GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 4

Answer 1

Argument 1:

Let, K: Kerry errs; M: Myrna fails to show; B: Kerry does break record.

Forming Argument:

$K\vee M$

$K\rightarrow \neg B$

M

$\therefore K$

Assuming conclusion is False and trying to make all premises True. If we become succeed then argument is not valid otherwise argument is valid.

K=False (Assuming conclusion is False)
M=True (made third premise True)

$K \rightarrow \neg B \equiv False\rightarrow \neg B \equiv True$ (2nd Premise is True)

$K \vee M \equiv False\vee True \equiv True$ (1st Premise is True)

Hence, Argument 1 is not valid.

Argument 2:

Let, L: Tasha leaves; C: Carol moves in; S: Sam is happy; J: Josh laughs

Forming Argument:

$L\rightarrow C$

$C\rightarrow \neg S$

$\neg S\rightarrow J$

$S$

$\therefore \neg L$

Assuming conclusion is False and trying to make all premises True. If we become succeed then argument is not valid otherwise argument is valid.

With L=True, Conclusion becomes false. 

S=True (made 4th premise True)

$\neg S\rightarrow J \equiv \neg True\rightarrow J \equiv False\rightarrow J\equiv True$ (3rd premise is True)

$C\rightarrow \neg S \equiv C\rightarrow \neg True \equiv C\rightarrow False$

To make 2nd premise True, take C=False. $False\rightarrow False\equiv True$ (2nd premise True)

$L\rightarrow C \equiv True \rightarrow False \equiv False$ (unable to make 1st premise True)
Hence, Argument 2 is Valid.

Answer: Option B: Only Argument 2 is valid.

Answer 2

Lets check Argument 1 first.

Let kerry errs = K

Let Myra shows = M

Let break the record = B

 

Therefore, the arguments are,

K V M’

K → B’

M’

---

B’

 

Now, we can try to make the premises True and the conclusion False. If we are able to do so, it will be invalid.

(Solving)

B’ = F

M’ = T so M = F

K V M’ = T as k V T = T

K → B’ = T so K → F = T only when K is F which is possible.

 

So, we are able to make all premises True and conclusion as False. Hence, it is an invalid argument.

 

 

Similarly checking Argument 2 now.

Let tasha leaves = L [PS : do not take propositional variable as T, as it might be confusing later!]

Let Carol moves in = M

Let sam is happy = H

Let Josh laughs = J

 

Therefore, the arguments are,

L → M

M → H’

H’ → J

H

---

L’

 

Now, we can try to make the premises True and the conclusion False. If we are able to do so, it will be invalid.

(Solving)

L’ = F

L → M = M’ → L’ so M’ → F = T only when M’ = F so M = T

M → H’ = T as T → H’ = T only when H’ = T so H = F

Now if H = F then H != T as a premise

So, we are not able to make all premises True and conclusion as False. Hence, it is a valid argument.

Comments

In argument 1: conclusion is kerry break the record then how B'?