Abstract
We present a logical system sound and complete with respect to the class of conditional probability spaces. The universal modality has an important role in our language: conditional probabilities have a peculiar behavior when conditionalized over empty events, so we use the universal modality to express nonemptiness of events. We add public announcement to our language in a way that we are able to express sentences like: “the agent believes the probability of p is zero and after the announcement of p he/she believes the probability of p is greater than zero.” Because of the Conditional Probability Language we are able to express reduction axioms for the probability Public Announcement Logic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that an agent believes with probability 1 that a formula is false is equivalent to the the agent beliefs in the proposition with probability zero.
- 2.
A probabilistic function that sums up at most to 1.
- 3.
We abbreviate \(\lnot \Box \lnot A\) by \(\Diamond A\).
- 4.
A function \(\eta \) is \(\sigma \)-additive if for a family of pairwise disjoint sets \((A_i)_{i \in \mathbb {N}}\) in \( \Sigma \) holds that \(\eta (\bigcup _{i\in \mathbb {N}}A_i) = \sum _{i = 0}^{\infty }\eta (A_i) \), for \(i \ne j\).
- 5.
\(\mathbb {Q}\) is the set of rational numbers.
- 6.
\(T(w, A \cap B | C) = T(w,A\mid B\cap C) \cdot T(w, B\mid C)\).
- 7.
Since \(r_1>r_2\) implies \((1-r_2) + r_1 > 1\), an instance of (A5) is \(L_{r_1}(\phi ||\psi ) \rightarrow \lnot L_{1-r_2}(\lnot \phi ||\psi ).\)
- 8.
A logic is decidable if there is an algorithm that given any formula of the language decides whether it is a theorem.
References
R. J. Aumann, ‘Interactive Epistemology II: Probability’, International Journal of Game Theory 28, 301–314, 1999.
A. Baltag and S. Smets, ‘Probabilistic Dynamic Belief Revision’, TARK 2007.
P. Battigalli and M. Siniscalchi, ‘Hierarchies of conditional beliefs and interactive epistemology in dynamic games’, Journal of Economic Theory 88:188–230, 1999.
O. Board, ‘Dynamic interactive epistemology’, Games and Economic Behavior 49–80, 2004.
M.J. Cresswell and G.E. Hughes, ‘A New Introduction to Modal Logic’, Routledge, 1996.
R. Fagin and J.Y. Halpern, ‘Reasoning about knowledge and probability’, Journal of the Association for computing machinery vol 41 N 2, 340–367, 1994.
R. Fagin, J.Y. Halpern and R. Megiddo, ‘A Logic for Reasoning about Probabilities’, Information and Computation, 87 78–128, 1990.
P.R. Halmos, ‘Measure theory’, Springer-Verlag New York-Heidelberg-Berlin, 1950.
J.C. Harsanyi, ‘Games with incomplete information played by Bayesian players’, parts I–III, Management Sci. 14, 159–182, 320–334, 486–502, 1967–1968.
A. Heifetz and P. Mongin, ‘Probability Logic for Type Spaces’, Games and Economic Behaviour 35, 31–53, 2001.
B. P. Kooi, ‘Probabilistic Epistemic Logic’, Journal of Logic, Language and Information 12: 381–408, 2003.
R.B. Myerson, ‘Multistage games with communication’, Econometrica 54:323–358, 1985.
A. Renyi, ‘On a new axiomatic theory of probability’, 1955.
J. Sack, ‘Extending Probability Dynamic Epistemic Logic’, Synthese 169, 241–257, 2009.
R. Stalnaker, ‘Extensive and strategic forms: Games and models for games’, Research in Economics 53, 293–319, 1999.
J. van Benthem, ‘Conditional Probability meets Update Logic’, Journal of Logic, Language and Information 12, 409–421, 2003.
J. van Benthem, ‘Modal Logic for Open Minds’, CSLI, 2010.
W. van der Hoek and M. Pauly, ‘Modal logic for games and information’, in Handbook of Modal Logic, 2006.
H. van Ditmarsch, W. van der Hoek and B. Kooi, ‘Dynamic Epistemic Logic’, Synthese Library, vol. 337, 2007.
Weatherson, B, ‘Stalnaker on sleeping beauty’, Philosophical studies. 155 issue 3, 445–456, 2011.
C. Zhou, ‘A complete deductive System for probability logic’, Journal of Logic and Computation, 19 issue 6, 1427–1454, 2009.
C. Zhou, ‘Complete Deductive Systems for Probability Logic with Application in Harsanyi Type Spaces’. PhD. Thesis, 2007.
C. Zhou, ‘Probability for Harsanyi Type Space,’ Logical Methods in Computer Science, 2014.
Acknowledgments
I am grateful to my Ph.D. advisor Makoto Kanazawa for his helpful comments and ideas to simplify some of the proofs in many ways.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hernandes, M.S.C. (2015). Conditional Probability Logic over Conditional Probability Spaces. In: Ju, S., Liu, H., Ono, H. (eds) Modality, Semantics and Interpretations. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47197-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-47197-5_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-47196-8
Online ISBN: 978-3-662-47197-5
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)