# What are the prerequisites for studying modal logic?

A book I'm currently reading briefly mentioned epistemic logic, but didn't say what it was. Since I've never heard of this logic I decided to look it up and saw that it was a type of modal logic. I have no experience with this logic either, but I'm curious to learn more about it. So before I go out and grab a book on this topic, I'd like some feedback on what I should already know before tackling this subject. I have some experience with first-order logic and I am a mathematics student at the undergraduate level. Your help is greatly appreciated.

An understanding of classical propositional logic and first-order logic should suffice. Some notes:

1. Modal languages look very much like non-modal ones. For example, if you have a non-modal propositional language generated by the following grammar:

(L1) φ := p | ¬φ | (φ ∧ φ),

you can obtain a modal propositional language by adding a new clause for the modal operator □:

(L2) φ := p | ¬φ | (φ ∧ φ) | □φ.

Based on this necessity operator □ you can then define the possibility operator ◇φ as ¬□¬φ, etc.. There are more complex modal languages, of course (e.g. the language of quantified modal logic).

2. Formulas of modal languages, like formulas of non-modal ones, are defined inductively, so if you have some familiarity with compositional definitions and mathematical induction, you can define useful metrics for modal formulas (e.g. modal depth) that can help you prove various things about all modal formulas in a given language (e.g. unique readability/parsing).

3. Modal models are graph-like structures called relational or Kripke models. They're structures consisting of a set of worlds W, a binary accessibility relation on W, and a valuation V from propositional letters and worlds to truth-values. The key difference between non-modal and modal semantics is that in modal semantics the truth of a formula is world-relative, so while we specify:

(M1) M |= p iff V(p) = 1;

as the truth-conditions of 'p' in a non-modal propositional logic, in a modal one we say instead:

(M2) M, w |= p iff V(p, w) = 1.

There are lots of other points of similarity and contrast between modal and non-modal languages. Familiarity with graphs, trees and induction would be helpful, but since you have some experience with first-order logic, I would recommend that you just jump into the study of modal logic without preparing for it. (Keep Enderton on the side if you come across topics you haven't seen before.)

van Benthem, J. (2010) Modal Logic for Open Minds, Stanford, CSLI Lecture Notes #199.
Blackburn, P., de Rijke, M., Venema, Y. (2002) Modal Logic, Cambridge, CTTCS #53.
Holliday, W.H. (2012) Modal Reasoning, Lecture Course (Spring), UC Berkeley.

• NB: That Van Benthem book is intended for advanced undergraduates/beginning graduates. – user3164 May 8 '14 at 4:52
• This is an excellent answer, I want to suggest one more book though: Fitting and Mendelsohn's First-Order Modal Logic is an excellent text, geared at philosophers. It's a bit more advanced than some of the other modal logic textbooks (like Hughes and Creswell). But it will give you the full apparatus you need for reading contemporary papers in metaphysics, for instance. One question: Does anybody know a good textbook specifically on Epistemic Logic? That seems most relevant to the OP's question, and I'm also curious. – user5172 May 8 '14 at 11:51
• A good textbook is: Meyer, J.-J.Ch and Hoek, W. van der. Epistemic Logic for AI and Computer Science. Cambridge, 1995. – sequitur May 8 '14 at 13:56
• You can see also in SEP the entry on Epistemic logic with biblio, and the article on Epistemic logic into Dov Gabbay (editor), Handbook of the History of Logic. Vol 7 : Logic and the modalities in the twentieth century (2006). – Mauro ALLEGRANZA May 8 '14 at 16:49
• And of course Jaakko Hintikka, Knowledge and Belief (1962). – Mauro ALLEGRANZA May 8 '14 at 16:53