Transcript DEE: If you think we're wax-works, you ought to pay, you know.
DEE: If you think we're wax-works, you ought to pay, you know. Wax-works weren't made to be looked at for nothing. Nohow.
DUM: Contrariwise, if you think we're alive, you ought to speak. DUM: I know what you're thinking about, but it isn't so, nohow. DEE: Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.
MA 110: Finite Math
Dr. Maria Byrne Instructional Laboratory 0345 Lecture 1/21/2009 Homework: 1.2: 1-21 odd (Skip 7c, 9c); (
36 for a BP
) Due Monday 1/26
Today We Will 1. Collect Homework 1.1
2. Begin 1.2 Symbolic Logic We will learn about : • • • Statements Negations Compound Statements
1.2 Symbolic Logic
Symbolic logic
involves the use of symbols to represent the quantities and relationships of statements.
These quantities and relationships are
abstracted away
from their original content.
A Statement • A
statement
is a sentence that can be objectively determined TRUE or FALSE.
• A statement is denoted by a lowercase letter, usually
p
,
r
or
s
. • A statement can be identified by whether or not it can be assigned a truth value.
A Statement • A
statement
is a sentence that can be objectively determined TRUE or FALSE.
• A statement is denoted by a lowercase letter, usually
p
,
q
or
r
. • A statement can be identified by whether or not it can be assigned a truth value.
A Statement • A
statement
is a sentence that can be objectively determined TRUE or FALSE.
• A statement is denoted by a lowercase letter, usually
p
,
r
or
s
. • A statement can be identified by whether or not it can be assigned a truth value.
Identifying Statements
Abraham Lincoln was the best president.
Abraham Lincoln was the best president.
Opinion Not a statement.
Abraham Lincoln was the best president.
Opinion Not a statement.
2+2=4
2+2=4 Is a statement.
2+2=5
2+2=5 Is a statement.
It is a lousy day.
It is a lousy day.
Opinion.
Not a statement.
There are exactly 100,400,327 stars in the universe.
There are exactly 100,400,327 stars in the universe.
Is a statement.
How many stars are in the universe?
How many stars are in the universe?
Question.
Not a statement.
The sun will rise tomorrow.
The sun will rise tomorrow.
Is a statement.
Grab that fish!
Grab that fish!
Instructions Not a statement.
Ouch!
Ouch!
Not a statement.
Classify the sentences below: "Socrates is a man.”…………………………….
Statement "A triangle has three sides.”…………………… "Paris is the capital of England.”……………….
"Who are you?”……………………………… Statement Statement Question "Run!" ………………………………………… "I had one grunch but the eggplant over there.”..
Instuction Nonsense "Red is a pretty color.”………………………… “This is not a statement.”……………………… Opinion Paradox
Classify the sentences below: "Socrates is a man.”…………………………….
Statement "A triangle has three sides.”…………………… "Paris is the capital of England.”……………….
"Who are you?”……………………………… Statement Statement Question "Run!" ………………………………………… "I had one grunch but the eggplant over there.”..
Instruction Nonsense "Red is a pretty color.”………………………… “This is not a statement.”……………………… Opinion Paradox
Classify the sentences below: "Socrates is a man.”…………………………….
Statement "A triangle has three sides.”…………………… "Paris is the capital of England.”……………….
"Who are you?”……………………………… Statement Statement Question "Run!" ………………………………………… "I had one grunch but the eggplant over there.”..
Instruction Nonsense "Red is a pretty color.”………………………… “This is not a statement.”……………………… Opinion Paradox
Negations (Not
not negations
)
Negation • The
negation
of a statement
p
is the statement
q
that is a
denial
of the statement
p
. • A denial has the opposite truth value of
p
.
• The symbol for the negation is:
~
Negation • The
negation
of a statement
p
is the statement
q
that is a
denial
of the statement
p
. • A denial has the opposite truth value of
p
.
• The symbol for the negation is:
~
Negation • The
negation
of a statement
p
is the statement
q
that is a
denial
of the statement
p
. • A denial has the opposite truth value of
p
.
• The symbol for the negation is:
~
Example Statement: Her dress is red.
Negation: Her dress is not red.
Example Statement: Her dress is red.
Negation: Her dress is not red.
Example Statement: Today is January 21st. Negation: Today is not January 21st.
Example Statement: Today is January 21st. Negation: Today is not January 21st.
Example Statement: Today is January 21st. Negation: Today is not January 21st. Statement: True (T) Negation: False (F)
Example Statement: Lizards are mammals. Negation: Lizards are not mammals
Example Statement: Lizards are mammals. Negation: Lizards are not mammals
Example Statement: Lizards are mammals. Negation: Lizards are not mammals Statement: False (F) Negation: True (T)
Example Statement: We are not mice. Negation: We are not not mice.
Example Statement: We are not mice. Negation: We are not not mice.
Example Statement: We are not mice. Negation: We are not not mice.
Example Statement: We are not mice. Negation: We are mice.
Example Statement: We are not mice. Negation: We are mice. Statement: True (T) Negation: False (F)
The Negation of a Negation • “Not not p” is “p” • In symbols: to class
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– Negation: It is not the case that her dress is red.
• We are mice.
– Negation: It is not the case that we are mice.
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– Negation: It is not the case that her dress is red.
• We are mice.
– Negation: It is not the case that we are mice.
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– Negation: It is not the case that her dress is red.
• We are mice.
– Negation: It is not the case that we are mice.
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– Negation: It is not the case that her dress is red.
• We are mice.
– Negation: It is not the case that we are mice.
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– Negation: It is not the case that her dress is red.
• We are mice.
– Negation: It is not the case that we are mice.
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– Negation: It is not the case that her dress is red.
• We are mice.
– Negation: It is not the case that we are mice.
“It is not the case that” • Always negates a sentence.
• Her dress is red.
– It is not the case that her dress is red.
• We are mice.
– It is not the case that we are mice.
Equivalent Ways to Negate • Negations of “We are not mice.” • 1. We are mice.
• 2. It is not the case that we are not mice.
Equivalent Ways to Negate • Negations of “We are not mice.” • 1. We are mice.
• 2. It is not the case that [we are not mice].
Equivalent Ways to Negate • Negations of “We are not mice.” • 1. We are mice.
• 2. It is not the case that [we are not mice].
• 3. To class.
Equivalent Statements • If two statements have the same meaning, we say that they are equivalent.
• If
p
is equivalent to
q
, we write
p
≡
q
Tricky Negation Statement: Some cats have tails.
Tricky Negation Statement: Some cats have tails.
Negation: It is not the case that some cats have tails.
Tricky Negation Statement: Some cats have tails.
Negation: It is not the case that some cats have tails.
What does this mean?
Tricky Negation Statement: Some cats have tails.
Negation: It is not the case that some cats have tails.
What does this mean?
No cat has a tail!
Negate the Following Sentences • Some of the beverages contain caffeine. • Some of the fruits are not red. • All candy promotes tooth decay.
Compound Statements
Compound Statements
Compound statements
are statements connected by AND, OR or “IF … THEN”
Compound Statements
Compound statements
are statements connected by AND, OR or “IF … THEN” p AND q p OR q IF p, THEN q.
Conjunction • AND • The conjunction of p and q is “p AND q”.
• Symbolized by: • Example: p: I’m a dog. q: You’re a cat.
p q: I’m a dog and you’re a cat.
Disjunction • OR • The disjunction of p and q is “p OR q”.
• Symbolized by: • Example: p: I’m a dog. q: You’re a cat.
p q: I’m a dog or you’re a cat.
Conditional • IF…THEN • The condition of p and q is “IF p THEN q”.
• Symbolized by: • Example: p: I’m a dog. q: You’re a cat.
p q: If I’m a dog then you’re a cat.
AND OR