Basic Concepts in Modal Logic

These lecture notes were composed while teaching a class at Stanford and studying the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Companion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text influenced me the most, though the order of presentation is inspired more by Goldblatt.2 My goal was to write a text for dedicated undergraduates with no previous experience in modal logic. The text had to meet the following desiderata: (1) the level of difficulty should depend on how much the student tries to prove on his or her own—it should be an easy text for those who look up all the proofs in the appendix, yet more difficult for those who try to prove everything themselves; (2) philosophers (i.e., colleagues) with a basic training in logic should be able to work through the text on their own; (3) graduate students should find it useful in preparing for a graduate course in modal logic; (4) the text should prepare people for reading advanced texts in modal logic, such as Goldblatt, Chellas, Hughes and Cresswell, and van Benthem, and in particular, it should help the student to see what motivated the choices in these texts; (5) it should link the two conceptions of logic, namely, the conception of a logic as an axiom system (in which the set of theorems is constructed from the bottom up through proof sequences) and the conception of a logic as a set containing initial ‘axioms’ and closed under ‘rules of inference’ (in which the set of theorems is constructed from the top down, by carving out the logic from the set of all formulas as the smallest set closed under the rules); finally, (6) the pace for the presentation of the completeness theorems should be moderate—the text should be intermediate between Goldblatt and Chellas in this regard (in Goldblatt, the completeness proofs come too quickly for the undergraduate, whereas in Chellas, too many unrelated....
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