Prove Logical Operations Using Inference Rules

Question

Premise 1: p ∧ q

Premise 2: q → r

Premise 3: s → ¬r

Premise 4: ¬r → ¬u

Premise 5: t ∨ s

Premise 6: t → ¬p ∨ U

Prove: u ∧ q

Does anybody know how to solve this proof using rules of inference? I know the rules of inference like modus ponens/tollens but I am not sure how to use them here. I am still beginning to learn these types of proofs.

Can anybody show me how to complete this? Thanks.

You could always set up truth tables, but I suspect you are looking for formal mathematical proofs. — Tim Biegeleisen, Oct 02 '15 at 02:27
There are several different possible sets of inference rules you could be using, including Hilbert’s or Natural Reasoning. What are your axioms? — Davislor, Oct 02 '15 at 03:08

score 2 · Accepted Answer · answered Oct 02 '15 at 02:54

2

Since p ∧ q → p and p ∧ q → q, by Premise 1 both p and q are true.

By Premise 2 we now know that r is true.

By Premise 3, r → ¬s, so s is false.

Then, by Premise 5, t must be true.

Now, by Premise 6, ¬p ∨ u is true, but since p is true, it is u which must be true.

Finally, both q and u are true and so it is u ∧ q.

(Also note that Premise 4 is not needed)

answered Oct 02 '15 at 02:54

Leandro Caniglia

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Thanks man! This is exactly what I was looking for. I was able to figure out the laws and make a nice proof! – Oct 02 '15 at 03:17

Prove Logical Operations Using Inference Rules

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