true

2.  P(x) is the property that x is a plant, and the domain of
interpretation is the collection of all flowers:

true

3.  P(x) is the property that x is positive, and the domain of
interpretation consists of integers:

not true

■  Can you find one interpretation in which both ∀(x)P(x) is true and
∃(x)P(x) is false?

Not possible

■  Can you find one interpretation in which both ∃(x)P(x) is true and
∀(x)P(x) is false?

Case 1 as mentioned above

●  Predicates involving properties of single variables : unary
predicates

●  Binary, ternary and n-ary predicates are also possible.

■  (∀x) (∃y)Q(x,y) is a binary predicate. This expression reads as
“for every x there exists a y such that Q(x,y).”


Interpretation
●  Formal deﬁnition: An interpretation for an expression involving
predicates consists of the following:

A collection of objects, called the domain of interpretation,
which must include at least one object.

An assignment of a property of the objects in the domain to each
predicate in the expression.

An assignment of a particular object in the domain to each
constant symbol in the expression.

●  Predicate wffs can be built similar to propositional wffs using
logical connectives with predicates and quantiﬁers.

●  Examples of predicate wffs

■  (∀x)[P(x) → Q(x)]

■  (∀x) ((∃y)[P(x,y) V Q(x,y)] → R(x))

■  S(x,y) Λ R(x,y)


Scope of a variable in an expression

●  Brackets are used wisely to identify the scope of the variable.

■  (∀x) [∃(y)[P(x,y) V Q(x,y)] → R(x)]

■  Scope of (∃y) is P(x,y) V Q(x,y) while the scope of (∀x) is the entire
expression.

■  (∀x)S(x) V (∃y)R(y)

■  Scope of x is S(x) while the scope of y is R(y).

■  (∀x)[P(x,y) → (∃y) Q(x,y)]

■  Scope of variable y is not deﬁned for P(x,y) hence y is called a free
variable. Such expressions might not have a truth value at all.

●  What is the truth of the expression

■  ∃(x)[A(x) Λ ∀(y)[B(x,y) → C(y)]] in the interpretation

■  A(x) is “x > 0” , B(x, y) is “x > y” and C(y) is “y ≤ 0” where the
domain of x is positive integers and the domain of y is all integers

True, x=1 is a positive integer and any integer less than x is ≤ 0


Translation: Verbal statements to symbolic form
●  “Every person is nice” can be rephrased as “For any thing, if it
is a person, then it is nice.” So, if P(x) is “x is a person” and Q
(x) be “x is nice,” the statement can be symbolized as

■  (∀x)[P(x) → Q(x)]

■  “All persons are nice” or “Each person is nice” will also have the
same symbolic form.

●  “There is a nice person” can be rewritten as “There exists
something that is both a person and nice.”

■  In symbolic form, (∃x)[P(x) Λ Q(x)].

■  Variations: “Some persons are nice” or “There are nice persons.”

●  What would the following form mean for the example above?

(∃x)[P(x) → Q(x)]

■  This will only be true if there are no persons in the world but that
is not the case.

■  Hence such a statement is false, so almost always, ∃ goes with Λ
(conjunction) and ∀ goes with → (implication).


Translation
●  Hint: Avoid confusion by framing the statement in different forms as
possible.

●  The word “only” can be tricky depending on its presence in the
statement.

■  X loves only Y ⇔ If X loves anything, then that thing is Y.

■  Only X loves Y ⇔ If anything loves Y, then it is X.

■  X only loves Y ⇔ If X does anything to Y, then it is love.

●  Example for forming symbolic forms from predicate symbols

■  D(x) is “x is dog”; R(x) is “x is a rabbit”; C(x,y) is “x chases y”

■  All dogs chase all rabbits ⇔ For anything, if it is a dog, then for any
other thing, if it is a rabbit, then the dog chases it ⇔ (∀x)[D(x) → (∀y)(R
(y) → C(x,y)]

■  Some dogs chase all rabbits ⇔ There is something that is a dog and for
any other thing, if that thing is a rabbit, then the dog chases it ⇔ (∃x)[D
(x) Λ (∀y)(R(y) → C(x,y)]

■  Only dogs chase rabbits ⇔ For any two things, if one is a rabbit and the
other chases it, then the other is a dog ⇔ (∀y) (∀x)[R(y) Λ C(x,y) → D
(x)]


Negation of statements
●  A(x): Everything is fun

●  Negation will be “it is false that everything is fun,” i.e. “something
is nonfun.”

●  In symbolic form, [(∀x)A(x)]ʹ′ ⇔ (∃x)[A(x)]ʹ′

●  Similarly negation of “Something is fun” is “Nothing is fun” or
“Everything is boring.”

●  Hence, [(∃x)A(x)]ʹ′ ⇔ (∀x)[A(x)]ʹ′


Class Exercise
●  What is the negation of “Everybody loves somebody sometime.”

■  Everybody hates somebody sometime

■  Somebody loves everybody all the time

■  Everybody hates everybody all the time

■  Somebody hates everybody all the time

√

●  What is the negation of the following statements?

●  Some pictures are old and faded.

Every picture is neither old nor faded.

●  All people are tall and thin.

Someone is short or fat.

●  Some students eat only pizza.

Every student eats something that is not pizza.

●  Only students eat pizza.

There is a non-student who eats pizza.


Class exercise
●  S(x): x is a student; I(x): x is intelligent; M(x): x likes music

●  Write wffs that express the following statements:

■  All students are intelligent.

For anything, if it is a student, then it is intelligent ⇔
(∀x)[S(x) → I (x)]
■  Some intelligent students like music.

There is something that is intelligent and it is a student and it likes music ⇔
(∃x)[I(x) Λ S(x) Λ M(x)]
■  Everyone who likes music is a stupid student.

For anything, if that thing likes music, then it is a student and it is not intelligent ⇔
(∀x)(M(x) → S(x) Λ [I (x)]ʹ′)

■  Only intelligent students like music.

For any thing, if it likes music, then it is a student and it is intelligent ⇔
(∀x)(M(x) → S(x) Λ I(x))


Validity
●  Analogous to a tautology of propositional logic.

●  Truth of a predicate wff depends on the interpretation.

●  A predicate wff is valid if it is true in all possible interpretations
just like a propositional wff is true if it is true for all rows of the
truth table.

●  A valid predicate wff is intrinsically true.

Propositional Wffs

Predicate Wffs

Truth values

True or false – depends on True, false or neither (if the
the truth value of statement wff has a free variable)

letters

Intrinsic truth

Tautology – true for all truth Valid wff – true for all
values of its statements

interpretations

Methodology

Truth table to determine if it No algorithm to determine
is a tautology

validity


Validity examples
●  (∀x)P(x) → (∃x)P(x)

■  This is valid because if every object of the domain has a certain
property, then there exists an object of the domain that has the
same property.

●  (∀x)P(x) → P(a)

■  Valid – quite obvious since is a member of the domain of x.

●  (∃x)P(x) → (∀x)P(x)

■  Not valid since the property cannot be valid for all objects in the
domain if it is valid for some objects of than domain. Can use a
mathematical context to check as well.

■  Say P(x) = “x is even,” then there exists an integer that is even but
not every integer is even.

●  (∀x)[P(x) V Q(x)] → (∀x)P(x) V (∀x)Q(x)

■  Invalid, can prove by mathematical context by taking P(x) = x is
even and Q(x) = x is odd.

■  In that case, the hypothesis is true but the conclusion is false
because it is not the case that every integer is even or that every
integer is odd.


Class Exercise
●  What is the truth of the following wffs where the domain consists
of integers:

●  (∀x)[L(x) → O(x)] where O(x) is “x is odd” and L(x) is “x < 10”?

●  (∃y)(∀x)(x + y = 0)?

●  (∃y)(∃x)(x2 = y)?

●  (∀x)[x < 0 → (∃y)(y > 0 Λ x + y = 0)]?

●  Using predicate symbols and appropriate quantiﬁers, write the
symbolic form of the following English statement:

●  D(x) is “x is a day”

M is “Monday”

T is “Tuesday”

●  S(x) is “x is sunny”

R(x) is “x is rainy”

●  Some days are sunny and rainy.

●  It is always a sunny day only if it is a rainy day.

●  It rained both Monday and Tuesday.

●  Every day that is rainy is not sunny.


CPSC 125 Ch 1 sec 3

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CPSC 125 Ch 1 sec 3