Mathematical Logic

Answer the following questions orally.
1. Divide 30 by half and add 10. What do you get?
2. Imagine you are in a sinking rowboat surrounded
by sharks. How would you survive?
3. Eskimos are very good hunters, but they never
hunt penguins. Why not?
4. There was an airplane crash. Every single person
died, but two people survived. How is this
possible?
5. If John’s son is my son’s father, what am I to
John?

6. A clerk in a butcher shop stands five feet and ten inches
tall. He wears size 10 shoes. What does he weigh?
7. In British Columbia, you cannot take a picture of a man
with a wooden leg. Why not?
8. 0, 3, 6, 9, 12, 15, 18 … 960. How many numbers are in
the series if all terms are included?
9. Tina’s granddaughter is my daughter’s daughter. Who is
Tina?
10.I recently returned from a trip. Today is Saturday. I
returned three days before the day after the day before
tomorrow. On what day did I return?

What is Logic?
Logic n.1. The branch of philosophy concerned
with analysing the patterns of reasoning by
which a conclusion is drawn from a set of
premises, without reference to meaning or
context
(Collins English Dictionary)

Foundations of Logic
Mathematical Logic is a tool for working with
complicated compound statements. It includes:
• A language for expressing them.
• A concise notation for writing them.
• A methodology for objectively reasoning about
their truth or falsity.
• It is the foundation for expressing formal proofs
in all branches of mathematics.

Propositional Logic
• Propositional Logic is the logic of compound
statements built from simpler statements using
so-called Boolean connectives.
Some applications in computer science:
• Design of digital electronic circuits.
• Expressing conditions in programs.
• Queries to databases & search engines.

Definition of a Proposition
• A proposition is simply a statement (i.e., a
declarative sentence) with a definite meaning,
having a truth value that’s either true (T) or
false (F) (never both, neither, or somewhere in
between).
• A proposition (statement) may be denoted by a
variable like P, Q, R,…, called a proposition
(statement) variable.

Examples of a Proposition
1. Manila is the capital city of the Philippines.
2. A is a consonant.
3. Virgil wrote Iliad and Odyssey.
4. The Catholic Bible has 66 books.
5. 6 + 9 = 15
6. 𝜋 is an irrational number.
7. Pneumonoultramicroscopicsilicovolcanoconiosis is
the longest word in a major dictionary.

Proposition or Not?
1. Clean the room.
2. Scientifically, eggplant is not a vegetable.
3. Can’t leopard remove its own spots?
4. What a magnificent building!
5. Litotes is an exaggerated statement.
6. The Passion of the Christ directed by Mel
Gibson.
7. x=3
8. x +y >4
9. Verb is a word that denotes an action.
10.This statement is false.

Operators / Connectives
• An operator or connective combines one or
more operand expressions into a larger
expression. (E.g., “+” in numeric expressions)
• Unary operators take 1 operand (e.g., −3)
• Binary operators take 2 operands (e.g., 3 × 4).
• Propositional or Boolean operators operate on
propositions or truth values instead of on
numbers.

Some Popular Boolean Operators
Formal Name Nickname Arity Symbol
• Negation operator NOT Unary ¬
• Conjunction operator AND Binary 
• Disjunction operator OR Binary 
• Exclusive-OR operator XOR Binary 
• Implication operator IMPLIES Binary 
• Biconditional operator IFF Binary

The Negation Operator
The unary negation operator “¬” (NOT)
transforms a property into its logical negation.
E.g. If p = “I have brown hair.”
then ¬p = “I do not have brown hair.”
Truth table for NOT:
p ¬p
T F
F T
Operand Column Result Column

The Conjunction Operator
The binary conjunction operator “” (AND)
combines two propositions to form their logical
conjunction.
E.g.
If p=“I will have salad for lunch.” and q=“I will
have steak for dinner.”, then pq=“I will have
salad for lunch and I will have steak for dinner.”

Conjunction Truth Table
• Note that a conjunction p1p2… pn of n
propositions will have 2n rows
• in its truth table.
Conjunction Truth Table
. p q pq
F F F
F T F
T F F
T T T

The Disjunction Operator
The binary disjunction operator “” (OR)
disjunction.
p = “My car has a bad engine.”
q = “My car has a bad carburetor.”
pq = “Either my car has a bad engine, or my car
has a bad carburetor.”

Disjunction Truth Table
Note that pq means that p is true, or q is true, or
both are true! So, this operation is also called
inclusive or, because it includes the possibility
that both p and q are true.
p q pq
F F F
F T T
T F T
T T T

Nested Propositional Expressions
• Use parentheses to group sub-expressions:
“I just saw my old friend, and either he’s grown or
I’ve shrunk.” = f (g s)
− (f g) s would mean something different
− f g s would be ambiguous
• By convention, “¬” takes precedence over both
“” and “”.
− ¬s f means (¬s) f , not ¬ (s f)

A Simple Exercise
Let
p = “It rained last night”,
q = “The sprinklers came on last night,”
r = “The lawn was wet this morning.”
Translate each of the following into English:
¬p = “It didn’t rain last night.”
r ¬p = “The lawn was wet this morning, and it
didn’t rain last night.”
¬ r p q = “Either the lawn wasn’t wet this morning,
or it rained last night, or the
sprinklers came on last night.”

The Exclusive Or Operator
The binary exclusive-or operator “” (XOR)
“exclusive or”
p = “I will earn an A in this course,”
q = “I will drop this course,”
p q = “I will either earn an A in this course, or I
will drop it (but not both!)”

Exclusive-Or Truth Table
• Note that pq means that p is true, or q is true,
but not both!
• This operation is called exclusive or, because it
excludes the possibility that both p and q are
true.
• “¬” and “” together are not universal.
p q pq
F F F
F T T
T F T
T T F

Natural Language is Ambiguous
• Note that English “or” can be ambiguous regarding
the “both” case!
“Pat is a singer or Pat is a writer.” – 
“Pat is a man or Pat is a woman.” – 
p q pq
F F F
F T T
T F T
T T ?
Need context to disambiguate the meaning!
For this class, assume “or” means inclusive.

The Implication Operator
antecedent consequence
The implication p q states that p implies q.
i.e., If p is true, then q is true; but if p is not true,
then q could be either true or false.
E.g., let p = “You study hard.”
q = “You will get a good grade.”
p q = “If you study hard, then you will get a
• good grade.” (else, it could go either way)

Implication Truth Table
• p q is false only when p is true but q is not
true.
• p q does not say that p causes q!
• p q does not require that p or q are ever true!
• E.g. “(1=0) pigs can fly” is TRUE!
p q pq
F F T
F T T
T F F
T T T

Examples of Implications
• “If this lecture ends, then the sun will rise
tomorrow.” True or False?
• “If Tuesday is a day of the week, then I am a
penguin.” True or False?
• “If 1+1=6, then Obama is president.” True or False?
• “If the moon is made of green cheese, then I am
richer than Bill Gates.” True or False?

Why does this seem wrong?
• Consider a sentence like,
− “If I wear a red shirt tomorrow, then the U.S. will
attack Iraq the same day.”
• In logic, we consider the sentence True so long
as either I don’t wear a red shirt, or the US
attacks.
• But in normal English conversation, if I were to
make this claim, you would think I was lying.
− Why this discrepancy between logic &
language?

Resolving the Discrepancy
• In English, a sentence “if p then q” usually really implicitly
means something like,
− “In all possible situations, if p then q.”
That is, “For p to be true and q false is impossible.”
Or, “I guarantee that no matter what, if p, then q.”
• This can be expressed in predicate logic as:
− “For all situations s, if p is true in situation s, then q is also true
in situations”
− Formally, we could write: "s, P(s) Q(s)
• This sentence is logically False in our example, because for
me to wear a red shirt and the U.S. not to attack Iraq is a
possible (even if not actual) situation.
− Natural language and logic then agree with each other.

English Phrases Meaning p  q
• “p implies q”
• “if p, then q”
• “if p, q”
• “when p, q”
• “p only if q”
• “p is sufficient for q”
• “q is necessary for p”
• “q follows from p”
• “whenever p, q”
• “q if p”
• “q when p”
• “q whenever p”
• “q is implied by p”
We will see some
equivalent logic
expressions later.

Converse, Inverse, Contrapositive
Some terminology, for an implication p q:
• Its converse is: q p.
• Its inverse is: ¬p¬q.
• Its contrapositive: ¬q ¬ p.
• One of these three has the same meaning (same
truth table) as p q. Can you figure out which?

How do we know for sure?
Proving the equivalence of p q and its
contrapositive using truth tables:

The biconditional operator
The biconditional p q states that p is true if and
only if (IFF) q is true.
p = “You can take the flight.”
q = “You buy a ticket”
p q = “You can take the flight if and only if you buy a
ticket.”

Biconditional Truth Table
• p q means that p and q have the same truth value.
• Note this truth table is the exact opposite of ’s!
− p q means ¬(p q)
• p q does not imply p and q are true, or cause each
other. p q p q
F F T
F T F
T F F
T T T

Boolean Operations Summary
We have seen 1 unary operator and 5 binary operators.
Their truth tables are below.
p q ¬p pq pq pq pq pq
F F T F F F T T
F T T F T T T F
T F F F T T F F
T T F T T F T T

Well-formed Formula (WFF)
A well-formed formula (Syntax of compound
proposition)
1. Any statement variable is a WFF.
2. For any WFF α, ¬α is a WFF.
3. If α and β are WFFs, then (α β), (α β), (α 
β) and (α β) are WFFs.
4. A finite string of symbols is a WFF only when it
is constructed by steps 1, 2, and 3.

Example of well-formed formula
• By definition of WFF
− WFF: ¬(PQ), (P(PQ)), (¬PQ), ((P Q)
(QR))(PR)), etc.
− not WFF:
1.(PQ) (Q) : (Q) is not a WFF.
2. (PQ: but (PQ) is a WFF. etc..

Tautology
• Definition:
A well-formed formula (WFF) is a tautology if for
every truth value assignment to the variables
appearing in the formula, the formula has the
value of true.
• Ex. p p (ㅑp p)

Substitution instance
• Definition:
A WFF A is a substitution instance of another
formula B if A is formed from B by substituting
formulas for variables in B under condition that the
same formula is substituted for the same variable
each time that variable is occurred.
• Ex. B: p(j p), A: (rs)(j (rs))
• Theorem:
A substitution instance of a tautology is a
tautology
• Ex. B: p p, A: (q r) (q r)

Contradiction
• Definition:
A WFF is a contradiction if for every truth value
assignment to the variables in the formula, the
formula has the value of false.
Ex. (p p)

Valid consequence (1)
• Definition:
A formula (WFF) B is a valid consequence of a
formula A, denoted by A ㅑB, if for all truth value
assignments to variables appearing in A and B,
the formula B has the value of true whenever the
formula A has the value of true.

• Definition:
A formula (WFF) B is a valid consequence of a
formula A1,…, An,(A1,…, Anㅑ B) if for all truth
value assignments to the variables appearing in
A1,…, An and B, the formula B has the value of
true whenever the

• Theorem: Aㅑ B iff ㅑ (A B)
• Theorem: A1,…, Anㅑ B iff (A1 …An)ㅑB
• Theorem: A1,…, Anㅑ B iff (A1 …An-1) ㅑ(AnB)

Logical Equivalence (Part 1)
Logical Equivalence Part 1.mp4

Logical Equivalence (Part 2)
Logical Equivalences - EECS 203.mp4

Logical Equivalence
• Definition:
Two WFFs, p and q, are logically equivalent
IFF p and q have the same truth values for every
truth value assignment to all variables contained in
p and q.
Ex. ¬ ¬p, p : ¬ ¬p p
p p, p : p p p
(p ¬ p) q, q : (p ¬ p) q q
p ¬ p, q ¬ q : p ¬ p q ¬ q

Logical Equivalence
• Theorem:
If a formula A is equivalent to a formula B then
ㅑAB (A B)
• Theorem:
If a formula D is obtained from a formula A by
replacing a part of A, say C, which is itself a
formula, by another formula B such that CB,
then AD

Proving Equivalence via Truth Tables
Ex. Prove that pq ¬(¬p ¬q).
p q p q ¬p ¬q ¬ p ¬q ¬((¬p ¬q))
F F F T T T F
F T T T F F T
T F T F T F T
T T T F F F T

Equivalence Laws - Examples
• Identity: pT p pF p
• Domination: pT T pF F
• Idempotent: pp p pp p
• Double negation: ¬¬p p
• Commutative: pq qp pq qp
• Associative: (pq)rp(qr) (pq)rp(qr)

More Equivalence Laws
• Distributive:
p(qr) (pq)(pr) p(qr) (pq)(pr)
• De Morgan’s:
¬(pq) ¬p ¬q
¬(pq) ¬p ¬q
• Trivial tautology/contradiction:
p ¬pT p ¬pF

Defining Operators via Equivalences
Using equivalences, we can define operators in
terms of other operators.
• Exclusive or:
pq(pq)¬ (pq)
pq(p¬q)(¬p q)
• Implies:
pq ¬p q
• Biconditional:
pq (pq) (qp)
pq ¬ (pq)

Exercise 1
• Let p and q be the proposition variables denoting
p: It is below freezing.
q: It is snowing.
Write the following propositions using variables, p and q,
and logical connectives.
a) It is below freezing and snowing.
b) It is below freezing but not snowing.
c) It is not below freezing and it is not snowing.
d) It is either snowing or below freezing (or both).
e) If it is below freezing, it is also snowing.
f) It is either below freezing or it is snowing, but it is not snowing if it is
below freezing.
g) That it is below freezing is necessary and sufficient for it to be
snowing

Predicate Logic
• Predicate logic is an extension of propositional
logic that permits concisely reasoning about
whole classes of entities.
• Propositional logic (recall) treats simple
propositions (sentences) as atomic entities.
• In contrast, predicate logic distinguishes the
subject of a sentence from its predicate.
− Ex. arithmetic predicates: x=3, x>y, x+y=z
− propositions: 4=3, 3>4, 3+4=7 if (x>3) then y=x;

Universes of Discourse (U.D.s)
• The power of distinguishing objects from
predicates is that it lets you state things about
many objects at once.
• E.g., let P(x)=“x+1>x”. We can then say,
“For any number x, P(x) is true” instead of
(0+1>0) (1+1>1) (2+1>2) ...
• Definition:
− The collection of values that a variable x can
take is called x’s universe of discourse.

Quantifier Expressions
• Definition:
− Quantifiers provide a notation that allows us to
quantify (count) how many objects in the universe
of discourse satisfy a given predicate.
− “"” is the FOR "LL or universal quantifier.
"x P(x) means for all x in the u.d., P holds.
− “$” is the $XISTS or existential quantifier.
$x P(x) means there exists an x in the u.d. (that is,
1 or more) such that P(x) is true.

The Universal Quantifier "
• Example:
Let the u.d. of x be parking spaces at SNU.
Let P(x) be the predicate “x is full.”
Then the universal quantification of P(x), "x P(x),
is the proposition:
− “All parking spaces at DMMMSU are full.”
− i.e., “Every parking space at DMMMSU is full.”
− i.e., “For each parking space at SNU, that space
is full.”

The Existential Quantifier $
• Example:
Let the u.d. of x be parking spaces at SNU.
Let P(x) be the predicate “x is full.”
Then the existential quantification of P(x), $x P(x),
is the proposition:
− “Some parking space at SNU is full.”
− “There is a parking space at SNU that is full.”
− “At least one parking space at SNU is full.”

Free and Bound Variables
• Definition:
− An expression like P(x) is said to have a free
variable x (meaning, x is undefined).
A quantifier (either "or $) operates on an
− expression having one or more free variables,
and binds one or more of those variables, to
produce an expression having one or more bound
variables.
− Ex. $x [x+y=z], x is bound but y and z are free
variables

Example of Binding
• P(x,y) has 2 free variables, x and y.
• "x P(x,y) has 1 free variable y, and one bound
variable x.
• “P(x), where x=3” is another way to bind x.
− Ex. “x+y=3”, T if x=1 and y=2, F if x=2 and y=6
• An expression with zero free variables is a bona-
fide (actual) proposition.
• An expression with one or more free variables is
still only a predicate: "x P(x,y)

Nesting of Quantifiers
Example: Let the u.d. of x & y be people.
Let L(x,y)=“x likes y” (a predicate w. 2 f.v.’s)
Then $y L(x,y) = “There is someone whom x
likes.”
(A predicate w. 1 free variable, x)
Then "x $y L(x,y) =
“Everyone has someone whom they like.”
(A _proposition_ with _o_ free variables.)

Example of Binding
• "xI [x<x+1] : T
• "xI [x=3] : F
• "xI "yI [x+y>x]: F
• "xI+ "yI+ [x+y>x]: T
• $xI [x<x+1] : T
• $xI [x=3] : T
• $xI [x=x+1] : F

WFF for Predicate Calculus
A WFF for (the first-order) calculus
1.Every predicate formula is a WFF.
2.If P is a WFF, ¬P is a WFF.
3.Two WFFs parenthesized and connected by ,
,
, form a WFF.
4.If P is a WFF and x is a variable then ("x )P and
($x)P are WFFs.
5.A finite string of symbols is a WFF only when it
is constructed by steps 1-4.

Quantifier Exercise
If R(x,y)=“x relies upon y,” express the following in
unambiguous English:
"x $y R(x,y) = Everyone has someone to rely on.
$y "x R(x,y) = There’s a poor overburdened soul whom
everyone relies upon (including himself)!
$x "y R(x,y) = There’s some needy person who relies upon
everybody (including himself).
"y $x R(x,y) = Everyone has someone who relies upon them.
"x "y R(x,y) = Everyone relies upon everybody. (including
themselves)!

Natural language is ambiguous!
• “Everybody likes somebody.”
− For everybody, there is somebody they like,
• "x $y Likes(x,y)
or, there is somebody (a popular person) whom
[Probably more likely.]
− everyone likes?
• $y "x Likes(x,y)
• “Somebody likes everybody.”
− Same problem: Depends on context, emphasis.

More to Know About Binding
• "x $x P(x) - x is not a free variable in $x P(x),
therefore the "x binding isn’t used.
• ("x P(x)) Q(x) - The variable x in Q(x) is outside of
the scope of the "x quantifier, and is therefore free.
Not a proposition!
"x P(x) Q(x) ≠ "x (P(x) Q(x))
("x P(x)) Q(x)
"x P(x) Q(y) : clearer notation
• ("x P(x)) ($x Q(x)) – This is legal, because there
are 2 different x’s!

Quantifier Equivalence Laws
• Definitions of quantifiers: If u.d.=a,b,c,…
"x P(x) P(a) P(b) P(c) …
$x P(x) P(a) P(b) P(c) …
• From those, we can prove the laws:
¬"x P(x) $x ¬P(x)
¬$x P(x) "x ¬P(x)
• Which propositional equivalence laws can be used to
prove this?
Ex. ¬$x"y"z P(x,y,z)"x¬"y"z P(x,y,z)
"x$y¬"z P(x,y,z)
"x$y$z ¬P(x,y,z)

More Equivalence Laws
• "x "y P(x,y) "y "x P(x,y)
$x $y P(x,y) $y $x P(x,y)
• "x $y P(x,y) <≠> $y "x P(x,y)
• "x (P(x) Q(x))("x P(x)) ("x Q(x))
$x (P(x) Q(x))($x P(x)) ($x Q(x))
• "x (P(x) Q(x)) <≠> ("x P(x)) ("x Q(x))
$x (P(x) Q(x)) <≠> ($x P(x)) ($x Q(x))

Defining New Quantifiers
• Definition:
− $!x P(x) to mean “P(x) is true of exactly one x in
the universe of discourse.”
• Note that $!x P(x) $x (P(x) $y (P(y) (y
x)))
“There is an x such that P(x), where there is no y
such that P(y) and y is other than x.”

Exercise 2
Let F(x, y) be the statement “x loves y,” where the universe of
discourse for both x and y consists of all people in the world.
Use quantifiers to express each of these statements.
a) Everybody loves Jerry.
b) Everybody loves somebody.
c) There is somebody whom everybody loves.
d) Nobody loves everybody.
e) There is somebody whom Lydia does not love.
f) There is somebody whom no one loves.
g) There is exactly one person whom everybody loves.
h) There are exactly two people whom Lynn loves.
i) Everyone loves himself or herself
j) There is someone who loves no one besides himself or
herself.

Exercise
1. Let p, q, and r be the propositions
p: You have the flu.
q: You miss the final examination
r: You pass the course
Express each of these propositions as an English
sentence.
(a) (p￢r)∨(q￢r)
(b) (p∧q) ∨(￢q∧r)

Exercise (cont.)
2. Assume the domain of all people.
Let J(x) stand for “x is a junior”,
S(x) stand for “x is a senior”, and
L(x, y) stand for “x likes y”.
Translate the following into well-formed formulas:
(a) All people like some juniors.
(b) Some people like all juniors.
(c) Only seniors like juniors.

Exercise (cont.)
3. Let B(x) stand for “x is a boy”, G(x) stand for “x is a
girl”, and T(x,y) stand for “x is taller than y”.
Complete the well-formed formula representing the
given statement by filling out the missing part.
(a) Only girls are taller than boys: (?)(∀y)((? ∧T(x,y)) ?)
(b) Some girls are taller than boys: (∃x)(?)(G(x) ∧(? ?))
(c) Girls are taller than boys only: (?)(∀y)((G(x) ∧?) ?)
(d) Some girls are not taller than any boy: (∃x)(?)(G(x) ∧(? ?))
(e) No girl is taller than any boy: (?)(∀y)((B(y) ∧?) ?)

Mathematical Logic

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Mathematical Logic