document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167696>Question 1167696</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Setting up a prove in Fitch notation\r<BR>\n" );
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document.write( "Premises:<BR>\n" );
document.write( "PvQ<BR>\n" );
document.write( "~QvR\r<BR>\n" );
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document.write( "Conclusion; PvR </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852318 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852318\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ## Fitch Proof Setup for $P \lor R$\r<BR>\n" );
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document.write( "Here is the setup for a proof of $P \lor R$ from the premises $P \lor Q$ and $\neg Q \lor R$ using Fitch notation. This proof requires a **proof by cases** (Disjunction Elimination).\r<BR>\n" );
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document.write( "### Goal: $\mathbf{P \lor R}$\r<BR>\n" );
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document.write( "| Line | Formula | Justification |<BR>\n" );
document.write( "| :--- | :--- | :--- |<BR>\n" );
document.write( "| 1 | $P \lor Q$ | Premise |<BR>\n" );
document.write( "| 2 | $\neg Q \lor R$ | Premise |<BR>\n" );
document.write( "| 3 | $\qquad \mathbf{P}$ | **Assumption** ($\lor$ Elim) |<BR>\n" );
document.write( "| 4 | $\qquad P \lor R$ | $\lor$ Intro: 3 |<BR>\n" );
document.write( "| 5 | $\qquad$ **Q** | **Assumption** ($\lor$ Elim) |<BR>\n" );
document.write( "| 6 | $\qquad \neg Q \lor R$ | Repetition (or simply refer to 2) |<BR>\n" );
document.write( "| 7 | $\qquad \qquad \mathbf{\neg Q}$ | **Assumption** ($\lor$ Elim) |<BR>\n" );
document.write( "| 8 | $\qquad \qquad \perp$ | $\neg$ Elim: 5, 7 |<BR>\n" );
document.write( "| 9 | $\qquad \qquad \mathbf{R}$ | **Assumption** ($\lor$ Elim) |<BR>\n" );
document.write( "| 10 | $\qquad \qquad P \lor R$ | $\lor$ Intro: 9 |<BR>\n" );
document.write( "| 11 | $\qquad P \lor R$ | $\lor$ Elim: 6, 7-8, 9-10 |<BR>\n" );
document.write( "| 12 | $P \lor R$ | $\lor$ Elim: 1, 3-4, 5-11 |\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "## Completed Proof\r<BR>\n" );
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document.write( "The key insight is that $\neg Q \lor R$ serves as a crucial intermediate step within the second case of the main proof.\r<BR>\n" );
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document.write( "| Line | Formula | Justification |<BR>\n" );
document.write( "| :--- | :--- | :--- |<BR>\n" );
document.write( "| 1 | $P \lor Q$ | Premise |<BR>\n" );
document.write( "| 2 | $\neg Q \lor R$ | Premise |<BR>\n" );
document.write( "| 3 | $\qquad \mathbf{P}$ | **Assumption (Case 1)** |<BR>\n" );
document.write( "| 4 | $\qquad P \lor R$ | $\lor$ Intro: 3 |<BR>\n" );
document.write( "| 5 | $\qquad \mathbf{Q}$ | **Assumption (Case 2)** |<BR>\n" );
document.write( "| 6 | $\qquad \neg Q \lor R$ | Repetition: 2 |<BR>\n" );
document.write( "| 7 | $\qquad \qquad \mathbf{\neg Q}$ | **Assumption (Subcase 2a)** |<BR>\n" );
document.write( "| 8 | $\qquad \qquad \perp$ | $\neg$ Elim: 5, 7 |<BR>\n" );
document.write( "| 9 | $\qquad \qquad R$ | $\perp$ Elim: 8 |<BR>\n" );
document.write( "| 10 | $\qquad \qquad P \lor R$ | $\lor$ Intro: 9 |<BR>\n" );
document.write( "| 11 | $\qquad \qquad \mathbf{R}$ | **Assumption (Subcase 2b)** |<BR>\n" );
document.write( "| 12 | $\qquad \qquad P \lor R$ | $\lor$ Intro: 11 |<BR>\n" );
document.write( "| 13 | $\qquad P \lor R$ | $\lor$ Elim: 6, 7-10, 11-12 |<BR>\n" );
document.write( "| 14 | $P \lor R$ | $\lor$ Elim: 1, 3-4, 5-13 | </SPAN>\n" );
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