Humanities
Basic Logic

Question Description

I need to do the exercises 12 & 13. they are basic logic let me know if you can help.

thanks,

Linnet

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Week III Deadline for lessons 12 and 13: Friday January 19. Lesson 12 We learned thus far the following rules of valid inference: MP Modus Ponens MT Modus Tollens DS Disjunctive Syllogism HS Hypothetical Syllogism Exercise 1. Below you see conclusions derived by four valid rules of inference. Write, next to each of the arguments, which method was used to infer the conclusion. Again, your choices are: MP, MT, DS, and HS. a) 1. A→ B 2. B→ C 3. A→ C b) 1. A 2. A→ B 3. A→ B c) 1. - B 2. A→ B 3. - A d) 1. A→ B 2. - B 3. - A e) 1. - B 2. C→ B 3. - C f) 1. - B 2. A v B 3. A Exercise 2. Create your own imaginative arguments in English for each of the 4 logical formulas. For example, for Modus Tolles you can write something like this: MT: 1. If you are a bird, then you can fly. 2. But you clearly cannot fly. 3. So, you are not a bird. B→ F – F –B Remember to use two premises and a conclusion for each argument. Now we will learn a 5th logical formula. Our 5th rule of inference concerns conjunctions and is called Simplification, abbreviated as Simp. This rule means that from a conjunction, you can infer one of its conjuncts. This is very simple and straightforward. From R & C , infer R or infer C. 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, she will be able to buy a new computer. 2. Ann got a birthday card from her friend and she got the check 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 2. B & G 3. G 2, Simp 4. C 3, 1, MP (If she Gets the check, she will be able to buy Computer) (Here, in line 3, we obtain G from line 2 by means of Simplification , abbreviate Simp.) ( We obtained C from lines 3 and 1 by means of Modus Ponens, MP) Let’s try another example. 1. 2. 3. 4. A  ( B v C) A&-D A 2, Simp BvC 1, 3 MP Exercises 3. Your turn: a) 1. K  ( A v B) 2. - S & K 3. 4. b) 1. A v B 2. - B & - C 3. 4. (I derived A from line 2 by Simplification) ( I derived B v C from lines 1 and 3 by Modus Ponens ) c) 1. - A & - B 2. A v D 3. 4. d) 1. A & (B  C) 2. A  M 3. 4. Lesson 13 Recall from last week, De Morgan’s Laws (DM): From - ( A v B), infer From - (A & B), infer -A&-B -Av–B You should memorize these formulas, more specifically, know that -(AvB) is logically equivalent to -A&-B and that - (A & B ) is logically equivalent to ( -Av -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) 4. C 1 DM 2, 3 MP (I derived – (A v B) from line 1 by De Morgan’s Laws ) ( I derived C from lines 2 and 3 by Modus Ponens) Next example: 1. – ( A v B ) 2. (– A & - B )  D _________________________ 3. 4. –A&-B D 1, DM 2, 3 MP Since – ( A v B ) is the same as – A & - B ( 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 4. 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. 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) Exercise 1. Identify the inference method used in the following arguments: (The first one is an example with an answer in red; this is how you have to answer the remaining problems.) 1. 2. 3. Q QL L Answer: This is Modus Ponens. Here the conclusion in line 3 was derived by means of Modus Ponens (MP) from lines 1 and 2. a) 1. A v B 2. – B 3. A Answer: This is _______________ b) 1. A  B 2. B  - C 3. A  - C Answer: This is ________ c) 1. – ( A v B ) _________________ 2. – A & - B Answer: This is…… d) 1. A  B 2. – B 3. – A Answer : e) 1. A & B 2. A Answer: f) 1. A v (B & C) 2. – (B & C) 3. A Answer : Line 2 was derived from ….. by …… g) 1. - - A 2. A Answer : h) 1. A  (B & C) 2. (B & C )  M 3. A  M Answer: Exercise 2. Continue solving. The first 4 are for practice. First solve by yourself, then check with the answers. a) 1. P→ (C v D) 2. (A v B) → P Solution: 1. P→ (C v D) 2. (A v B) → P 3. ( A v B ) → (C v D) 2, 1 HS b) 1. A→ B 2. C → A 3. – B 4. 5. Solution: 1. A→ B 2. C → A 3. – B 4. - A 5. – C Or: 1, 3 MT 2, 4 MT 4. C → B 5. – C d) 1. A v B 2. A  - C 3. – B 4. – C  D 5. D E ____________ 6. 7. 8. 9. Solution: 2, 1 HS 4, 3 MT 1. A v B 2. A  - C 3. – B 4. – C  D 5. D E ____________ 6. 7. 8. 9. A A D D E 1, 3 DS 2, 4 HS 6, 7 MP 5, 8 MP Now your turn, do it yourself: e) 1. 2. 3. –( P&–Q) P QL f) 1. D v – E 2. B  E 3. – D ___________ 4. 5. g) 1. 2. 3. 4. A v C AB CD –D __________ 5. 6. 7. h) 1. A  B 2. ( A v C ) v D 3. – C & - D _______ 4. 5. 6. 7. 8. i) 1. (M v B) v G 2. – (B v G) ____________ 3. 4. 5. 6. 7. I give you the answer below for the i) problem. Check with your answer above for practice. 1. M v B) v G 2. – (B v G) ____________ 3. – B & - G 4. – G 5. M v B 6. – B 7. M j) 1. – ( A v B) 2. B v C _______ 3. 4. 5. k) 1. – T 2. ( R & S )  T 3. R ______ 4. 5. 6. 2, DM 3, Simpl 1, 4 DS 3, Simpl 5, 6 DS ...
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Final Answer

Attached.

Running head: RULES OF INFERENCE

Rules of Inference
Student’s Name
Institution Affiliation

RULES OF INFERENCE

1

Rules of Inference

Lesson 12
Exercise 1
a)
1. A→ B used MP rule
2. B→ C used MP rule
3. A→ C used DS rule
b)
1. A in this case A is a premise
2. A→ B used MP rule
3. A→ B used MP rule
c)
1. - B used MT rule
2. A→ B used MP rule
3. - A used MP rule
d)
1. A→ B used MP rule
2. - B used DS rule
3. – A in this case –A is a premise
e)
1. - B used DS rule
2. C→ B used DS rule
3. – C used MT rule
f)
1. - B used DS rule

RULES OF INFERENCE

2

2. A v B used MP rule
3. A in this case A is a premise
Exercise 3
a)
If James receives the parcel before I go, he will share with me whatever it contains.
The generosity of James helped me in saving the few cents I had planned to buy a book; James
shared the parcel items with me.
1. K( A v B)
2. - S & K
3. K

I derived K from the second line through Simplification,

4. A v B

I derived A v B from lines 1 & 2 by Modus Ponens, MP

b)
I can decide to take salad or I can decide to take soup.
I will not decide salad and therefore I will decide soup.
1. A v B

(It is either I decide to take salad or soup)

2. - B & - C
3. –C
4.

I derived –C from the first and third line using the Simplification method

AvB

I derived A v B from lines 3 and 1 using the Disjunction Syllogism, DS

c)
1. - A &- B
2.

AvD

3. –B
4.

2, Simp

–A v –B 1, 2

HS
d)
1. A& (B C)

(I derived –B from line 4 through Simplfication)
(I derived –A v –B from lines 1 and 2 using Hypothetical Syllogism,

RULES OF INFERENCE

3

2. A M
3. A&

1, 2 Simp

4. B C 1 & 2

(I derived A& from the first and second by Simlification
(I used Modus Tollens, MT in deriving B C from lines 1 and 2.

Lesson 13
Exercise 1
a)
1. A v B
2. – B
3. A
In this case, A was derived from line 1 by Modus Ponens
b)
1. A  B
2. B  - C
3. A  - C
-C was derived from line 1 and 2 by Disjunction Syllogism
c)
1. – ( A v B )
_________________
2. – A & - B
-A & -B was derived from line 1 by De Morgan’s Laws because –A v –B is logically equivalent
to –(A B)
d)
1. A  B
2. – B

RULES OF INFERENCE

3. – A
-B was derived from line 2 by Modus Tulles and –A was derived from line 1 by Modus Ponens
e)
1. A & B
2. A
A was derived from line 1 through Modus Ponens
f)
1. A v (B& C)
2. – (B& C)
3. A
-(B &C) was derived from line 2 a...

montesamaris (867)
Carnegie Mellon University

Anonymous
Return customer, been using sp for a good two years now.

Anonymous
Thanks as always for the good work!

Anonymous
Excellent job

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