Propositional Logic

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chipdavis

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Jun 11, 2005
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I have a two part problem that is giving me a lot of trouble. Mostly because the rules I would have used to solve these two are ruled out.

Problem - Use propositional logic to prove the following:

Part 1 - (Don't use exportation rule of inference)
[(P ^ Q) -> R)] -> [P -> (Q -> R)]

I've got

(P ^ Q) -> R) hyp

Part 2 - (Don't use contraposition rule of inference)
(P -> Q) -> (Q' -> P')

I've got

(P -> Q) hyp
P' v Q 1, imp

Any suggestions / help would be much appreciated. Thanks in advance!
 
We know by Material Equivalence that (A → B)≡ (~A∨B).
[(P^Q) → R] ≡
[~(P^Q)∨R] ≡
[(~Pv~Q)vR] ≡
[~P∨(~Q∨R)]≡
[P → (Q → R)]
 
Thanks for responding pka..

I'm afraid I can't quite follow what you've done though. I don't believe we have dicussed Material Equivalence yet. Can you baby step me through the process? :(
 
“don't believe we have discussed Material Equivalence yet.”
Well that is the problem!
With these kinds of questions, everything depends upon definitions.
I follow Irving Copi’s notation for pure logic.
Your author/text may have a totally different idea as to the definitions.

In other words, there is no standard for definitions.
So unless we know the exact text (and have access to it), there is little we can do to help you.
That may seem odd to you! But that is the way it is!
 
thanks anyway pka..

I understand what you mean. I am using the 5th edition of "Mathematical Structures for Computer Science" if that helps, but most likely you do not have access to this text. I believe I need to solve the problem using a combination of Equivalence & Inference rules as that is what the book is covering in this chapter. I guess Material Equivalence could be the same as Equivalence, but the book does not use this term.

What does the ~ mean in you answer? I have not seen this symbol either. Perhaps that will help me understand your steps.
 
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