# ⊢ ((AvB) -> C) -> (A -> C) using simple derivation rules

My thought process:

``````- This derivation has no premises.
- The desired conclusion is a conditional, therefore assume
the antecedent and derive the conditional.
``````

What I have so far:

``````1   1) (AvB) -> C   P
``````

I don't know what to do next since there's no easy way to use MP(Modus Ponens) from there.

• What derivation rules are allowed? Natural deduction? – Eliran Oct 16 '16 at 22:04
• P, Add, Simp, Conj, MP, PC, RAA @EliranH – K.Wong Oct 16 '16 at 22:07
• @K.Wong in your comment, you say MP can be used ... in your question you say it can't ... – virmaior Oct 16 '16 at 22:35
• @virmaior, in my comment I stated the rules that are ALLOWED. In my question, it is obvious you can't use MP since the rule states that "if we have the sentence A -> B, and we have A, then we can use MP to derive B." We have the sentence "A -> B: (AvB) -> C" but we don't have "A: (AvB)." Therefore the rule MP cannot be used yet. – K.Wong Oct 16 '16 at 22:42
• @K.Wong okay, I've edited your language to make that clearer... we do from time to time get derivation questions that do not allow MP, and I read your question as saying that. – virmaior Oct 16 '16 at 23:59

Given that your goal is ⊢ ((AvB) -> C) -> (A -> C) , you have two different ways of getting there:

(A) Assume (AvB) -> C assume the left side of your conditional and get to your end point via CP

(B) Assume ~ ( ((AvB) -> C) -> (A -> C) ) negate the entire of your object then do RAA]

In this case, I think you've made the right choice:

1. | (AvB) -> C) P

To get to A -> C we can again:

(A) assume the left side and CP

(B) negate the entire expression and RAA

In this case, it's a lot easier to negate -- since we can use addition to show that the negation we make is invalid:

1. | | ~( A -> C) P
2. | | | A P
3. | | | A v B Add 3
4. | | | C MP 1,4
5. | | A -> C CP 3-5
6. | A -> C RAA 2-6
7. ⊢ ((AvB) -> C) -> (A -> C)

(If your proof system requires it, you may need to:

(1) repeat the assumption at line 1 before line 5.

(2) add a conjunction after 7 of 4 and 7 to show the contradiction).

• What are these lines " | ", that are between the numbering and the lettering? – K.Wong Oct 17 '16 at 5:35
• Layer of assumption. If there were no lines, those would be assumptions given in the problem. (so line 8 has no lines, line 1 has the new assumption (= one), line 2 has a new assumption (= two), line 3 has a new assumption (= three), lines 4-5 still have three, line 6 discharges one by CP (=two), line 7 discharges one by RAA (=one), line discharges one by CP (=zero)). – virmaior Oct 17 '16 at 7:33