Do Now : What do you believe is meant by the term “logic”?

Dictionary.com defines the term “logic” as

the science that investigates the principles governing correct or reliable inference.

Aim : What exactly are logical proofs?

Logic

If Something (true or false)

Then Something else (true or false)

Q

Based on weather the first term, “P”, or the second term, “Q” is true depends on the outcome of the logical setup.

P

P -> Q

Aim : What exactly are logical proofs?

Conditional

Inverse

Converse

Contrapositive

Continue To“THE RULES”

Using a setup similar to the one in the earlier slide

Set up this phrase as terms such as “P” or “Q”

P -> Q

_______P_______

∴Q

If it rains, I will need a jacket

It started to rain

So I took my jacket

Aim : What exactly are logical proofs?

~P -> ~Q

If the temperature is not below 0 degrees,

then it is not cold

P -> Q

If the temperature is below 0 degrees,

then it is cold

Aim : What exactly are logical proofs?

Back toTHE BASICS

Q -> P

If it is Cold, then it is below 0 degrees

Aim : What exactly are logical proofs?

~Q -> ~ P

If it is not Cold,

then it is not below 0 degrees

Back toTHE BASICS

Law of Disjunctive InferenceLaw of Detachment

De Morgan's LawChain Rule

Law of Disjunctive Addition Law of Simplification

Aim : What exactly are logical proofs?Aim : What exactly are logical proofs?

CONTINUE LESSONBack to

THE BASICS

Aim : What exactly are logical proofs?

Back toTHE RULES

P -> Q

____P___

∴QIf it rains, I will need a jacket

It started to rain

So I need my jacket

P -> Q

_____Q_____

∴PIf it rains, I will need a jacket

I need my jacket

So it is raining

Aim : What exactly are logical proofs?

Back toTHE RULES

P \/ Q

_ ~Q______

∴ PIt either rained or snowed

It did not snow

So it rained

Aim : What exactly are logical proofs?

Back toTHE RULES

___P /\ Q___

∴ P

___P /\ Q___

∴ Q

I read and I write

So I read

I read and I write

So I write

Aim : What exactly are logical proofs?

Back toTHE RULES

___P___

∴ P \/ Q

I am sleeping

So I am either sleeping or doing homework

Aim : What exactly are logical proofs?

Back toTHE RULES

_~(~P /\ ~Q)_

∴ P \/ Q

_~(P /\ Q)_

∴ ~P \/ ~Q

_~(P \/ Q)_

∴ ~P /\ ~Q

_~(~P \/ ~Q)_

∴ P /\ Q

I like math and science

~(I like math and science)

I don’t like math or I don’t like science

I like math or science

~(I like math or science)

I don’t like math and I don’t like science

Aim : What exactly are logical proofs?

Back toTHE RULES

P -> Q

_ Q -> R______

∴ P -> R

I want things If they are green

If it is green, then it was made with green dye

So I want things made from green dye

1. Assume the statement "Michael is takingsociology and biology" is true. Which of the

following statements must be true?[A] Michael is taking both sociology and

biology.[B] Michael is taking neither sociology, nor

biology.[C] Michael is taking only sociology.[D] Michael is taking only biology.

2. Assume the statement "Cheryl is takingfencing or sculpture" is true. Which of the

following statements must be false?[A] Cheryl is taking only sculpture.

[B] Cheryl is taking neither fencing, norsculpture.

[C] Cheryl is taking both fencing andsculpture.

[D] Cheryl is taking only fencing.

3. Assume the statement "James is taking dramaand physics" is true. Which of the following

statements must be true?[A] James is taking both drama and physics.

[B] James is taking only drama.[C] James is taking neither drama, nor

physics.[D] James is taking only physics.

4. Assume the statement "James is taking historyor tennis" is true. Which of the following

statements must be false?[A] James is taking only tennis.

[B] James is taking neither history, nortennis.

[C] James is taking only history.[D] James is taking both history and tennis.

5. Assume the statement "James is taking dramaand physics" is true. Which of the following

statements must be true?[A] James is taking both drama and physics.

[B] James is taking only physics.[C] James is taking only drama.

[D] James is taking neither drama, norphysics.

6. Assume the statement "Pamela is takingmusic or physics" is true. Which of thefollowing statements must be false?

[A] Pamela is taking only music.[B] Pamela is taking only physics.

[C] Pamela is taking neither music, norphysics.

[D] Pamela is taking both music and physics.

p q ~p ~q p/\q p\/q p->q

~p\/q

p<->q p\/~p

T T F F T T T T T

F F F T T F F F T

F T T F T F T F T

F F T T F F T T T

HW: Fill Out the Below Table Based on the given information

Aim : What exactly are logical proofs?