Assertion, Denial and Non-classical Theories

In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naïve theories of classes and truth. 1 assertion, denial & sequents Friends of truth-value gaps and truth-value gluts both must distinguish the assertion of a negation (asserting p¬pq) and denial (denying ppq). If you take there to be a truth-value glut at ppq the appropriate claim to make (when asked) is to assert p¬pq without thereby denying ppq. If you take there to be a truthvalue gap at ppq the appropriate claim to make (when asked) is to deny ppq without thereby asserting p¬pq. Comments from readers on this paper are very welcome. Please check the webpage http: //consequently.org/writing/adnct for the latest version of the paper, to post comments and to read comments left by others. { Thanks to the Logic Seminar at the University of Melbourne and the audience at wcp4 (including Jc Beall, Patrick Girard and Jerry Seligman), and to Michael De, for comments on this material, and to an anonymous referee, who helpfully pressed on certain points, and asked me to write up the little proof (in footnote 22) that there are at most two truth values. { This research is supported by the Australian Research Council, through grant dp0343388, and Max Richter’s 24 Postcards in Full Colour.