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3.1 Statements Negations, and Quantified Statements. Sentences can be factual statements, opinions, commands or questions. Symbolic logic only works with factual statements. A statement is a declarative sentence that is either true or false, but not both simultaneously.

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slide2

Sentences can be factual statements, opinions, commands or questions. Symbolic logic only works with factual statements.

A statement is a declarative sentence that is either true or false, but not both simultaneously.

Shakespeare wrote sonnets, and the poem exhibits iambic pentameter.

You can pay me now, or you can pay me later.

If he said it, then it must be true.

My pistol was made by Smith and Wesson.

slide3

Commands, questions and opinions are not statements because they are not either true or false.

Stand on your head.

Will you do it now?

He is an idiot.

Symbolic logic uses lower case letters to represent statements.

p: Pigs can fly.

q: kumquats are orange.

slide4

Lewis Carroll’s corral problem

Arrange 16 cows in the four corrals so that if you enter the lower right corral and move counter-clockwise, the number of cows in a corral will always get closer to ten.

slide5

Negations

The sun is a not a star.

The sun is a star.

Today is not Monday.

Today is Monday.

7x − 13y ≥ 27

7x − 13y < 27

Politicians lie.

Politicians do not lie.

The state has a governor.

The state does not have a governor.

A < B

A ≥ B

A ≤ B

A > B

P < 17

P≥ 17

slide6

Symbolically the negation is represented by~

Let p represent “It is raining.” and let q represent “The grass is not wet.”

~ p

It is not raining.

It is not true that the grass is not wet. (The grass is wet.)

~q

slide8

Quantifiers

all , each, every, no and none are universal quantifiers.

All A are B is a universal statement.

There are no A that are not B is an equivalent universal statement.

B

A

“All dogs are mammals.” is a universal statement.

“There are no dogs that are not mammals.” is an equivalent universal statement.

mammals

dogs

slide9

Quantifiers

all , each, every, no and none are universal quantifiers.

No A are B is a universal statement.

All A are not B is an equivalent universal statement.

B

A

“No dog is a reptile.” is a universal statement.

“All dogs are not reptiles.” is an equivalent universal statement.

reptiles

dogs

slide10

Quantifiers

some, there exists, and at least one are existential quantifiers.

Some A are B is an existential statement.

There exists at least one A that is a B is an equivalent existential statement.

B

A

“Some marbles are red.” is a existential statement.

“There exists at least one marble that is red.” is an equivalent existential statement.

red things

marbles

slide11

Negations of Quantified Statements

all , each, every, no and none are universal quantifiers.

some, there exists, and at least one are existential quantifiers.

The negation of a universal is an existential.

The negation of an existential is a universal.

Some A are not B

All A are B

All A are not B. (No A is B)

Some A are B

slide12

Quantifiers

all , each, every, no and none are universal quantifiers.

some, there exists, and at least one are existential quantifiers.

All girls in the group are named Mary

Which group would make the statement true?

Group 1 Mary Smith, Mary Jones, Mary Peterson

Group 2 Mary Worth, Sally Forth, Ginger Rogers

Group 3 Belle Star, Lisa Smith, Harriet Nelson

slide13

Quantifiers

all , each, every, no and none are universal quantifiers.

some, there exists, and at least one are existential quantifiers.

What is the negation of “All girls in the group are named Mary.”?

At least one girl in the group is not named Mary.

Which group or groups would make the statement true?

Group 1 Mary Smith, Mary Jones, Mary Peterson

Group 2 Mary Worth, Sally Forth, Ginger Rogers

Group 3 Belle Star, Lisa Smith, Harriet Nelson

slide14

Quantifiers

all , each, every, no and none are universal quantifiers.

some, there exists, and at least one are existential quantifiers.

There is a dog in the group named Rover.

Which group would make the statement true?

Group 1 Lassie, Fido, Bullet

Group 2 Spot, Milo, Rover

slide15

Quantifiers

all , each, every, no and none are universal quantifiers.

some, there exists, and at least one are existential quantifiers.

What is the negation of “There is a dog in the group named Rover.”?

None of the dogs in the group is named Rover.

Which group would make the statement true?

Group 1 Lassie, Fido, Bullet

Group 2 Spot, Milo, Rover