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Deadline for Week III: Friday January 18. Email me your work by midnight.

My email: kamillasmith27@gmail.com

Week III

Day 9.

Exercise 1.

Write which of these arguments are valid and which are invalid.

1. A B Premise 1.

2. A Premise 2.

_____________________

3. B Conclusion

1. A B Premise 1.

2. -B Premise 2.

_____________________

3. -A Conclusion

1. A B Premise 1.

2. -A Premise 2.

_____________________

3. -B Conclusion

1. A B Premise 1.

2. B Premise 2.

_____________________

3. A Conclusion

Now, write again the two valid and two invalid argument forms using, this time, letters of your choice.

Do not forget to state which are valid and which are invalid forms of reasoning.

II.

III.

IV.

The two valid forms of reasoning have names.

The first one is called Modus Ponens. The Second is called Modus Tollens.

For invalid reasoning we will simply say “invalid.”

Modus Ponens we will abbreviate as MP

Modus Tollens we will abbreviate as MT

From now on, whenever you solve an argument of this form:

1. A B

2. A

__________

3. B 1, 2 MP

you will write to the right of your conclusion 1,2 MP to let the reader know that you derived the conclusion from the Modus Ponens rule of inference. The number 1 and 2 indicate the lines from which you derived line 3, which is here the conclusion line.

Likewise, whenever you solve an argument of the Modus Tollens form, you will add: 1, 2 MT to signify that you derived the conclusion from the premises by means of Modus Tollens, as in:

1. A B

2. - B

________

3. -A 1, 2 MT

Is the conclusion in the argument below derived by MP or MT?

1. If roses are red, then violets are blue.

2. Roses are red.

3. Therefore, Violets are blue.

Your answer:

If we were to put in symbolic notation the argument about roses, it would look like this:

1. R B

2. R

______________

3. B

Exercise 2. Convert the arguments into symbols, then say whether the arguments are valid or invalid.

1. If Ptolemaic astronomy is true, then the sun orbits the Earth.

2. But the sun does not orbit the Earth (everybody knows, the Earth orbits the sun)

3. Therefore, Ptolemaic astronomy is not true.

1. If this is a rattlesnake, then this is a poisonous reptile.

2. This is a poisonous reptile.

3. Therefore, this is a rattlesnake.

Exercise 3. Identify the two arguments below as valid or invalid.

1. A D

2. D

_____________

3. A

1. M N

2. – M

_____________

3. – N

____________________________________________________________

Consider now this simple (true) premise:

1. If you are the Pope, then you are Catholic.

As you remember from the earlier notes, a conditional sentence consists of the if –part which we call antecedent, and the then-part , called the consequent.

antecedent consequent

Now rewrite the first premise and then in the second premise deny the consequent. (To deny means to negate a proposition, write a negation). What can you conclude? First, write the argument in full sentences.

1. If you are the Pope, then you are Catholic.

Now write the same argument in symbols.

1. P C

_______

______________________________________________

Examine again how the conclusion follows the premises. This time the premises are a bit more complicated but the rules are the same.

1. A ( B v C) (If A is true, then B or C is true.)

2. A (It turns out A is indeed true.)

__________

3. B v C 1, 2 MP (Therefore, we can conclude that B or C is true. We concluded

this by using the Modus Ponens rule of inference.)

1. ( A v D ) B

2. - B

________

3. – (A v D) 1, 2 MT

Exercise 4.

Derive valid conclusions from the premises.

Do not forget to write MP or MT to the right! Do not forget to write the number lines!):

1. A Q

2. A

_____

1. A ( Q v B)

2. – (Q v B)

_____

1. A M

2. - M

_____

1. B C

2. B

_____

Day 10.

You already know that premises are pieces of given information. For example:

1. I have $20.

2. If I have $20, I can buy oranges.

3. If I buy oranges, I can make orange juice.

What can I conclude from these 3 premises?

Your answer: ______________________________________

Now let’s put this in logical terms:

1. H (This will stand for “I have $20.)

2. H -> O

3. O - > J

___________

What can I conclude here? What will be my first step?

I have H as given, and I am told that If H then O , So I should be able to conclude O , meaning that I can buy oranges. I will conclude that from line 1 and 2 and the method is called Modus Ponens. So I will write it like this:

4. O 1, 2 MP

What else can I conclude from what is given? In line 3, the sentences says that If O then J, meaning if I buy oranges I can make orange juice. In line 4 I already established that I can buy orange juice. So I can clearly conclude that I can now make orange juice. I will write this in this way:

5. J 3, 4 MP

But there is an alternative method of inference called Hypothetical Syllogism (HS).

I could derive the same conclusion from the 3 premises by using the HS method, like this:

1. H

2. H -> O

3. O - > J

___________

4. H - > J 2, 3 HS

5. J 1, 4, MP

Let’s consider one more scenario.

1. She can do math.

2. If she can do math, she can major in business.

3. If she majors in business, she might get a job on Wall Street.

___________

1. M

2. M -> B

3. B -> W

___________

4. B 1, 2 MP (I concluded she can major in business, since she can do math.)

5. W 3, 4 MP (I concluded she might get a job on Wall Street since she can major in math.)

Or :

1. M

2. M -> B

3. B -> W

______

4. M –> W 2, 3 HS

5. W 1, 4.

Hypothetical Syllogism (HS), has this form:

1. A B

2. B C

____________

3. A C

This formula fits the following argument:

“If Alex graduates, he will get a job. If he gets a job, he will have money. So, if he graduates, he will have money.”

We are not told in this argument that Alex is definitely going to graduate, but if he is, then he will also get a job, and that will lead to him getting money. Ultimately:

If A, then C.

Exercise 1

Derive conclusions by means of Hypothetical Syllogism (HS):

1. A→ B

2. B→C

1. H→ - O

2. - O→P

1. - A→ B

2. B→ -C

d) * challenging one:

1. - P→ A

2. Q→ - P

It is rather easy to see why the two premises lead to the conclusion in:

1. A B

2. BC

3. A C

But the conclusion may be more difficult to see when the premises are not in a convenient order as in:

1. B C

2. AB

3. A C

In this example, you need to think of the second premise as the first one, If A, then B, then you should think of the first premise as the second one, If B, then C.

So, if A then B, and if B then C, therefore, If A then C.

In the example below, you should reverse (in your head) the second and first premise to see that what is being said is: “If Q, then – P” and “If –P, then A.”

1. - P→ A

2. Q→ - P

So:

1. Q → – P

2. – P → A

So far we learned:

1.Modus Ponens

2.Modus Tollens

3. Hypothetical Syllogism

Modus Ponens we abbreviated as MP

Modus Tollens we abbreviated as MT

Hypothetical Syllogism as HS

Now let’s look at arguments containing disjunctions.

Arguments with Disjunctions

Remember, an argument is not just one sentence. It is a sequence of interconnected statements.

Let’s say our first statement is this:

I have a coin in either my left hand or my right hand.

In symbols: L v R

Now, you just checked my left hand and it turns out that there is no coin in my left hand. But, given that I did not lie, and that it is 100% true that I have a coin is either my left hand or my right hand, then we can conclude that the coin must be for sure in my right hand.

The argument looks like this:

1. L v R

2. -L

_________

3.R

Likewise, if the coin is not in my right hand then it must be in my left hand.

1. L v R

2. -R

_________

3. L

These two argument forms are valid.

Are the following arguments valid?

1. A v B

2. - B

_____

3. A

II.

1. A v (B C)

2. – (B C)

_______

3. A

However, this is not valid:

1. T v C

2. T

_______

3. C

Why? Because nowhere does it say that the two options are mutually exclusive. In other words we must assume in deductive logic that the disjunction is not mutually exclusive, unless specified otherwise. One of the choices must be true for the sentence to be true, but it may well be that both choices are the case.

Consider again the argument form:

1. L v R

2. – L

_________

3. R

This argument form also has a name. It is called Disjunctive Syllogism. We will abbreviate this logical inference as DS and use it in this way:

1. L v R

2. – L

_________

3. R 1, 2 DS

And likewise:

1. L v R

2. – R

_________

3. L 1, 2 DS

Exercise 2.

Write the letter that you can conclude from the two premises, and specify what method you used, just like in the two examples above.

1. A v B

2. – A

________

1. (A & B) v C

2. – (A & B)

________

1. M B

2. M

____________

1. A v B

2. – A

__________

1. A v (P & Q )

2. – A

_________________

1. A ( B v C)

2. A

_________________

3. B v C

1. (A M) v N

2 – N

____________

1. M v N

2. – N

__________

1. (A v B ) v (P & Q )

2. – (A v B)

_________________

1. (P & Q) ( B v C)

2. P & Q

_________________

Day 11

Keep in mind, an argument can have more than two premises, as in:

1. If today is Monday, then tomorrow is Tuesday.

2. If tomorrow is Tuesday, then tomorrow is the deadline.

3. Today is indeed Monday

4. Therefore: Tomorrow is the deadline.

In symbolic terms:

1. M T

2. T D

3. M

4. D

Why do we conclude D here? Because first we are given the condition that if M is true then T is true, and we also are told in premise 3 that M is indeed true, so we can ascertain that T is true. Once we know that T is true we can conclude that D is true because we are told that if T is true then D is true.

Another example:

1. A (B C)

2. A

3. B

4. C

We obtained C, because in premise 1 we are told that If A it true then the statement in the parenthesis is true. Then, premise 2 asserts that A is indeed true, in which case the sentence in the parenthesis (B C) must also be true. In line 3, the premise states that B is also true. Now, since we now have the condition that if B is true then C is true, and also we are told that B is indeed true, we can logically conclude that C is true.

Exercise 1.

Look at the following arguments and derive a valid conclusion from the premises (just like in the two examples above).

1. C ( Q R)

2. C

3. Q

1. B ( P v Q)

2. B

1. A ( B v Q)

2. A

3. – B

Now, we will derive the conclusion in a much more systematic way. This means we will write down the names of all logical operations and the number lines from which we derived a given statement. First, examine carefully the examples below.

1. P (L v R)

2. P

3. – L

________

We know that the conclusion here based on the 3 given premises will be R. But we must write down all the steps to show how exactly we obtained this R.

1. P (L v R)

2. P

3. – L

________

4. L v R 1, 2 MP (I am saying in this line, that I derived the

phrase L v R from line 1 and 2 by

means of MP, Modus Ponens)

5. R 4, 3 DS (In line 5, I am indicating that I derived R

from line 4 and line 3 by means of

Disjunctive Syllogism; what this means is

that if I know that L or R is true and I also

know from line 3 that L is negated so that it is not true that L, then my only option is R. Again, think of this phrase as: I have a coin in my Left or Right hand for sure. It turns out that the coin is not in my Left hand, so I conclude that it is in my Right hand. Hence, R is my conclusion.

More practice examples:

(Try to solve the problems by yourself, then check your answer with the solution given below in blue.)

1. P (L v R) Premise 1 (line 1)

2. P Premise 2 (line 2)

3. – R Premise 3 (line 3)

Solution:

3. P (L v R) Premise 1 (line 1)

4. P Premise 2 (line 2)

5. – R Premise 3 (line 3)

________

4. L v R 1, 2 MP (MP, Modus Ponens, from line 1 and 2)

5. L 4, 3 DS (Meaning: I obtained the conclusion L by

means of Disjunctive Syllogism from line 4 and 3)

1. (L R) v C

2. – C

3. L

________

4. L R 1, 2 DS (I obtained L R by Disjunctive Syllogism)

5. R 4, 3 MP (I obtained the conclusion R by means of

MP, Modus Ponens, from line 4 and line 3.)

Exercise 2. Derive conclusions from the premises. Be sure to write down the method you used and the lines from which you derived the letters.

1. A ( P v Q)

2. – ( P v Q)

1. D ( P v Q)

2. D

3. – Q

1. (P Q) v ( A & B)

2. – ( A & B )

1. (P Q) v ( A & B)

2. – ( A & B )

3. P

Day 12.

We begin today with more practice exercises. First cover the derivations in blue, then try to derive conclusions, then check with the blue answers if you got everything correctly.

1. (L R) v C

2. – C

3. – R

________

4. L R 1, 2 DS

5. - L 4, 3 MT

1. (A B) v [(C D) v ( M L) ]

2. – (A B)

3. – (C D)

4. M

5. (C D) v ( M L) 1, 2 DS

6. M L 5, 3 DS

7. L 6, 4 MP

Here he first 4 lines are the four given premises. Everything below the line is what I have to do to derive, step by step, my conclusion, which is the last line.

In line 5, I derive (C D) v ( M L) . How could I do so? Well, in line 2 I’m told that it is not the case that (A B), in other words this conditional is negated. But this conditional was really my first disjunct in line 1 and if this disjunct is negated then I am left only with the second disjunct, which is everything in the big parenthesis.

But then I look at premise 3 and I see that (C D) is also negated. So, from this elaborate disjunction in line 5 I’m now left with only the ( M L). In line 6 to the right I am stating that from line 5 and 3 I derived ( M L) by means of Disjunctive Syllogism (DS). But since I am also told, in line 4, that M is true I can conclude that L is true, since I am told in line 6 that If M is true then L is true.

M is indeed true, says line 4. Therefore, L is true. This I based on the rule of inference called Modus Ponens (MP).

Exercise 1.

Derive conclusions from the premises. Be sure to write the method you are using and the lines from which you derived the letters.

1. (M N) v A

2. – (M N)

1. (P Q) v (A B)

2. – (P Q)

3. A

1. (P Q) v [(A B) v ( M L) ]

2. – (P Q)

1. P v [(A B) v ( M L) ]

2. – P

3. – (A B)

1. {(A B) v [(C D) v ( E F) ]} v G

2. – G

Today we will add one more rule.

Our 5th rule of inference concerns conjunctions. We will call this rule Simplification, abbreviated as Simp.

Simplification means that from a conjunction you can infer one of its conjuncts. This is very simple and straightforward.

From R & C , we can infer R but also be we infer C.

1. R & C 1. R & C

______ _________

2. R 2. C

For example, from the sentence:

“It is raining and it is cold” we can infer that it is raining. We can also infer that it is cold.

Suppose we have the following information:

1. If Ann gets the check from the office, she will be able to buy a new computer.

2. Ann got a birthday card from her sister and she got the check from the office too!

We can conclude correctly that Ann will be able to buy the computer. Even thought the information about the birthday card is here irrelevant, we can separate the second sentence and deduce from it that she got the check. Now that she has the check, she is able to buy a new computer.

In symbols:

1. G → C (If she Gets the check, she will be able to buy Computer)

2. B & G

3. G 2, Simp (Here, in line 3, we obtain G from line 2 by means of

Simplification, abbreviated Simp.)

4. C 3, 1, MP ( We obtained C from lines 3 and 1 by means of Modus

Ponens, MP)

Let’s examine another example.

1. A ( B v C)

2. A & - D

3. A 2, Simp (I derived A from line 2 by Simplification)

4. B v C 1, 3 MP ( I derived B v C from lines 1 and 3 by Modus

Ponens )

Exercise 2. Derive valid conclusions:

1. K ( A v B)

2. - S & K

1. A v B

2. - B & - C

Now create 3 arguments of your own. Try to use all these methods (if possible):

MP Modus Ponens

MT Modus Tollens

HS Hypothetical Syllogism

DS Disjunctive Syllogism

Simp Simplification

Argument 1:

Argument 2:

Argument 3:

Day 13

Recall from week 4, De Morgan’s Laws (DM):

From - ( A v B), infer - A & - B

From - (A & B), infer - A v – B

You should memorize these formulas, more specifically, know that

- ( A v B ) is logically equivalent to - A & - B

and that

- (A & B ) is logically equivalent to ( - A v - B )

We will add these two laws of inference in our exercises in deriving conclusions from premises, as in:

1. – A & - B

2. – ( A v B) C

_______________

3. – (A v B) 1 DM (I derived – (A v B) from line 1 by De Morgan’s Laws )

4. C 2, 3 MP ( I derived C from lines 2 and 3 by Modus Ponens)

Next example:

1. – ( A v B )

2. (– A & - B ) D

_________________________

3. – A & - B 1, DM Since – ( A v B ) is the same as – A & - B

4. D 2, 3 MP ( I derived D from line 2 and 3 by means of

Modus Ponens

One more example, extra difficult (A+ level) :

1. - ( - A & B)

2. – A

___________

3. - (- A) v - B 1, DM ( Be careful here, from a negation of a conjunction

I get a disjunction where each disjunct has to be negated.

This is the first of the De Morgan Laws. But since my first letter

A has a “minus” I need to keep that “minus” and then I will see that

this will amount to a double negation, in which case I will be

left with an A instead of – A.

4. A v – B 3, DN ( Here I have to write that I obtained the A from – (- A)

by means of Double Negation DN)

*Note here that if you have something like

- (-P), that means you can derive P from it by DN

5. – B 2, 4 DS ( I obtained – B from lines 2 and 4 by Disjunctive

Syllogism)

You have now finished your work for week III.

Please double check all your answers and email your document to

Kamillasmith27@gmail.com

_____________________________________________________________________________________

Here is additional work for students at A+ level. This is not required but recommended for students who like logic.

We will learn how to “prove” symbolic arguments without using full English sentences.

Suppose we have to “prove” the validity of the following abstract argument:

F, F B / B

As you can see, this argument has 2 premises. The first premise is in blue, the second premise is in green. The conclusion is in red.

1. F Premise 1

2. FB Premise 2

The conclusion is simply B.

So, to prove that the argument is valid we will use the method of Modus Ponens.

1. F Premise 1

2. FB Premise 2

________________________

3. B 1, 2 MP

How would we prove the validity of this argument:

– (B & G) (Q v R), - A - (B & G), - A / Q v R

First, we have to number of all the premises. There are clearly 3 premises (separated by commas). The first one in orange, the second is in green, the third is in blue.

1. – (B & G) (Q v R) Premise 1

2. - A - (B & G), Premise 2

3. - A Premise 3

________

What do we do next? We will write in line 4 what we can obtain from the premises. Let’s start with something simple. How about line 3?

Given that it is the case that – A, what can I obtain from this line and line 2? I can obtain – (B & G) by Modus Ponens, like this:

1. – (B & G) (Q v R) Premise 1

2. - A - (B & G), Premise 2

3. - A Premise 3

________

4. – ( B & G ) 2, 3 MP

But the conclusion in this argument is Q v R , so we are clearly not done. We have to do something else to end with the conclusion

Q v R.

Let’s now look at premise 1. Premise 1 states that if it is the case that – (B & G) , then it is the case that Q v R . In line 5, therefore, we can conclude by MP, that it is Q v R. Like this:

1. – (B & G) (Q v R) Premise 1

2. - A - (B & G), Premise 2

3. - A Premise 3

________

4. – ( B & G ) 2, 3 MP

5. Q v R 1, 4 MP

It turns out that the conclusion here Q v R is exactly the same as the conclusion given in the argument Q v R. So we just proved that this argument:

– (B & G) (Q v R), - A - (B & G), - A / Q v R

is indeed valid.

Now your turn.

Prove that the following arguments are valid.

(A G) M, A G / M

b) – ( A & P) , A v C, - C / - P