My premise is ((P v R) → S) and the desired conclusion is R-->S. I cannot understand how to get disjunctions out in problems like this.
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A good rule of thumb when your conclusion is a conditional is to assume the antecedent, derive the consequent, then discharge the assumption.– BumbleCommented Feb 28 at 2:05
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^ They probably can't use the Deduction Theorem here, and are required to apply syllogisms to solve the problem.– Michael CareyCommented Feb 28 at 3:44
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@DavidGudeman , that was in fact the first thing i did. I'm not exactly sure what you get out of telling me that its a "simple problem". Writing apple is simple to you now, yet you wouldn't belittle a kid learning how to write by telling him that its a simple word to write. I suggest that rather than commenting useless comment like this you look inward and ask yourself why you feel the need to belittle someone trying to understand something.– Oscar NolfCommented Feb 28 at 3:46
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@Bumble thank you! that actually makes a lot of sense. Ill apply that to the rest of the problems– Oscar NolfCommented Feb 28 at 3:52
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@MichaelCarey We've just learned this, which is probably why im a little confused by it– Oscar NolfCommented Feb 28 at 3:52
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1 Answer
You may have to move things around first in order to apply the syllogisms that you have, but this is the idea:
(P ∨ R) → S :: 1 Premise
(¬ P ∧ ¬ R) ∨ S :: 2 Logical Equivalence + De Morgans
(S ∨ ¬ P ) ∧ (S ∨ ¬ R) :: 3 Distribution Law
(S ∨ ¬ R) :: 4 Simplification
(R → S) :: 5 Logical Equivalence
Note: I skipped over some applications of commutativity laws.
Also, Step (2) and (5) follow from the equivance between p → q and ¬ p ∨ q
If you are skeptical of the equivalence or haven't been taught it. Compare the truth tables of both statements
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That makes sense. Thanks to @Bumble's comment I actually got the solution. Preciate you taking the time to write this out though! Commented Feb 28 at 4:01
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Of course! I'm glad you solved it! It's good to see multiple versions, and it's best to have solved it yourself without using my posted solution anyways. Commented Feb 28 at 4:03