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Page 1: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Propositional Logic

Page 2: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Conditional Statement

If p then q

p is called the hypothesis; q is called the conclusion

“If your GPA is 4.0, then you don’t need to pay tuition fee.”The department says:

When is the above sentence false?

• It is false when your GPA is 4.0 but you still have to pay tuition fee.

• But it is not false if your GPA is below 4.0.

Another example: “If there is a bandh today, then there is no class.”

When is the above sentence false?

Page 3: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

IMPLIES::

Logic Operator

T

T

F

T

P Q

FF

TF

FT

TT

QP

Convention: if we don’t say anything wrong, then it is not false, and thus true.

Page 4: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Logical Equivalence

If you see a question in the above form,

there are usually 3 ways to deal with it.

(1)Truth table

(2)Use logical rules

(3)Intuition

Page 5: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

If-Then as Or

T

T

F

T

P Q

FF

TF

FT

TT

QPIdea 2: Look at the false rows,

negate and take the “and”.

• If you don’t give me all your money, then I will kill you.

• Either you give me all your money or I will kill you (or both).

• If you talk to her, then you can never talk to me.

• Either you don’t talk to her or you can never talk to me (or both).

Page 6: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Negation of If-Then

• If your GPA is 4.0, then you don’t need to pay tuition fee.

• Your term GPA is 4.0 and you still need to pay tuition fee.

• If my computer is not working, then I cannot finish my homework.

• My computer is not working but I can finish my homework.

previous slide

DeMorgan

Page 7: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Contrapositive

The contrapositive of “if p then q” is “if ~q then ~p”.

Statement: If x2 is an even number,

then x is an even number.

Statement: If you are a CS year 1 student,

then you are taking CTS002.

Contrapositive: If x is an odd number,

then x2 is an odd number.

Contrapositive: If you are not taking CTS002,

then you are not a CS year 1 student.

Fact: A conditional statement is logically equivalent to its contrapositive.

Page 8: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Proofs

Statement: If P, then Q

Contrapositive: If Q, then P.

F F T

T F F

F T T

T T T

T T T

T F F

F T T

F F T

Page 9: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

If, Only-If

• You will succeed if you work hand.

• You will succeed only if you work hard.

R if S means “if S then R” or equivalently “S implies R”

We also say S is a sufficient condition for R.

You will succeed if and only if you work hard.

P if and only if (iff) Q means P and Q are logically equivalent.

R only if S means “if R then S” or equivalently “R implies S”

We also say S is a necessary condition for R.

That is, P implies Q and Q implies P.

Page 10: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Math vs English

Parent: if you don’t clean your room, then you can’t watch a DVD.

C D

This sentence says

In real life it also meansSo

Mathematician: if a number x greater than 2 is not an odd number,

then x is not a prime number.

This sentence says

But of course it doesn’t mean

Page 11: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Necessary, Sufficient Condition

Mathematician: if a number x greater than 2 is not an odd number,

then x is not a prime number.

This sentence says

But of course it doesn’t mean

Being an odd number > 2 is a necessary condition for this number to be prime.

Being a prime number > 2 is a sufficient condition for this number to be odd.

Page 12: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Necessary AND Sufficient Condition

IFF::

T

F

F

T

P Q

FF

TF

FT

TT

QP

Note: P Q is equivalent to (P Q) (Q P)

Note: P Q is equivalent to (P Q) ( P Q)

Is the statement “x is an even number if and only if x2 is an even number” true?

Page 13: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Argument

An argument is a sequence of statements.

All statements but the final one are called assumptions or hypothesis.

The final statement is called the conclusion.

An argument is valid if:

whenever all the assumptions are true, then the conclusion is true.

If today is Wednesday, then yesterday was Tuesday.

Today is Wednesday.

Yesterday was Tuesday.

Page 14: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Modus Ponens

If p then q.pq

p q p→q p q

T T T T T

T F F T F

F T T F T

F F T F F

Modus ponens is Latin for “method of affirming”.

assumptions conclusion

If bandh, then class cancelled.Bandh.Class cancelled.

Page 15: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Modus Tollens

If p then q.~q~p

p q p→q ~q ~p

T T T F F

T F F T F

F T T F T

F F T T T

Modus tollens is Latin for “method of denying”.

assumptions conclusion

If Bandh, then class cancelled.Class not cancelled.No Bandh.

Page 16: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

( ), ( ), ( )P Q Q R R P

P Q R

Equivalence

A student is trying to prove that propositions P, Q, and R are all true.

She proceeds as follows.

First, she proves three facts:

• P implies Q

• Q implies R

• R implies P.

Then she concludes,

``Thus P, Q, and R are all true.''

Proposed argument:

Is it valid?

assumption

conclusion

Page 17: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Valid Argument?

assumptions conclusion

P Q R

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

T T T

T F T

F T T

F T T

T T F

T F T

T T F

T T T

OK?

T yes

F yes

F yes

F yes

F yes

F yes

F yes

F no

To prove an argument is not valid, we just need to find a counterexample.

( ), ( ), ( )P Q Q R R P

P Q R

Is it valid?

Page 18: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Valid Arguments?

If p then q.qp

If you are a fish, then you drink water.You drink water.You are a fish.

p q p→q q p

T T T T T

T F F F T

F T T T F

F F T F F

assumptions conclusion

Assumptions are true, but not the conclusion.

Page 19: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Valid Arguments?

If p then q.~p~q

If you are a fish, then you drink water.You are not a fish.You do not drink water.

p q p→q ~p ~q

T T T F F

T F F F T

F T T T F

F F T T T

assumptions conclusion

Page 20: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Exercises

Page 21: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

More Exercises

Valid argument True conclusion

True conclusion Valid argument

Page 22: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Contradiction

If you can show that the assumption that the statement

p is false leads logically to a contradiction,

then you can conclude that p is true.

You are wearing a jacket.

If it was warm, then you would not have worn a jacket.

It is not warm.

Page 23: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Knights and Knaves

Knights always tell the truth.

Knaves always lie.

A says: B is a knight.

B says: A and I are of opposite type.

Suppose A is a knight.

Then B is a knight (because what A says is true).

Then A is a knave (because what B says is true)

A contradiction.

So A must be a knave.

So B must be a knave (because what A says is false).

Page 24: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

Quick Summary

Conditional Statements• The meaning of IF and its logical forms

• Contrapositive

• If, only if, if and only if

Arguments• definition of a valid argument

• method of affirming, denying, contradiction

Key points:

(1)Make sure you understand conditional statements and contrapositive.

(2)Make sure you can check whether an argument is valid.

Page 25: Propositional Logic. Conditional Statement If p then q p is called the hypothesis; q is called the conclusion “If your GPA is 4.0, then you don’t need

“The sentence below is false.”

“The sentence above is true.”

Which is true?

Which is false?


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