UNIT-1
MATHEMATICAL LOGIC
Introduction: Logic?
• We will study
– Propositional Logic (PL)
– First-Order Logic (FOL)
• Logic
– is the study of the logic relationships between
objects and
– forms the basis of all mathematical reasoning and
all automated reasoning
Introduction: PL?
• Propositional Logic (PL) = Propositional
Calculus = Sentential Logic
• In Propositional Logic, the objects are called
propositions
• Definition: A proposition is a statement that
is either true or false, but not both
• We usually denote a proposition by a letter: p,
q, r, s, …
Outline
• Defining Propositional Logic
–
–
–
–
Propositions
Connectives
Precedence of Logical Operators
Truth tables
• Usefulness of Logic
– Bitwise operations
– Logic in Theoretical Computer Science (SAT)
– Logic in Programming
• Logical Equivalences
– Terminology
– Truth tables
– Equivalence rules
Introduction: Proposition
• Definition: The value of a proposition is called
its truth value; denoted by
– T or 1 if it is true or
– F or 0 if it is false
• Opinions, interrogative, and imperative are
not propositions
• Truth table
p
0
1
Propositions: Examples
• The following are propositions
– Today is Monday
– The grass is wet
– It is raining
M
W
R
• The following are not propositions
– C++ is the best language
– When is the pretest?
– Do your homework
Opinion
Interrogative
Imperative
Are these propositions?
• 2+2=5
• Every integer is divisible by 12
• Microsoft is an excellent company
Logical connectives
• Connectives are used to create a compound
proposition from two or more propositions
Negation (e.g., a or !a or ā) $\neg$, $\bar$
And or logical conjunction (denoted )
$\wedge$
OR or logical disjunction (denoted ) $\vee$
XOR or exclusive or (denoted )
$\oplus$
Impli ion (denoted  or )
$\Rightarrow$, $\rightarrow$
– Biconditional (denoted  or )
$\LeftRightarrow$, $\leftrightarrow$
–
–
–
–
–
• We define the meaning (semantics) of the logical
connectives using truth tables
Precedence of Logical Operators
• As in arithmetic, an ordering is imposed on the use of logical
operators in compound propositions
• However, it is preferable to use parentheses to disambiguate
operators and facilitate readability
 p  q   r  (p)  (q  (r))
• To avoid unnecessary parenthesis, the following precedences
hold:
1.
2.
3.
4.
Negation ()
Conjunction ()
Disjunction ()
Implication ()
5.
Biconditional ()
Logical Connective: Negation
• p, the negation of a proposition p, is also a
proposition
• Examples:
– Today is not Monday
– It is not the case that today is Monday, etc.
• Truth table
p
p
0
1
1
0
Logical Connective: Logical And
• The logical connective And is true only when both of the
propositions are true. It is also called a conjunction
• Examples
– It is raining and it is warm
– (2+3=5) and (1<2)
– Schroedinger’s cat is dead and Schroedinger’s cat is not dead.
• Truth table
p
q
0
0
0
1
1
0
1
1
pq
Logical Connective: Logical OR
• The logical disjunction, or logical OR, is true if one or
both of the propositions are true.
• Examples
– It is raining or it is the second lecture
– (2+2=5)  (1<2)
– You may have cake or ice cream
• Truth table
p
q
pq
0
0
0
0
1
0
1
0
0
1
1
1
pq
Logical Connective: Exclusive Or
• The exclusive OR, or XOR, of two propositions is true when
exactly one of the propositions is true and the other one is
false
• Example
– The circuit is either ON or OFF but not both
– Let ab<0, then either a<0 or b<0 but not both
– You may have cake or ice cream, but not both
• Truth table
p
q
pq
pq
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
1
pq
Logical Connective: Implication (1)
• Definition: Let p and q be two propositions. The
implication pq is the proposition that is false when
p is true and q is false and true otherwise
– p is called the hypothesis, antecedent, premise
– q is called the conclusion, consequence
• Truth table
p
q
pq
pq
pq
0
0
0
0
0
0
1
0
1
1
1
0
0
1
1
1
1
1
1
0
pq
Logical Connective: Implication (2)
• The implication of pq can be also read as
–
–
–
–
–
–
–
–
–
–
If p then q
p implies q
If p, q
p only if q
q if p
q when p
q whenever p
q follows from p
p is a sufficient condition for q (p is sufficient for q)
q is a necessary condition for p (q is necessary for p)
Logical Connective: Implication (3)
• Examples
– If you buy you air ticket in advance, it is cheaper.
– If x is an integer, then x2  0.
– If it rains, the grass gets wet.
– If the sprinklers operate, the grass gets wet.
– If 2+2=5, then all unicorns are pink.
Exercise: Which of the following implications is true?
• If -1 is a positive number, then 2+2=5
True. The premise is obviously false, thus no matter what the
conclusion is, the implication holds.
• If -1 is a positive number, then 2+2=4
True. Same as above.
• If sin x = 0, then x = 0
False. x can be a multiple of . If we let x=2, then sin x=0 but x0.
The implication “if sin x = 0, then x = k, for some k” is true.
Logical Connective: Biconditional (1)
• Definition: The biconditional pq is the
proposition that is true when p and q have the
same truth values. It is false otherwise.
• Note that it is equivalent to (pq)(qp)
• Truth table
p
q
pq
pq
pq pq pq
0
0
0
0
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
0
1
Logical Connective: Biconditional (2)
• The biconditional pq can be equivalently read
as
–
–
–
–
p if and only if q
p is a necessary and sufficient condition for q
if p then q, and conversely
p iff q (Note typo in textbook, page 9, line 3)
• Examples
– x>0 if and only if x2 is positive
– The alarm goes off iff a burglar breaks in
– You may have pudding iff you eat your meat
Exercise: Which of the following biconditionals is true?
• x2 + y2 = 0 if and only if x=0 and y=0
True. Both implications hold
• 2 + 2 = 4 if and only if 2<2
True. Both implications hold.
• x2  0 if and only if x  0
False. The implication “if x  0 then x2  0” holds.
However, the implication “if x2  0 then x  0” is false.
Consider x=-1.
The hypothesis (-1)2=1  0 but the conclusion fails.
Converse, Inverse, Contrapositive
• Consider the proposition p  q
– Its converse is the proposition q  p
– Its inverse is the proposition p  q
– Its contrapositive is the proposition q  p
Truth Tables
• Truth tables are used to show/define the
relationships between the truth values of
– the individual propositions and
– the compound propositions based on them
p
q
pq
pq
pq
pq
pq
0
0
0
0
0
1
1
0
1
0
1
1
1
0
1
0
0
1
1
0
0
1
1
1
1
0
1
1
Constructing Truth Tables
• Construct the truth table for the following
compound proposition
(( p  q ) q )
p
q
pq
q
(( p  q ) q )
0
0
0
1
1
0
1
0
0
0
1
0
0
1
1
1
1
1
0
1
Outline
• Defining Propositional Logic
–
–
–
–
Propositions
Connectives
Precedence of Logical Operators
Truth tables
• Usefulness of Logic
– Bitwise operations
– Logic in Theoretical Computer Science (SAT)
– Logic in Programming
• Logical Equivalences
– Terminology
– Truth tables
– Equivalence rules
Usefulness of Logic
• Logic is more precise than natural language
– You may have cake or ice cream.
• Can I have both?
– If you buy your air ticket in advance, it is cheaper.
• Are there or not cheap last-minute tickets?
• For this reason, logic is used for hardware and
software specification
– Given a set of logic statements,
– One can decide whether or not they are satisfiable
(i.e., consistent), although this is a costly process…
Bitwise Operations
•
•
•
•
Computers represent information as bits (binary digits)
A bit string is a sequence of bits
The length of the string is the number of bits in the string
Logical connectives can be applied to bit strings of equal
length
• Example
0110 1010 1101
0101 0010 1111
_____________
Bitwise OR
0111 1010 1111
Bitwise AND
...
Bitwise XOR
…
Logic in TCS
• What is SAT? SAT is the problem of determining
whether or not a sentence in propositional logic
(PL) is satisfiable.
– Given: a PL sentence
– Question: Determine whether or not it is satisfiable
• Characterizing SAT as an NP-complete problem
(complexity class) is at the foundation of
Theoretical Computer Science.
• What is a PL sentence? What does satisfiable
mean?
Logic in TCS: A Sentence in PL
• A Boolean variable is a variable that can have a value 1
or 0. Thus, Boolean variable is a proposition.
• A term is a Boolean variable
• A literal is a term or its negation
• A clause is a disjunction of literals
• A sentence in PL is a conjunction of clauses
• Example: (a  b  c  d)  (b  c)  (a  c  d)
• A sentence in PL is satisfiable iff
– we can assign a truth value
– to each Boolean variables
– such that the sentence evaluates to true (i.e., holds)
SAT in TCS
• Problem
– Given: A sentence in PL (a complex proposition),
which is
• Boolean variables connected with logical connectives
• Usually, as a conjunction of clauses (CNF = Conjunctive
Normal Form)
– Question:
• Find an assignment of truth values (0/1)
• That makes the sentence true, i.e. the sentence holds
Logic in Programming: Example 1
• Say you need to define a conditional
statement as follows:
– Increment x if the following condition holds
(x > 0 and x < 10) or x=10
• You may try: If (0<x<10 OR x=10) x++;
• Can’t be written in C++ or Java
• How can you modify this statement by using
logical equivalence
• Answer: If (x>0 AND x<=10) x++;
Logic in Programming: Example 2
• Say we have the following loop
While
((i<size AND A[i]>10) OR
(i<size AND A[i]<0) OR
(i<size AND (NOT (A[i]!=0 AND NOT (A[i]>=10)))))
• Is this a good code? Keep in mind:
– Readability
– Extraneous code is inefficient and poor style
– Complicated code is more prone to errors and difficult
to debug
– Solution? Comes later…
Outline
• Defining Propositional Logic
–
–
–
–
Propositions
Connectives
Precedence of Logical Operators
Truth tables
• Usefulness of Logic
– Bitwise operations
– Logic in Theoretical Computer Science (SAT)
– Logic in Programming
• Logical Equivalences
– Terminology
– Truth tables
– Equivalence rules
Propositional Equivalences: Introduction
• To manipulate a set of statements (here, logical
propositions) for the sake of mathematical
argumentation, an important step is to replace
one statement with another equivalent
statement (i.e., with the same truth value)
• Below, we discuss
– Terminology
– Establishing logical equivalences using truth tables
– Establishing logical equivalences using known laws (of
logical equivalences)
Terminology:
Tautology, Contradictions, Contingencies
• Definitions
– A compound proposition that is always true, no
matter what the truth values of the propositions that
occur in it is called a tautology
– A compound proposition that is always false is called a
contradiction
– A proposition that is neither a tautology nor a
contradiction is a contingency
• Examples
– A simple tautology is p  p
– A simple contradiction is p  p
Logical Equivalences: Definition
• Definition: Propositions p and q are logically
equivalent if p  q is a tautology.
• Informally, p and q are equivalent if whenever
p is true, q is true, and vice versa
• Notation: p  q (p is equivalent to q), p  q,
and p  q
• Alert:  is not a logical connective $\equiv$
Logical Equivalences: Example 1
• Are the propositions (p  q) and (p  q)
logically equivalent?
• To find out, we construct the truth tables for
each:
p
q
pq
p
pq
0
0
0
1
1
0
1
1
The two columns in the truth table are identical, thus we conclude that
(p  q)  (p  q)
Logical Equivalences: Example 1
• Show that
(Exercise 25 from Rosen)
(p  r)  (q  r)  (p  q)  r
p
q
r
0
0 0
0
0 1
0
1 0
0
1 1
1
0 0
1
0 1
1
1 0
1
1 1
p r
q r
(p r)  (q  r)
pq
(p  q)  r
Propositional Equivalences: Introduction
• To manipulate a set of statements (here, logical
propositions) for the sake of mathematical
argumentation, an important step is to replace
one statement with another equivalent
statement (i.e., with the same truth value)
• Below, we discuss
– Terminology
– Establishing logical equivalences using truth tables
– Establishing logical equivalences using known laws (of
logical equivalences)
Logical Equivalences: Cheat Sheet
• Table of logical equivalences can be found in
Rosen (page 24)
• These and other can be found in a handout on
the course web page:
http://www.cse.unl.edu/~cse235/files/LogicalEquivalences.pdf
• Let’s take a quick look at this Cheat Sheet
Using Logical Equivalences: Example 1
• Logical equivalences can be used to construct
additional logical equivalences
• Example: Show that (p  q) q is a tautology
0. (p  q) q
1.  (p  q)  q
2.  (p  q)  q
3.  p  (q  q)
4.  p  1
5.  1
Implication Law on 0
De Morgan’s Law (1st) on 1
Associative Law on 2
Negation Law on 3
Domination Law on 4
My Advice
•
•
•
•
Remove double implication
Replace implication by disjunction
Push negation inwards
Distribute
Using Logical Equivalences: Example 2
• Example (Exercise 17)*: Show that (p  q)  (p  q)
• Sometimes it helps to start with the second proposition (p  q)
0. (p  q)
1.  (p  q)  (q  p)
Equivalence Law on 0
2.  (p  q)  (q  p)
Implication Law on
1
3.  (((p  q)  (q  p)))
Double negation on 2
4.  ((p  q)  (q  p))
De Morgan’s Law…
5.  ((p  q)  (q  p))
De Morgan’s Law
6.
 ((p  q)  (p  p)  (q  q)  (q  p))
Distribution Law
7.  ((p  q)  (q  p))
Identity Law
8.  ((q  p )  (p  q))
Implication Law
9.  (p  q)
Equivalence Law
*See Table 8 (p 25) but you are not allowed to use the table for the proof
Using Logical Equivalences: Example 3
• Show that (q  p)  (p  q)  q
0. (q  p)  (p  q)
1.  (q  p)  (p  q)
Implication Law
2.  (q  p)  (p  q)
De Morgan’s
& Double negation
3.  (q  p)  (q  p)
4.  q  (p  p)
5.  q  1
q
Commutative Law
Distributive Law
Identity Law
Identity Law
Proving Logical Equivalences: Summary
• Proving two PL sentences A,B are equivalent using TT + EL
1.
2.
3.
Verify that the 2 columns of A, B in the truth table are the same (i.e.,
A,B have the same models)
Verify that the column of (AB  BA) in the truth table has all-1
entries (it is a tautology)
Put A,B in CNF, they should be the same
•
4.
Sequence of equivalence laws: Biconditional, implication, moving
negation inwards, distributivity
Apply a sequence of inference laws
•
•
•
Starting from one sentence, usually the most complex one,
Until reaching the second sentence
Typical sequence: Biconditional, implication, moving negation inwards,
distributivity
Logic in Programming: Example 2
(revisited)
• Recall the loop
While
((i<size AND A[i]>10) OR
(i<size AND A[i]<0) OR
(i<size AND (NOT (A[i]!=0 AND NOT (A[i]>=10)))))
• Now, using logical equivalences, simplify it!
• Using De Morgan’s Law and Distributivity
While ((i<size) AND
((A[i]>10 OR A[i]<0) OR
(A[i]==0 OR A[i]>=10)))
• Noticing the ranges of the 4 conditions of A[i]
While ((i<size) AND (A[i]>=10 OR A[i]<=0))
Programming Pitfall Note
• In C, C++ and Java, applying the commutative
law is not such a good idea.
• For example, consider accessing an integer
array A of size n:
if (i<n && A[i]==0) i++;
is not equivalent to
if (A[i]==0 && i<n) i++;
Predicate Logic and Quantifies
Outline
• Introduction
• Terminology:
– Propositional functions; arguments; arity; universe of
discourse
• Quantifiers
– Definition; using, mixing, negating them
• Logic Programming (Prolog)
• Transcribing English to Logic
• More exercises
Introduction
• Consider the statements:
x>3, x=y+3, x+y=z
• The symbols >, +, = denote relations between x and 3, x, y, and
4, and x,y, and z, respectively
• These relations may hold or not hold depending on the values
that x, y, and z may take.
• A predicate is a property that is affirmed or denied about the
subject (in logic, we say ‘variable’ or ‘argument’) of a statement
• Consider the statement : ‘x is greater than 3’
– ‘x’ is the subject
– ‘is greater than 3’ is the predicate
Propositional Functions (1)
• To write in Predicate Logic ‘x is greater than 3’
– We introduce a functional symbol for the predicate and
– Put the subject as an argument (to the functional symbol):
P(x)
• Terminology
– P(x) is a statement
– P is a predicate or propositional function
– x as an argument
Propositional Functions (2)
• Examples:
–
–
–
–
Father(x): unary predicate
Brother(x.y): binary predicate
Sum(x,y,z): ternary predicate
P(x,y,z,t): n-ary predicate
Propositional Functions (3)
• Definition: A statement of the form P(x1,x2,…, xn) is
the value of the propositional symbol P.
• Here: (x1,x2,…, xn) is an n-tuple and P is a predicate
• We can think of a propositional function as a
function that
–
–
–
–
Evaluates to true or false
Takes one or more arguments
Expresses a predicate involving the argument(s)
Becomes a proposition when values are assigned to the
arguments
Propositional Functions: Example
• Let Q(x,y,z) denote the statement ‘x2+y2=z2’
– What is the truth value of Q(3,4,5)?
Q(3,4,5) is true
– What is the truth value of Q(2,2,3)?
Q(2,3,3) is false
– How many values of (x,y,z) make the predicate
true?
There are infinitely many values that make the proposition true,
how many right triangles are there?
Universe of Discourse
• Consider the statement ‘x>3’, does it make
sense to assign to x the value ‘blue’?
• Intuitively, the universe of discourse is the set
of all things we wish to talk about; that is the
set of all objects that we can sensibly assign to
a variable in a propositional function.
• What would be the universe of discourse for
the propositional function below be:
EnrolledCSE235(x)=‘x is enrolled in CSE235’
Universe of Discourse: Multivariate functions
• Each variable in an n-tuple (i.e., each argument) may
have a different universe of discourse
• Consider an n-ary predicate P:
P(r,g,b,c)= ‘The rgb-values of the color c is (r,g,b)’
• Example, what is the truth value of
– P(255,0,0,red)
– P(0,0,255,green)
• What are the universes of discourse of (r,g,b,c)?
Outline
• Introduction
• Terminology:
– Propositional functions; arguments; arity; universe of
discourse
• Quantifiers
– Definition; using, mixing, negating them
• Logic Programming (Prolog)
• Transcribing English to Logic
• More exercises
Quantifiers: Introduction
• The statement ‘x>3’ is not a proposition
• It becomes a proposition
– When we assign values to the argument: ‘4>3’ is false, ‘2<3’ is true, or
– When we quantify the statement
• Two quantifiers
– Universal quantifier 
$\forall$
the proposition is true for all possible values in the universe of
discourse
– Existential quantifier
$\exists$
the proposition is true for some value(s) in the universe of discourse
Universal Quantifier: Definition
• Definition: The universal quantification of a
predicate P(x) is the proposition ‘P(x) is true for all
values of x in the universe of discourse.’ We use the
notation:  x P(x), which is read ‘for all x’.
• If the universe of discourse is finite, say {n1,n2,…,nk},
then the universal quantifier is simply the
conjunction of the propositions over all the elements
 x P(x)  P(n1)  P(n2)  …  P(nk)
Universal Quantifier: Example 1
• Let P(x): ‘x must take a discrete mathematics course’ and Q(x):
‘x is a CS student.’
• The universe of discourse for both P(x) and Q(x) is all UNL
students.
• Express the statements:
– “Every CS student must take a discrete mathematics course.”
 x Q(x)  P(x)
– “Everybody must take a discrete mathematics course or be a CS
student.”  x ( P(x)  Q(x) )
– “Everybody must take a discrete mathematics course and be a CS
student.”  x ( P(x)  Q(x) )
Are these statements true or false?
Universal Quantifier: Example 2
• Express the statement: ‘for every x and every
y, x+y>10’
• Answer:
– Let P(x,y) be the statement x+y>10
– Where the universe of discourse for x, y is the set
of integers
– The statement is: x y P(x,y)
• Shorthand: x,y P(x,y)
Existential Quantifier: Definition
• Definition: The existential quantification of a
predicate P(x) is the proposition ‘There exists a value
x in the universe of discourse such that P(x) is true.’
We use the notation:  x P(x), which is read ‘there
exists x’.
• If the universe of discourse is finite, say {n1,n2,…,nk},
then the existential quantifier is simply the
disjunction of the propositions over all the elements
 x P(x)  P(n1)  P(n2)  …  P(nk)
Existential Quantifier: Example 1
• Let P(x,y) denote the statement ‘x+y=5’
• What does the expression x y P(x,y) mean?
• Which universe(s) of discourse make it true?
Existential Quantifier: Example 2
• Express the statement: ‘there exists a real solution to ax2+bxc=0’
• Answer:
– Let P(x) be the statement x= (-b(b2-4ac))/2a
– Where the universe of discourse for x is the set of real numbers. Note
here that a, b, c are fixed constants.
– The statement can be expressed as x P(x)
• What is the truth value of x P(x)?
– It is false. When b2<4ac, there are no real number x that can satisfy
the predicate
• What can we do so that x P(x)is true?
– Change the universe of discourse to the complex numbers, C
Quantifiers: Truth values
• In general, when are quantified statements
true or false?
Statement True when…
x P(x)
x P(x)
False when...
There is an x for which
P(x) is true for every x
P(x) is false
There is an x for which
P(x) is false for every x
P(x) is true
Mixing quantifiers (1)
• Existential and universal quantifiers can be used
together to quantify a propositional predicate. For
example:
x y P(x,y)
is perfectly valid
• Alert:
– The quantifiers must be read from left to right
– The order of the quantifiers is important
– x y P(x,y) is not equivalent to y x P(x,y)
Mixing quantifiers (2)
• Consider
– x y Loves (x,y): Everybody loves somebody
– y x Loves(x,y): There is someone loved by everyone
• The two expressions do not mean the same thing
• (y x Loves(x,y))  (x y Loves (x,y)) but the
converse does not hold
• However, you can commute similar quantifiers
– x y P(x,y) is equivalent to y x P(x,y) (thus, x,y P(x,y))
– x y P(x,y) is equivalent to y x P(x,y) (thus x,y P(x,y))
Mixing Quantifiers: Truth values
Statement
True when...
False when...
P(x,y) is true for every
xy P(x,y)
pair x,y
There is at least one pair
x,y for which P(x,y) is false
xy P(x,y)
For every x, there is a y
for which P(x,y) is true
There is an x for which
P(x,y) is false for every y
xy P(x,y)
There is an x for which
P(x,y) is true for every y
For every x, there is a y for
which P(x,y) is false
xyP(x,y)
There is at least one pair P(x,y) is false for every pair
x,y for which P(x,y) is true x,y
Mixing Quantifiers: Example (1)
• Express, in predicate logic, the statement that
there is an infinite number of integers
• Answer:
‒ Let P(x,y) be the statement that x<y
‒ Let the universe of discourse be the integers, Z
‒ The statement can be expressed by the following
x y P(x,y)
Mixing Quantifiers: Example (2)
• Express the commutative law of addition for R
• We want to express that for every pair of
reals, x,y, the following holds: x+y=y+x
• Answer:
‒Let P(x,y) be the statement that x+y
‒ Let the universe of discourse be the reals, R
‒ The statement can be expressed by the following
x y (P(x,y)  P(y,x))
Alternatively, x y (x+y = y+x)
Mixing Quantifiers: Example (3)
• Express the multiplicative law for nonzero reals R \
{0}
• We want to express that for every real number x,
there exists a real number y such that xy=1
• Answer:
x y (xy = 1)
Mixing Quantifiers: Example (4)
false mathematical statement
• Does commutativity for substraction hold over
the reals?
• That is: does x-y=y-x for all pairs x,y in R?
• Express using quantifiers
x y (x-y = y-x)
Mixing Quantifiers: Example (5)
• Express the statement as a logical expression: “There is a
number x such that when it is added to any number, the result
is that number and if it is multiplied by any number, the result
is x” as a logical expression
• Answer:
•
•
•
•
Let P(x,y) be the expression “x+y=y”
Let Q(x,y) be the expression “xy=x”
The universe of discourse is N,Z,R,Q (but not Z+)
Then the expression is:
x y P(x,y)  Q(x,y)
Alternatively: x y (x+y=y)  (xy = x)
Outline
• Introduction
• Terminology:
– Propositional functions; arguments; arity; universe of
discourse
• Quantifiers
– Definition; using, mixing, biding, negating them
• Logic Programming (Prolog)
• Transcribing English to Logic
• More exercises
Binding Variables
• When a quantifier is used on a variable x, we say that
x is bound
• If no quantifier is used on a variable in a predicate
statement, the variable is called free
• Examples
– In xyP(x,y), both x and y are bound
– In xP(x,y), x is bound but y is free
• A statement is called a well-formed formula, when all
variables are properly quantified
Binding Variables: Scope
• The set of all variables bound by a common
quantifier is called the scope of the quantifier
• For example, in the expression x,yzP(x,y,z,c)
– What is the scope of existential quantifier?
– What is the scope of universal quantifier?
– What are the bound variables?
– What are the free variables?
– Is the expression a well-formed formula?
Negation
• We can use negation with quantified expressions as
we used them with propositions
• Lemma: Let P(x) be a predicate. Then the followings
hold:
(x P(x))  x P(x)
 (x P(x))  x P(x)
• This is essentially the quantified version of De
Morgan’s Law (when the universe of discourse is
finite, this is exactly De Morgan’s Law)
Negation: Truth
Truth Values of Negated Quantifiers
Statement
True when…
False when...
P(x) is false for every x
There is an x for which
P(x) is true
x P(x)  There is an x for which
P(x) is false
x P(x)
P(x) is true for every x
x P(x) 
x P(x)
Outline
• Introduction
• Terminology:
– Propositional functions; arguments; arity; universe of
discourse
• Quantifiers
– Definition; using, mixing, negating them
• Logic Programming (Prolog)
• Transcribing English to Logic
• More exercises
Prolog (1)
• Prolog (Programming in Logic) is a
programming language based on (a restricted
form of) Predicate Logic (a.k.a. Predicate
Calculus and FOL)
• It was developed by the logicians of the
Artificial Intelligence community for symbolic
reasoning
Prolog (2)
• Prolog allows the users to express facts and rules
• Facts are propositional functions: student(mia),
enrolled(mia,cse235), instructor(patel,cse235), etc.
• Rules are implications with conjunctions:
teaches(X,Y) :- instructor(X,Z), enrolled(Y,Z)
• Prolog answers queries such as:
?enrolled(mia,cse235)
?enrolled(X,cse476)
?teaches(X,mia)
by binding variables and doing theorem proving (i.e., applying
inference rules) as we will see in Section 1.5
English into Logic
• Logic is more precise than English
• Transcribing English into Logic and vice versa can be tricky
• When writing statements with quantifiers, usually the correct
meaning is conveyed with the following combinations:
Use  with 
x Lion(x)  Fierce(x): Every lion is fierce
x Lion(x)  Fierce(x): Everyone is a lion and everyone is fierce
Use  with 
 x Lion(x)  Vegan(x): Holds when you have at least one vegan lion
 x Lion(x)  Vegan(x): Holds when you have vegan people in the
universe of discourse (even though there is no vegan lion in the
universe of discourse )
More exercises (1)
• Rewrite the following expression, pushing
negation inward:
 x ( y z P(x,y,z)   z y P(x,y,z))
• Answer:
x (y z P(x,y,z)  z y P(x,y,z))
More Exercises (2)
• Let P(x,y) denote ‘x is a factor of y’ where
– x  {1,2,3,…} and y  {2,3,4,…}
• Let Q(x,y) denote:
– x,y [P(x,y)  (x=y)  (x=1)]
• Question: When is Q(x,y) true?
Alert…
• Some students wonder if:
x,y P(x,y)  (x P(x,y))  (y P(x,y))
• This is certainly not true.
– In the left-hand side, both x,y are bound.
– In the right-hand side,
• In the first predicate, x is bound and y is free
• In the second predicate, y is bound and x is free
• Thus, the left-hand side is a proposition, but the right-hand side is not.
They cannot be equivalent
• All variables that occur in a propositional function must be
bound to turn it into a proposition
Rules of Inference
Proofs in mathematics are valid arguments
An argument is a sequence of statements that end in a conclusion
By valid we mean the conclusion must follow from the truth of the preceding
statements or premises
We use rules of inference to construct valid arguments
Valid Arguments in Propositional Logic
Is this a valid argument?
If you listen you will hear what I’m saying
You are listening
Therefore, you hear what I am saying
Let p represent the statement “you listen”
Let q represent the statement “you hear what I am saying”
pq
The argument has the form:
p
q
Valid Arguments in Propositional Logic
pq
(( p  q)  p)  q
p q
is a tautology (always true)
p
q
p  q ( p  q )  p (( p  q )  p )  q
0
0
1
0
1
0
1
1
0
1
1
0
0
0
1
1
1
1
1
1
This is another way of saying that
 therefore
Valid Arguments in Propositional Logic
When we replace statements/propositions with propositional variables
we have an argument form.
Defn:
An argument (in propositional logic) is a sequence of propositions.
All but the final proposition are called premises.
The last proposition is the conclusion
The argument is valid iff the truth of all premises implies the conclusion is true
An argument form is a sequence of compound propositions
Valid Arguments in Propositional Logic
The argument form with premises
and conclusion
is valid when
p1 , p2 ,, pn
q
( p1  p2    pn )  q
is a tautology
We prove that an argument form is valid by using the laws of inference
But we could use a truth table. Why not?
Rules of Inference for Propositional Logic
pq
p
q
modus ponens
aka
law of detachment
modus ponens (Latin) translates to “mode that affirms”
The 1st law
Rules of Inference for Propositional Logic
modus ponens
pq
p
q
If it’s a nice day we’ll go to the beach. Assume the hypothesis
“it’s a nice day” is true. Then by modus ponens it follows that
“we’ll go to the beach”.
Rules of Inference for Propositional Logic
modus ponens
pq
p
q
A valid argument can lead to an incorrect conclusion
if one of its premises is wrong/false!
2
3
3 2
2
2  3 ( 2 ) 2  3 2
3
2
2 
22 2 3 
(
2
)
22  ( 2 ) 22
 
3
2 3
2 2
2 2
3
2   
2
Rules of Inference for Propositional Logic
3
3
2   ( 2 )2   
2
2
3
2
2
9
2 
4
modus ponens
pq
2
p
q
A valid argument can lead to an incorrect
conclusion if one of its premises is wrong/false!
p:
3
2
2
3
q: 2 
2
pq
2
The argument is valid as it is constructed using modus ponens
But one of the premises is false (p is false)
So, we cannot derive the conclusion
The rules of inference
Rule of inference
pq
p
Tautology
Name
[ p  ( p  q )]  q
Modus ponens
[q  ( p  q )]  p
Modus tollen
q
q
pq
 p
pq
qr
[( p  q )  (q  r )]  ( p  r ) Hypothetic al syllogism
pr
pq
p
q
p
pq
pq
p
p
q
(( p  q )  p )  q
Disjunctiv e syllogism
p  ( p  q)
Addition
( p  q)  p
Simplifica tion
(( p )  (q ))  ( p  q )
Conjunctio n
[( p  q )  (p  r )]  ( p  r )
Resolution
pq
pq
p  r
q  r
Another view on what we are doing
You might think of this as some sort of game.
You are given some statement, and you want to see if it is a
valid argument and true
You translate the statement into argument form using propositional
variables, and make sure you have the premises right, and clear what
is the conclusion
You then want to get from premises/hypotheses (A) to the conclusion (B)
using the rules of inference.
So, get from A to B using as “moves” the rules of inference
Using the rules of inference to build arguments
It is not sunny this afternoon and it is colder than yesterday.
If we go swimming it is sunny.
If we do not go swimming then we will take a canoe trip.
If we take a canoe trip then we will be home by sunset.
We will be home by sunset
An example
Using the rules of inference to build arguments
1.
2.
3.
4.
5.
An example
It is not sunny this afternoon and it is colder than yesterday.
If we go swimming it is sunny.
If we do not go swimming then we will take a canoe trip.
If we take a canoe trip then we will be home by sunset.
We will be home by sunset
p It is sunny this afternoon
1.
p  q
q
It is colder tha n yesterday
2.
r p
r
We go swimming
s
We will take a canoe trip
3.  r  s
t
We will be home by sunset (the conclusion )
propositions
4.
st
5.
t
hypotheses
Using the rules of inference to build arguments
An example
1.
p  q
2.
r p
p It is sunny this afternoon
q
It is colder tha n yesterday
r
We go swimming
3.  r  s
s
We will take a canoe trip
4.
st
t
We will be home by sunset (the conclusion )
5.
t
Step
Step
Reason
Reason
Reason
1. 
pp  qq Hypothesis
Hypothesis
Hypothesis
Rule of inference
pq
p
2.  p
Simplifica
Simplifica tion
tion using
using (1)
(1)
3. r  p
Hypothesis
Hypothesis
4.  r
Modus
Modustollens
tollens using
using (2)
(2)and
and(3)
(3)
5. r  s Hypothesis
6. s
Modus ponens using (4) and (5)
7. s  t
Hypothesis
8. t
Modus ponens using (6) and (7)
Tautology
Name
[ p  ( p  q )]  q
Modus ponens
[q  ( p  q )]  p
Modus tollen
q
q
pq
 p
pq
qr
[( p  q )  (q  r )]  ( p  r ) Hypothetic al syllogism
pr
pq
p
q
p
pq
pq
p
p
q
(( p  q )  p )  q
Disjunctiv e syllogism
p  ( p  q)
Addition
( p  q)  p
Simplifica tion
(( p )  (q ))  ( p  q )
Conjunctio n
[( p  q )  (p  r )]  ( p  r )
Resolution
pq
pq
p  r
q  r
Using the resolution rule (an example)
1. Anna is skiing or it is not snowing.
2. It is snowing or Bart is playing hockey.
3. Consequently Anna is skiing or Bart is playing hockey.
We want to show that (3) follows from (1) and (2)
Using the resolution rule (an example)
1. Anna is skiing or it is not snowing.
2. It is snowing or Bart is playing hockey.
3. Consequently Anna is skiing or Bart is playing hockey.
hypotheses
propositions
1. p  r
p Anna is skiing
2. r  q
q
Bart is playing hockey
r
it is snowing
pq
p  r
Resolution rule
q  r
Consequently Anna is skiing or Bart is playing hockey
Rules of Inference & Quantified Statements
All men are £$%^$*(%, said Jane
John is a man
Therefore John is a £$%^$*(
Above is an example of a rule called “Universal Instantiation”.
We conclude P(c) is true, where c is a particular/named element
in the domain of discourse, given the premise
x P(x)
Rules of Inference & Quantified Statements
Rule of Inference
x P(x)
 P (c )
P(c) for an arbitrary c
 x P( x)
x P(x)
 P(c) for some element c
P(c) for some element c
 x P( x)
Name
Universal instantiat ion
Universal generalisa tion
Existentia l instantiat ion
Existentia l generalisa tion
Rules of Inference & Quantified Statements
All men are £$%^$*(%, said Jane
John is a man
Therefore John is a £$%^$*(
premises
M(x)
x is a man
B(x)
x is a £$%^$ * (
Step
Step
premises
x (M(x)  B(x))
Reason
Reason
1. 
xx (M(x)
(M(x) 
 B(x))
B(x)) Premise
Premise
2. M(John)  B(John)
Universal instantiat ion from (1.)
3. M(John)
Premise
4. B(John)
Modus ponens from (2.) and (3.)
Rule of Inference
x P(x)
 P (c )
P(c) for an arbitrary c
 x P( x)
x P(x)
 P(c) for some element c
P(c) for some element c
 x P( x)
Name
Universal instantiat ion
Universal generalisa tion
Existentia l instantiat ion
Existentia l generalisa tion
Rules of Inference & Quantified Statements
Maybe another example?