Method to derive conclusions

Suppose, you have been given the following assumptions:

1. (W ⊃ (F ∨ M))
2. ~F
3. (M ⊃ A)

In what way you can come up with conclusion from this? What kind of inferences rules can be used here? I know a few of them but don't see them being applied here.

• Consider clarifying your question by noting what kind of conclusion you are interested in reaching. As the question is written, there are infinitely many logical inferences that could be made from the above sentences. One could infer ~~~F or ~~~~~F, or ~MvA, or (W⊃F)v(W⊃M), but why would we do one of those rather than the other? You haven't said... – ChristopherE Oct 23 '14 at 14:59
• Isn't there supposed to be just one valid conclusion out these no matter which rule you follow? – cpx Oct 23 '14 at 15:27
• Absolutely not. There are infinitely many valid conclusions. ~~~F & (M⊃A) is a valid conclusion from these. So is (~MvA)vP. Etc. – ChristopherE Oct 23 '14 at 16:22
• I see. You're right. I just checked it on truth table. But I couldn't draw a single conclusion from inference rules I've read such as AND, NIF, CS, DS, MS, and MT. – cpx Oct 23 '14 at 18:38

Chris is right, of course, about the fact that infinitely many conclusions can be drawn from those premises (provided that there is at least one applicable inference rule). What I want to do here is to simplify one of the sentences to reveal an obvious conclusion that can be drawn from them.

Given that F is false, (1) is equivalent to (W → M). Here is one way of showing that:

• (01) (W → (F ∨ M))
• (02) ¬F
• (03) | W
• (04) | (F ∨ M) (by →-elimination from 1 and 3)
• (05) | | F
• (06) | | ⊥ (by ⊥-introduction from 2 and 5)
• (07) | | M (by ⊥-elimination from 6)
• (08) | | M
• (09) | | M (by reiteration from 8)
• (10) | M (by ∨-elimination from 4, 5-7, 8-9)
• (11) (W → M) (by →-introduction from 3-10).

So the three assumptions are:

1. (W → M)
2. ¬F
3. (M → A)

A nice conclusion now follows by the transitivity of →, viz. (W → A). This can be proven, for instance, by using → introduction and elimination rules:

1. | W
2. | M (by →-elimination from 1 and 4)
3. | A (by →-elimination from 5 and 3)
4. (W → A) (by →-introduction from 4 to 6)
• As I have only seen these rules being used to derive inferences. I wonder if the rule to get inference from `(W ⊃ (F ∨ M))` and `~F` was actually called Disjunctive syllogism? If so, then I wasn't aware that you can apply it on the sub-part in the same way and not just `(P ∨ Q), ~Q → P` only? – cpx Oct 23 '14 at 21:39
• @cpx I've added a derivation of (W -> M) using more basic rules. If you're allowed to use Disjunctive Syllogism (DS), then (A) Assume W. (B) Get (F v M) from 1. (C) Using A and premise 2 by DS get M. (D) Conclude (W -> M) by conditional introduction. – Hunan Rostomyan Oct 23 '14 at 21:57

Combining 1. and 2., you can derive:

4. W ⊃ M

Combining 3. and 4., you can derive:

5. W ⊃ A

I'm not sure the exact rules of inference that allow these derivations, but you can certainly prove them using truth tables.