<!DOCTYPE html><HTML lang="en"> <head><meta charset="utf-8"> <title>Pizzi (Claudio) & Williamson (Timothy) - Conditional Excluded Middle in Systems of Consequential Implication (Theo Todman's Book Collection - Paper Abstracts) </title> <link href="../../TheosStyle.css" rel="stylesheet" type="text/css"><link rel="shortcut icon" href="../../TT_ICO.png" /></head> <BODY> <CENTER> <div id="header"><HR><h1>Theo Todman's Web Page - Paper Abstracts</h1><HR></div><A name="Top"></A> <TABLE class = "Bridge" WIDTH=950> <tr><th><A HREF = "../../PaperSummaries/PaperSummary_07/PaperSummary_7661.htm">Conditional Excluded Middle in Systems of Consequential Implication</A></th></tr> <tr><th><A HREF = "../../Authors/P/Author_Pizzi (Claudio).htm">Pizzi (Claudio)</a> & <A HREF = "../../Authors/W/Author_Williamson (Timothy).htm">Williamson (Timothy)</a></th></tr> <tr><th>Source: Journal of Philosophical Logic 34, 4 (2005): 333-362</th></tr> <tr><th>Paper - Abstract</th></tr> </TABLE> </CENTER> <P><CENTER><TABLE class = "Bridge" WIDTH=400><tr><td><A HREF = "../../PaperSummaries/PaperSummary_07/PaperSummary_7661.htm">Paper Summary</A></td><td><A HREF="#ColourConventions">Text Colour-Conventions</a></td></tr></TABLE></CENTER></P> <hr><P><FONT COLOR = "0000FF"><u>Author s Abstract</u><FONT COLOR = "800080"><ol type="1"><li>It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. </li><li>Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius Thesis <a name="1"></a><A HREF="../../Notes/Notes_7/Notes_763.htm">(WBT</A><SUP>1</SUP>, opposite conditionals are not both true). </li><li>In the system CI.0 of consequential implication, which is intertranslatable with the <a name="2"></a><A HREF="../../Notes/Notes_1/Notes_121.htm">modal logic</A><SUP>2</SUP> KT, <a name="3"></a><A HREF="../../Notes/Notes_7/Notes_763.htm">WBT</A><SUP>3</SUP> is a theorem, so it is natural to ask which instances of CEM are derivable. </li><li>We also investigate the systems CIw and CI of consequential implication, corresponding to the <a name="4"></a><A HREF="../../Notes/Notes_1/Notes_121.htm">modal logics</A><SUP>4</SUP> K and KD respectively, with occasional remarks about stronger systems. </li><li>While unrestricted CEM produces <a name="5"></a><A HREF="../../Notes/Notes_1/Notes_121.htm">modal</A><SUP>5</SUP> collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. </li><li>It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. </li><li>The methods of the paper are those of propositional <a name="6"></a><A HREF="../../Notes/Notes_1/Notes_121.htm">modal logic</A><SUP>6</SUP> as applied to a special sort of conditional. </li></ol> </FONT><hr><FONT COLOR = "0000FF"><B>Comment: </B><BR><BR>See <a name="W484W"></a><A HREF = "http://www.philosophy.ox.ac.uk/faculty/members/docs/CEM.pdf" TARGET = "_top">Link</A> (Defunct).<BR><FONT COLOR = "0000FF"><HR></P><a name="ColourConventions"></a><p><b>Text Colour Conventions (see <A HREF="../../Notes/Notes_10/Notes_1025.htm">disclaimer</a>)</b></p><OL TYPE="1"><LI><FONT COLOR = "0000FF">Blue</FONT>: Text by me; &copy; Theo Todman, 2018</li><LI><FONT COLOR = "800080">Mauve</FONT>: Text by correspondent(s) or other author(s); &copy; the author(s)</li></OL> <BR><HR><BR><CENTER> <TABLE class = "Bridge" WIDTH=950> <TR><TD WIDTH="30%">&copy; Theo Todman, June 2007 - August 2018.</TD> <TD WIDTH="40%">Please address any comments on this page to <A HREF="mailto:theo@theotodman.com">theo@theotodman.com</A>.</TD> <TD WIDTH="30%">File output: <time datetime="2018-08-03T00:12" pubdate>03/08/2018 00:12:45</time> <br><A HREF="../../Notes/Notes_10/Notes_1010.htm">Website Maintenance Dashboard</A></TD></TR> <TD WIDTH="30%"><A HREF="#Top">Return to Top of this Page</A></TD> <TD WIDTH="40%"><A HREF="../../Notes/Notes_11/Notes_1140.htm">Return to Theo Todman's Philosophy Page</A></TD> <TD WIDTH="30%"><A HREF="../../index.htm">Return to Theo Todman's Home Page</A></TD> </TR></TABLE></CENTER><HR> </BODY> </HTML>