Carnap’s Problem for Modal Logic

Review of Symbolic Logic 16 (2):578-602 (2023)
  Copy   BIBTEX

Abstract

We take Carnap’s problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap’s problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to be interpreted as in relational Kripke semantics. Except when restricted to finite domains, no modal logic characterizes all the Kripkean interpretation of P. Moreover, we show that, in contrast with the case of first-order logic, the obvious requirement of permutation invariance is not adequate in the modal case. After pointing out some known facts about modal logics that nevertheless force the Kripkean interpretation, we focus on another feature often taken to embody the gist of modal logic: locality. We show that invariance under point-generated subframes (properly defined) does single out the Kripkean interpretations, but only among topological interpretations, not in general. Finally, we define a notion of bisimulation invariance — another aspect of locality — that, together with a reasonable closure condition, gives the desired general result. Along the way, we propose a new perspective on normal neighborhood frames as filter frames, consisting of a set of worlds equipped with an accessibility relation, and a free filter at every world.

Other Versions

No versions found

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 101,551

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Analytics

Added to PP
2021-03-08

Downloads
81 (#259,617)

6 months
7 (#722,178)

Historical graph of downloads
How can I increase my downloads?

Author Profiles

Denis Bonnay
Université Paris Nanterre
Dag Westerståhl
Stockholm University

Citations of this work

No citations found.

Add more citations

References found in this work

Some embedding theorems for modal logic.David Makinson - 1971 - Notre Dame Journal of Formal Logic 12 (2):252-254.
Logical constants across varying types.Johan van Benthem - 1989 - Notre Dame Journal of Formal Logic 30 (3):315-342.
Complete additivity and modal incompleteness.Wesley H. Holliday & Tadeusz Litak - 2019 - Review of Symbolic Logic 12 (3):487-535.

View all 9 references / Add more references