nLab possible worlds semantics

Idea

In the context of modal logic, by possible world semantics one means the interpretation of the possibility and necessity modalities as existential quantification and universal quantification over a context of “possible worlds”. Such a model is sometimes referred to as a Kripke frame.

For formalization of this in terms of hyperdoctrines/dependent type theory see at necessity and possibility – In first-order logic and dependent type theory.

Under categorical semantics such a context interprets as an object of some category whose interpretation is similar to that of a probability space in measure theory – which is one sensible formalization of possible worlds (e.g. Toronto-McCarthy 10, slide 23).

Related concepts

• Kripke frame

• probability space

• many-worlds interpretation of quantum mechanics

• multiverse

References

General

The original articles:

• Saul A. Kripke, A Completeness Theorem in Modal Logic, The Journal of Symbolic Logic 24 1 (1959) 1-14 [doi:10.2307/2964568, jstor:2964568, pdf]

• Saul A. Kripke, Semantical Considerations on Modal Logic, Acta Philosophical Fennica 16 (1963) 83-94 [pdf]

• Saul A. Kripke, Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi, Mathematical Logic Quaterly 9 5-6 (1963) 67-96 [doi:10.1002/malq.19630090502]

• Saul A. Kripke, Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi, in The Theory of Models (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [doi:10.1016/B978-0-7204-2233-7.50026-5]

Modern exposition:

• Patrick Blackburn, Johan van Benthem, Modal logic: a semantic perspective, Ch 1 in: The Handbook of Modal Logic, Studies in Logic and Practical Reasoning 3 (2007) [doi:10.1016/S1570-2464(07)80004-8, book webpage]

In general see the reference at modal logic, such as:

• Raymond D. Bradley, Norman Swartz, Possible Worlds – An Introduction to Logic and its Philosophy, Hackett Publishing (1979) [webpage, pdf]

• Rod Girle, Possible Worlds, McGill-Queen’s University Press (2003) [jstor:j.cttq48cx]

• Johan van Benthem, §2.2 of: Modal Logic for Open Minds (2010) [ISBN:9781575865980,pdf, webpage]

• Daniel Nolan, Topics in the Philosophy of Possible Worlds, Routledge (2002) [ISBN:9780415516303]

• Valentin Goranko, Martin Otto, Model Theory of Modal Logic, Section 5 in: Handbook of Modal Logic, Studies in Logic and Practical Reasoning 3 (2007) 249-329 [pdf, book webpage, publisher page]

and more philosophically (“modal realism”):

• David Lewis, On the Plurality of Worlds, Wiley (1986, 2001) [ISBN:978-0-631-22426-6, Wikipedia entry]

See also:

• Wikipedia, Possible world

• Stanford Encyclopedia of Philosophy, Possible Worlds

• Stanford Encyclopedia of Philosophy, Modal Logic – Possible worlds semantics

• Neil Toronto, Jay McCarthy, From Bayesian Notation to Pure Racket, 2010 (pdf)

Possible-worlds vs. Many-worlds

References which consider, in one way or another, the notions of

• possible worlds in epistemic modal logic

• many worlds in interpretations of quantum mechanics:

in relation to each other:

• Mario Bunge, Possibility and Probability, in: Foundations of Probability Theory, Statistical Interference, and Statistical Theories of Science, Reidel (1976) 17-34 $[$doi:10.1007/978-94-010-1438-0_2$]$

• Brian Skyrms, Part III of: Possible Worlds, Physics and Metaphysics, Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition 30 5 (1976) 323-332 $[$jstor:4319099$]$

• Paul Tappenden, p. 101 (4 of 17) in: Identity and Probability in Everett’s Multiverse, The British Journal for the Philosophy of Science, 51 1 (2000) 99-114 $[$jstor:3541750$]$

• Daniel Nolan, p. 22 of: Topics in the Philosophy of Possible Worlds, Routledge (2002) $[$ISBN:9780415516303$]$

• Rod Girle, Ch. 8 of: Possible Worlds, McGill-Queen’s University Press (2003) $[$jstor:j.cttq48cx$]$

• Simon Saunders, p. 196 in: Chance in the Everett Interpretation, in: Many Worlds?, Oxford University Press (2010) 181–205 $[$doi:10.1093/acprof:oso/9780199560561.003.0008$]$

• Nuriya Nurgalieva, Lídia del Rio, Inadequacy of Modal Logic in Quantum Settings, EPTCS 287 (2019) 267-297 $[$arXiv:1804.01106, doi:10.4204/EPTCS.287.16$]$

• Alastair Wilson, p. 20 of: Modal Metaphysics and the Everett Interpretation (2006) $[$philsci:2635, pdf$]$

• Vladislav Terekhovich, Modal Approaches in Metaphysics and Quantum Mechanics $[$arXiv:1909.10046$]$

• Alastair Wilson, The Nature of Contingency: Quantum Physics as Modal Realism, Oxford University Press (2020) $[$ISBN:9780198846215$]$

• Raoni W. Arroyo, Jonnas R. B. Arenhart, Whence deep realism for Everettian quantum mechanics?, Foundations of Physics 52 121 (2022) [arXiv:2210.16713, doi:10.1007/s10701-022-00643-0]

Beware that there is also

• Bas C. van Fraassen, Modal interpretation of repeated measurement, Philosophy of Science 64 4 (1997) 669-676 $[$doi:10.1086/392577, SEP review$]$

which, even if some vocabulary is superficially alike, does not refer either to modal logic nor to the many-worlds interpretation.