Use the eighteen rules of inference to derive the conclusion of the following symbolized argument.

Question

1.	C ⊃ (~L ⊃ Q)
2.	L ⊃ ~C
3.	~Q	/ ~C

Answer 1

Thought you might like to see a simpler validation process. This uses an "assumed indirect proof" method. The approach depends upon the axioms of the excluded middle and non-contradiction. The excluded middle is the concept that any clearly-expressed truth claim must be true or false, and that if it isn't one, then it must be the other. And the axiom of non-contradiction is the principle that something cannot be both the case and not the case (true and false) at the same time and in the same way.

Once we have written out the premises, we negate the conclusion. We assume that it is false. If doing so produces a contradiction, it indicates that the conclusion cannot be false -- it must be true. (There is a complex problem if the contradiction is in the premises, but that generally is not he case for textbook problems.)

1) C ⊃ (~L ⊃ Q) Premise

2) L ⊃ ~C Premise

3) ~Q Premise

4) C Negated conclusion (assumption)

5) ~L ⊃ Q Modus Ponens 1, 4

6) L Modus Tolens 5, 3

7) ~C Modus Ponens 2, 6

8) C & ~C Conjunction 4, 5 (and contradiction)

9) ~C Negate the assumption that led to the contradiction 4-8

This indicates that our assumption (C, 4) must be false, and therefore shows that the original conclusion (~C) is validly inferred from the premises.

This indirect assumption is a very powerful approach to validation.

Answer 2

A common thing that I like to do is convert everything to ∧ and ∨'s. Note that this may not be the most efficient algorithm, but gives a simple method. Some key reasons why it's typically easier are because of the Simplification, Conjunction and Addition inference laws. A big thing to remember for this method is the distribution rules/equivalence.

1) C ⊃ (~ L ⊃ Q) Hypothesis

2) L ⊃ ~ C Hypothesis

3) ~ Q Hypothesis

4) ~ C ∨ (~L ⊃ Q) Mat. I. on 1

5) ~C ∨ (L ∨ Q) Mat. I. on 4

6) ~L ∨ ~C Mat. I. on 3

7) Q ∨ (~C ∨ L) Comm. and Ass. on 5

8) ~C ∨ L M.T. on 7 & 3

9) ~ C ∨ ~L Comm. on 6

10) (~C ∨ L) ∧ (~C ∨ ~L) Conj. on 8 & 9

11) ~C ∧ (L ∨ ~L) Dist. on 10

12) ~C Simp.

On a curious note, I assume that you are a computer science student? I haven't seen ⊃ as the implication arrow in math since the early 1900's.

Good luck and study well!