Question 1204702
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The conclusion is ~C
With indirect proofs, or proofs by contradiction, we assume the opposite of the conclusion is the case. Then we show some contradiction arises from this assumption. Thereby proving the original conclusion to be the case.
<table border = "1" cellpadding = "5"><tr><td>Number</td><td></td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td></td><td>C --> (N & I)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>(N v P) --> (I --> ~C)</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>~C</td><td></td><td></td></tr><tr><td></td><td>3</td><td>~(~C)</td><td></td><td>Assumption for Indirect Proof</td></tr><tr><td></td><td>4</td><td>C</td><td>3</td><td>Double Negation</td></tr><tr><td></td><td>5</td><td>N & I</td><td>1, 4</td><td>Modus Ponens</td></tr><tr><td></td><td>6</td><td>I & N</td><td>5</td><td>Commutation</td></tr><tr><td></td><td>7</td><td>N</td><td>5</td><td>Simplification</td></tr><tr><td></td><td>8</td><td>I</td><td>6</td><td>Simplification</td></tr><tr><td></td><td>9</td><td>N v P</td><td>7</td><td>Addition</td></tr><tr><td></td><td>10</td><td>I --> ~C</td><td>2, 9</td><td>Modus Ponens</td></tr><tr><td></td><td>11</td><td>~C</td><td>10, 8</td><td>Modus Ponens</td></tr><tr><td></td><td>12</td><td>C & (~C)</td><td>4, 11</td><td>Conjunction</td></tr><tr><td>13</td><td></td><td>~C</td><td>3 - 12</td><td>Indirect Proof</td></tr></table>
More info:
<a href="https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson">Logic Rules of Inference and Replacement</a>


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