Question 1201675
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I'll do problem 1 to get you started.


This is what the 1st question looks like when each premise gets its own line.
1. R ⊃ (G v ~A) 
2. (G v ~A) ⊃ ~S 
3. G ⊃ S 
4. R
:. ~A


Here is one way to do the derivation.
<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Lines Used</td><td>Reason</td></tr><tr><td>1</td><td>R -> (G v ~A)</td><td></td><td></td></tr><tr><td>2</td><td>(G v ~A) -> ~S</td><td></td><td></td></tr><tr><td>3</td><td>G -> S</td><td></td><td></td></tr><tr><td>4</td><td>R</td><td></td><td></td></tr><tr><td>:.</td><td>~A</td><td></td><td></td></tr><tr><td>5</td><td>R -> ~S</td><td>1,2</td><td>Hypothetical Syllogism</td></tr><tr><td>6</td><td>~S</td><td>5,4</td><td>Modus Ponens</td></tr><tr><td>7</td><td>G v ~A</td><td>1,4</td><td>Modus Ponens</td></tr><tr><td>8</td><td>~G</td><td>3,6</td><td>Modus Tollens</td></tr><tr><td>9</td><td>~A</td><td>7,8</td><td>Disjunctive Syllogism</td></tr></table>
I used arrows in place of the horseshoe symbols.


For more information, check out the various rules of inference and rules of replacement as shown in the link below.
<a href = "https://logiccurriculum.com/2019/02/09/rules-for-proofs/">https://logiccurriculum.com/2019/02/09/rules-for-proofs/</a>


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