I Intro to Symbolic Logic: Replacement Rules

[Mikaelochi]

TL;DR Summary

In Virginia Klenk's book Understanding Symbolic Logic (5th edition), I am having trouble with problem 7b in Unit 8 which deals with the replacement rules.

Basically the problem starts with these given premises:
1. ~ (A ∨ (B⊃T))
2. (A ⋅ C) ∨ (W ⊃ ~D)
3. ~(P ∨ T) ⊃ D
4. ~P ≡ ~(T ⋅ S)
And from these premises, I must prove ∴ ~W. This is what I have done so far:
5. (~P ⊃ ~(T ⋅ S)) ⋅ (~(T ⋅ S) ⊃ ~P) B.E. 4
6. ~P ⊃ ~(T ⋅ S) Simp. 5
7. ~(T ⋅ S) ⊃ ~P Simp. 5
8. ~P ⊃ (~T ∨ ~S) DeM. 6
9. (~T ∨ ~S) ⊃ ~P DeM. 7
10. (~P ⋅ ~T) ⊃ D DeM. 3
11. (~A ⋅ ~(B ⊃ T)) DeM. 1
12. ~A Simp. 11
13. ~A ∨ ~C Add. 12
14. ~(A ⋅ C) DeM. 13
15. W ⊃ ~D D.S. 2, 14
To get ~W, all I need is D which I can restate as ~~D. But to get D, I need to get ~(P ∨ T). And the only way I know how to get ~(P ∨ T) is to get (~P ⋅ ~T). So, I would need ~P and ~T alone. I have no idea how to do that. Perhaps this approach is wrong. So, any help would be greatly appreciated. This problem feels borderline impossible.

 

[nuuskur]

Not familiar with the book and don't have it at hand. Are you given any leeway in the method you use to prove this? What you could do is show
<br /> 1. \land 2. \land 3 \land 4. \Rightarrow \neg W<br />
is a tautology. If it's not a tautology, then $\neg W$ doesn't follow from the premises.

 

[Stephen Tashi]

1. ~ (A ∨ (B⊃T))

Is "T" used to denote a proposition with an unspecified truth value? or does it denote a proposition that has the truth value "True"?

 

[WWGD]

Does the $\cdot$ stand for 'and'?

 

[TeethWhitener]

You’ve done DeMorgan’s on premise 1. What happens if you convert the conditional to a conjunction?

 

[Stephen Tashi]

Basically the problem starts with these given premises:
1. ~ (A ∨ (B⊃T))
2. (A ⋅ C) ∨ (W ⊃ ~D)
3. ~(P ∨ T) ⊃ D
4. ~P ≡ ~(T ⋅ S)

So, I would need ~P and ~T alone.

Take @TeethWhitener suggestion and get $(\sim A) \cdot B \cdot( \sim T)$ from 1.
Then use 4. to get $\sim P$.

 

[Mikaelochi]

Is "T" used to denote a proposition with an unspecified truth value? or does it denote a proposition that has the truth value "True"?

No, T is just a symbol representing a claim like A & B.

 

[Mikaelochi]

Does the $\cdot$ stand for 'and'?

Yeah, it stands for "and."

 

[Mikaelochi]

You’ve done DeMorgan’s on premise 1. What happens if you convert the conditional to a conjunction?

I've pieced it together now. Thank you!