SD Logics Problems: Derivability, Inconsistency, Equivalence, and Validity

[Kwiatkowski18]

My teacher threw us a couple of questions on SD that I know hardly anything about. The first one one to come up with correct answers receives an additional 10% on our next paper. I would really like to answer them all myself, but I don't have the information about SD to do them as I have never taken a Phil course regarding Logics. So please, if anyone could solve these for me, that would be wicked.

Legend: >=horeshoe = is triplebar

1. Prove that the following derivability claim holds in SD.
{(t > a) & (~t > b), a > ~l} deriviation: l > b

2. SHow that the following set of sentences is inconsistent in SD.
{(a v ~c) > ~b), ~b = (q & ~q), ~c v a}

3. Show that the members of the following pair of sentences are equivalent in SD.
a = b ~a = ~b

4. Show that the following argument is valid in SD.
(b & (e v g))
(b & g) = h
(h > f) & ~e

 

[Evo]

Do not post the same topic in more than one forum. We do not supply answers to schoolwork here. If you are having trouble solving a problem and show what you have done so far and where you are stuck, someone might give you suggestions on what you might do to try to solve it yourself.

I left your other thread open so that you can show what work you have done so far on each problem.

 

[Architeuthis Dux]

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1. To prove the derivability claim, we need to show that the conclusion, l > b, can be derived from the premises, {(t > a) & (~t > b), a > ~l}. Using the SD rule of hypothetical syllogism, we can combine the first two premises to get (t > b). Then, using the SD rule of disjunctive syllogism, we can combine this with the third premise, (a > ~l), to get (t > ~l). Finally, using the SD rule of contraposition, we can flip this statement to get (~l > t), and then combine it with the first premise, (~t > b), to get (~l > b). This shows that the conclusion, l > b, can be derived from the premises, and thus the derivability claim holds in SD.

2. To show that the set of sentences is inconsistent, we need to show that it is impossible for all of the statements to be true at the same time. Using the SD rule of hypothetical syllogism, we can combine the first two premises, {(a v ~c) > ~b) and ~b = (q & ~q)}, to get (a v ~c) > (q & ~q). Then, using the SD rule of disjunctive syllogism, we can combine this with the third premise, ~c v a, to get (q & ~q). However, this is a contradiction, as it is impossible for both q and ~q to be true at the same time. Therefore, the set of sentences is inconsistent in SD.

3. To show that the pair of sentences are equivalent, we need to show that they have the same truth value in all possible cases. Using the SD rule of contraposition, we can flip the first sentence, a = b, to get ~b = ~a. Then, using the SD rule of transposition, we can switch the two sides of the equation to get ~a = ~b. This shows that the two sentences have the same truth value in all possible cases, and thus they are equivalent in SD.

4. To show that the argument is valid, we need to show that the conclusion, f, follows logically from the premises, (b & (e v g)), (b & g) = h, and (h > f)