Modal and Justification Logic

Modal logic is the study of reasoning that involves various modalities.
  • Garson, James, Modal Logic, The Stanford Encyclopedia of Philosophy (Winter 2009 Edition) (Edward N. Zalta - editor)
  • Blackburn, P., Rijke, M. de, and Venema, Y., 2001, Modal Logic, Cambridge: Cambridge University Press
  • Chellas, Brian, 1980, Modal Logic: An Introduction, Cambridge: Cambridge University Press
  • Fitting, M. and Mendelsohn, R., 1998, First Order Modal Logic, Dordrecht: Kluwer.
  • Hughes, G. and Cresswell, M., 1996, A New Introduction to Modal Logic, London: Routledge.
It goes as far back as Aristotle, although the modern treatment of modal logic is usually attributed to C. I. Lewis's work at the beginning of 20th century, and encompasses subareas as diverse as alethic logic with modalities for necessity and possibility, deontic logic with modalities for obligations and permissions,
  • McNamara, Paul, Deontic Logic, The Stanford Encyclopedia of Philosophy (Fall 2010 Edition) (Edward N. Zalta - editor)
doxastic logic with modalities for beliefs, epistemic logic with modalities for knowledge,
  • Hendricks, Vincent and Symons, John, Epistemic Logic, The Stanford Encyclopedia of Philosophy (Spring 2009 Edition) (Edward N. Zalta - editor)
  • Hintikka, J., 1962, Knowledge and Belief: An Introduction to the Logic of the Two Notions, Ithaca, NY: Cornell University Press.
  • Meyer, J.-J.Ch and Hoek, W. van der, 1995, Epistemic Logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science 41, Cambridge: Cambridge University Press.
  • Fagin, R., Halpern, J. Y., Moses Y. and Vardi, M. Y., 1995, Reasoning about Knowledge, Cambridge: MIT Press.
temporal logic with time modalities such as in the nearest future,
  • Galton, Antony, Temporal Logic, The Stanford Encyclopedia of Philosophy (Fall 2008 Edition) (Edward N. Zalta - editor).
and provability logic with modalities for provability in a formal system
  • Verbrugge, Rineke (L.C.), Provability Logic, The Stanford Encyclopedia of Philosophy (Winter 2010 Edition) (Edward N. Zalta - editor)
  • Boolos, G., 1993, The Logic of Provability, Cambridge: Cambridge University Press.
Modal logics can be propositional or first-order, may involve one or several modalities; in the latter case, the modalities may be interdependent. In doxastic and epistemic logic, one may add the common belief/knowledge modality. Further, modal language can be extended by other interesting constructs such as public announcements or, more generally, action models in dynamic epistemic logic
  • Ditmarsch, H.P. van, van der Hoek, W., and Kooi, B., 2006, Dynamic Epistemic Logic, vol. 337 of Synthese Library, Dordrecht: Springer.
nominals in hybrid logic
  • Braüner, Torben, Hybrid Logic, The Stanford Encyclopedia of Philosophy (Winter 2011 Edition) (Edward N. Zalta - editor).
fixed-point operators in the modal mu-calculus
  • Julian Bradfield and Colin Stirling, Modal $\mu$-calculi, in Handbook of Modal Logic (P. Blackburn et al. - editors), Elsevier, Amsterdam 2006: pp.721-756.
Justification logic is a refinement of modal language that replaces each modality with a family of justification terms. It was introduced by S. Artemov as a refinement of provability logic with terms representing individual proofs.
  • Artemov, Sergei and Fitting, Melvin, Justification Logic, The Stanford Encyclopedia of Philosophy (Fall 2012 Edition) (Edward N. Zalta - editor),
  • Artemov, S., 1995, Operational modal logic, Technical Report MSI 95–29,Cornell University.
  • Artemov, S., 2001, Explicit provability and constructive semantics, The Bulletin of Symbolic Logic, 7(1): pp. 1–36.
The original aim was to solve a problem posed by Gödel of providing intuitionistic logic and modal logic S4 with a classical provability semantics. A similar proposal for solving this problem can be found in a lecture by Gödel that long remained unpublished:
  • Gödel, K. 1938. Vortrag bei Zilsel/Lecture at Zilsel's (*1938a), in Unpublished Essays and Lectures (Kurt Gödel Collected Works: Volume III) (S. Feferman, J. J. Dawson, W. Goldfarb, C. Parsons and R. Solovay - editors), Oxford: Oxford University Press, 1995, pp. 86–113.
The machinery of justification logic can be applied to other types of modal logics if justification terms are interpreted differently. For instance, doxastic/epistemic modalities can be replaced with justification terms understood as justifications for believing/knowing a statement. This provides a formal treatment of ``justified true belief'' from the famous definition of knowledge that can be found in two of Plato's dialogs and that has received much attention after Gettier's examples. Justification logic enables one to analyze these examples and other epistemic puzzles within the object language:
  • Artemov, S., 2008, The logic of justification, The Review of Symbolic Logic, 1(4): pp. 477–513.
Justification logic suggests a new approach to the logical omniscience problem:
  • Artemov, S. and R. Kuznets 2009, Logical omniscience as a computational complexity problem, in Theoretical Aspects of Rationality and Knowledge (A. Heifetz - editor), Proceedings of the Twelfth Conference (TARK 2009), ACM Publishers, pp. 14–23.
It enables one to study self-referentiality of justifications/proofs in modal reasoning.
  • Kuznets, R., 2010, Self-referential justifications in epistemic logic, Theory of Computing Systems, 46 (4): pp. 636–661.
Our main interest lies in extending the machinery of justification logics to new types of modalities. In particular, we are interested in the common knowledge modality and fixed-point operators, as well as in evidential treatment of dynamic epistemic logic, i.e., the formal analysis of reasons for belief change within the object language.