Working Group in the History and Philosophy of Logic, Mathematics, and Science

<!DOCTYPE html> <html> <head> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <title>Working Group in the History and Philosophy of Logic, Mathematics, and Science - Home</title>  <link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.2/css/bootstrap.min.css"> <link rel="stylesheet" type="text/css" media="screen" href="css/screen.css"> <link rel="stylesheet" type="text/css" media="print" href="css/print.css"> <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js" type="text/javascript"></script> <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.2/js/bootstrap.min.js"></script> <script src="js/nav.js"></script> <link rel="stylesheet" type="text/css" href="css/screen.css" /> <link rel="stylesheet" type="text/css" media="print" href="css/print.css" /> </head> <body> <div class="container-fluid"> <header> <img src="Trumpet.png" alt="Toricelli's Trumpet" title="Toricelli's Trumpet" id="trumpet" /> <h1>Working Group in the History and Philosophy of<br/> Logic, Mathematics, and Science</h1> </header> <div class="row"> <nav id="navbar" class="navbar navbar-default col-md-2"> <div class="navbar-header"> <button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> </div> <div class="navbar-collapse collapse"> <ul class="nav navbar-nav"><li class="active"><a href="index.html">Home</a></li><li><a href="events.html">Events</a></li><li><a class="dropdown-toggle" data-toggle="dropdown">Archives</a><ul class="dropdown-menu"><li><a href="archive00-01.html">2000-1</a></li><li><a href="archive01-02.html">2001-2</a></li><li><a href="archive02-03.html">2002-3</a></li><li><a href="archive03-04.html">2003-4</a></li><li><a href="archive04-05.html">2004-5</a></li><li><a href="archive05-06.html">2005-6</a></li><li><a href="archive06-07.html">2006-7</a></li><li><a href="archive07-08.html">2007-8</a></li><li><a href="archive08-09.html">2008-9</a></li><li><a href="archive09-10.html">2009-10</a></li><li><a href="archive10-11.html">2010-11</a></li><li><a href="archive11-12.html">2011-12</a></li><li><a href="archive12-13.html">2012-13</a></li><li><a href="archive13-14.html">2013-14</a></li><li><a href="archive14-15.html">2014-15</a></li><li><a href="archive15-16.html">2015-16</a></li><li><a href="archive16-17.html">2016-17</a></li><li><a href="archive17-18.html">2017-18</a></li><li><a href="archive18-19.html">2018-19</a></li><li><a href="archive19-20.html">2019-20</a></li><li><a href="archive22-23.html">2022-23</a></li><li><a href="archive23-24.html">2023-24</a></li></ul></li></ul> </div> </nav> <main role="main" class="col-md-10"> <p>We are a Working Group devoted to the discussion of historical and philosophical issues in symbolic logic, mathematics, and science. We meet on occasional Wednesday evenings for a talk and a lively discussion. The group is funded by the <a href="http://townsendcenter.berkeley.edu">Doreen B. Townsend Center for the Humanities</a> and the <a href="http://philosophy.berkeley.edu">Department of Philosophy</a>.</p> <p>All members of the academic community are welcome to attend. We have regular participants in many different fields, including philosophy, mathematics, history of science, and psychology.</p> <p>The group organizers are <a href="http://philosophy.berkeley.edu/holliday">Wesley Holliday</a>, <a href="http://philosophy.berkeley.edu/macfarlane">John MacFarlane</a>, <a href="http://philosophy.berkeley.edu/mancosu">Paolo Mancosu</a>, <a href="http://philosophy.berkeley.edu/yalcin">Seth Yalcin</a>, and <a href="https://philosophy.berkeley.edu/people/detail/583">Xueyin (Snow) Zhang</a>.</p> <h3 id="our-next-event">Our next event</h3> <div class="event"> <p>September 25, 2024, 6-7:30 PM <strong>(note special time)</strong> in 234 Philosophy</p> <p><em>Michael Mendler and Luke Burke (Otto-Friedrich University of Bamberg)</em></p> <p><span class="title"> The Došen Square under construction: A tale of four modalities</p> <p></span></p> <p>In classical modal logic, necessity <code>[]A</code>, possibility <code><>A</code>, impossibility <code>[]~A</code> and non-necessity <code><>~A</code> form a Square of Oppositions (SO) whose corners are interdefinable using classical negation. The relationship between these modalities in intuitionistic modal logic is a more delicate matter since negation is weaker. Intuitionistic non-necessity <code>[~]</code> and impossibility <code><~></code>, first investigated by Došen, have received less attention and — together with their positive counterparts <code>[]</code> and <code><></code> — form a square we call the Došen Square. Unfortunately, the core property of constructive logic, the Disjunction Property (DP), fails when the modalities are combined and, interpreted in birelational Kripke structures à la Došen, the Square partially collapses. We introduce the constructive logic CKD, whose four semantically independent modalities <code>[]</code>, <code><></code>, <code>[~]</code>, <code><~></code> prevent the Došen Square from collapsing under the effect of intuitionistic negation while preserving DP. The model theory of CKD involves a constructive Kripke frame interpretation of the modalities. A Hilbert deduction system and an equivalent cut-free sequent calculus are presented. Soundness, completeness and finite model property are proven, implying that CKD is decidable. The logics <code>HK[~]</code>, <code>HK[]</code>, <code>HK<></code> and <code>HK<~></code> of Došen and other known theories of intuitionistic modalities are syntactic fragments or axiomatic extensions of CKD.</p> </div> </main> </div> <footer> powered by <a href="https://github.com/jgm/yst">yst</a> </footer> </div> </body> </html>

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Working Group in the History and Philosophy of Logic, Mathematics, and Science - Home