SYMBOLIC LOGIC
Section 1.2
Statements
• This section we will study symbolic logic
which was developed in the late 17th
century.
• All logical reasoning is based on
statements.
• A statement is a sentence that is either
true or false.
Which of the following are
statements?
• The 2004 Summer Olympic Games were in
Athens, Greece.(Statement - true)
• Seinfeld was the best TV comedy of all time.
(Not a Statement – opinion)
• Did you watch The Godfather? (Not a statement
– a question)
• The Philadelphia Eagles won Super XV.
(statement – false)
• I am telling a lie. (not a statement – paradox)
Statements
• Traditionally, symbolic logic uses lower
case letters to denote statements. Usually
the letters p, q, r, s, t.
• Statements get labels.
– p: It is raining.
Compound Statements
• A compound statement is a statement
that contains one or more simpler
statements.
• Compound statements can be formed by
– inserting the word NOT,
– joining two or more statements with
connective words such as AND, OR,
IF…THEN, ONLY IF, IF AND ONLY IF.
Examples
• Steve did not do his homework.
– This is formed from the simpler statement, Steve did
his homework.
• Mr. D wrote the MAT114 notes and listened to a
Pink Floyd CD.
– This statement is formed from the simper statements:
Mr. D wrote the MAT114 notes.
Mr. D listened to a Pink Floyd CD.
• Compound statements are known as negations,
conjunctions, disjunctions, conditionals or
combinations of each.
NEGATION ~p
• The negation of a statement is the denial
of that statement. The symbolic
representation is a tilde ~.
• Negation of a simple statement is formed
by inserting not.
• Example: The senator is a Republican.
The negation is: The senator is not a
Republican.
Negation
• “All of Mr. D’s students are Philadelphia
Eagles fans.”
• The negation is: “Some of Mr. D’s students
are not Philadelphia Eagles fans.”
• To negate the first statement, we don’t
need to have all the students to be not
Eagles fans, we just need only one
student not to be an Eagles fan. Hence the
usage of some.
Negation
• “No students are math majors.”
• To deny this statement, we need at least
one instance in which a student does
major in math.
• “Some students are math majors.”
Negation
• To summarize negation:
• All p are q is negated by Some p are not q
• No p are q is negated by Some p are q
CONJUNCTION p ^ q
• A conjunction is a compound statement that
consists of 2 or more statements connected by
the word and.
• And is represented by the symbol ^.
• p ^ q represents “p and q”.
• Example:
p: Jerry Seinfeld is a comedian.
q: Jerry Seinfeld is a millionaire.
Express the following in symbolic form:
i. Jerry Seinfeld is a comedian and he is a millionaire.
ii. Jerry Seinfeld is a comedian and he is not a
millionaire.
Conjunction
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Using the symbolic representations
p: The lyrics are controversial.
q: The performance is banned.
Express the following in symbolic form:
a. “The lyrics are controversial and the performance is
banned.”
b. “The lyrics are not controversial and the performance is
not banned.”
Answers:
a. p ^ q
b. ~p ^ ~q
DISJUNCTION p v q
• When you connect statements with the word or
you form a disjunction.
• Or is represented by the symbol v.
• p v q is read as “p or q”.
• Using the p and q from the last slide, write out in
words p v q, and p v ~q.
• p v q is “the lyrics are controversial or the
performance is banned.”
• p v ~q is “the lyrics are controversial or the
performance is not banned.”
CONDITIONAL p  q
• A conditional is of the form “if p then q”. This is
also known as an implication. p is the hypothesis
(or premise), and q is the. conclusion.
• The representation of “if p then q” is p q.
• Again use the p and q from the previous 2
slides.
• “If the lyrics are not controversial, the
performance is not banned.”
• ~p  ~q
Sec. 1.2 #28
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Using the following symbolic representations
p: I am innocent.
q: I have an alibi.
express the following in words.
A. p ^ q
Answer: “I am innocent and I have an alibi.”
B. p  q
Answer: “If I am innocent, then I have an alibi.”
C. ~q  ~p
Answer: “If I do not have an alibi, then I am not
innocent.”
D. q v ~p
Answer: “I have an alibi or I am not innocent.”
Sec. 1.2 #30
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Using the symbolic representations
p: I am innocent.
q: I have an alibi.
r: I go to jail.
Express the following in words.
A. (p v q)  ~r
B. (p ^ ~q)  r
C. (~p ^ q) v r
D. (p ^ r)  ~q
Sec. 1.2 #30
A. If I am innocent or have an alibi, then I
do not go to jail.
B. If I am innocent and do not have an alibi,
then I go to jail.
C. I am not innocent and I have an alibi or I
go to jail.
D. If I am innocent and go to jail, then I do
not have an alibi.
Sec. 1.2 #23
• Translate the sentence to symbolic form.
• If you drink and drive, you are fined or you
go to jail.
• p: You drink.
• q: You drive.
• r: You are fined.
• s: You are jailed.
• Answer: (p ^ q)  (r v s).
Sec. 1.2 #14
• Translate into symbolic form.
• “No whole number is greater than 3 and
less than 4.”
• p: A whole number.
• q: A number greater than 3.
• r: A number less than 4.
• Answer: ~p  (q ^ r)