Question 1194113
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Please only post one problem at a time.


I'll do problem 55 to get you started. 
<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td>(L = N) -> C</td><td></td><td></td></tr><tr><td>2</td><td>(L = N) v (P -> ~E)</td><td></td><td></td></tr><tr><td>3</td><td>~E -> C</td><td></td><td></td></tr><tr><td>4</td><td>~C</td><td></td><td></td></tr><tr><td>:.</td><td>~P</td><td></td><td></td></tr><tr><td>5</td><td>~~E</td><td>3, 4</td><td>Modus Tollens</td></tr><tr><td>6</td><td>~(L = N)</td><td>1, 4</td><td>Modus Tollens</td></tr><tr><td>7</td><td>P -> ~E</td><td>2, 6</td><td>Disjunctive Syllogism</td></tr><tr><td>8</td><td>~P</td><td>7, 5</td><td>Modus Tollens</td></tr></table>
I used a standard equal sign in place of a triple equal sign. 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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