Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-dxfhg Total loading time: 0.273 Render date: 2021-03-06T15:44:17.063Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

MODEL THEORY OF MEASURE SPACES AND PROBABILITY LOGIC

Published online by Cambridge University Press:  08 April 2013

RUTGER KUYPER
Affiliation:
Department of Mathematics, Radboud University Nijmegen
SEBASTIAAN A. TERWIJN
Affiliation:
Department of Mathematics, Radboud University Nijmegen
Corresponding

Abstract

We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak models.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2013 

Access options

Get access to the full version of this content by using one of the access options below.

References

Aliprantis, C. D., & Border, K. C. (2006). Infinite-dimensional Analysis: A Hitchhiker’s Guide. Berlin, Germany: Springer.Google Scholar
Bageri, S.-M., & Pourmahdian, M. (2009). The logic of integration. Archive for Mathematical Logic, 48, 465492.CrossRefGoogle Scholar
Birkhoff, G. (1948). Lattice Theory (revised edition). Providence, RI: American Mathematical Society.Google Scholar
Bogachev, V. I. (2007). Measure Theory. Berlin, Germany: Springer.CrossRefGoogle Scholar
Carnap, R. (1945). On inductive logic. Philosophy of Science, 12, 7297.CrossRefGoogle Scholar
Fremlin, D. H. (1984). Consequences of Martin’s Axiom. Cambridge University Press.CrossRefGoogle Scholar
Gaifman, H. (1964). Concerning measures in first order calculi. Israel Journal of Mathematics, 2, 118.CrossRefGoogle Scholar
Halpern, J. Y. (1989). An analysis of first-order logics of probability. In: Sridharan, N. S., editor. Proceedings of 11th International Joint Conference on Artificial Intelligence (IJCAI-89). San Francisco, CA: Morgan Kaufmann Publishers, Inc. pp. 13751381.Google Scholar
Hansson, H., & Jonsson, B. (1994). A logic for reasoning about time and reliability. Formal Aspects of Computing, 6, 512535.CrossRefGoogle Scholar
Hodges, W. (1993). Model Theory. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Howson, C. (2009). Can logic be combined with probability? Probably. Journal of Applied Logic, 7, 177187.CrossRefGoogle Scholar
Jaeger, M. (2005). A logic for inductive probabilistic reasoning. Synthese, 144, 181248.CrossRefGoogle Scholar
Kearns, M. J., & Vazirani, U. V. (1994). An Introduction to Computational Learning Theory. Cambridge, MA: MIT Press.Google Scholar
Kechris, A. S. (1995). Classical Descriptive Set Theory. New York, NY: Springer.CrossRefGoogle Scholar
Keisler, H. J. (1985). Probability quantifiers. In Barwise, J., and Feferman, S., editors. Model-Theoretic Logics. New York, NY: Springer-Verlag, pp. 509556.Google Scholar
Kunen, K. (1983). Set Theory: An Introduction to Independence Proofs. Amsterdam, the Netherlands: North-Holland.Google Scholar
Loeb, P. A. (1975). Conversion from nonstandard to standard measure spaces and applications in probability theory. Transactions of the American Mathematical Society, 211, 113122.CrossRefGoogle Scholar
Scott, D., & Krauss, P. (1966). Assigning probabilities to logical formulas. In Hintikka, J., and Suppes, P., editors. Aspects of Inductive Logic. Amsterdam, the Netherlands: North Holland, pp. 219264.CrossRefGoogle Scholar
Steinhorn, C. I. (1985). Borel structures and measure and category logics. In Barwise, J., and Feferman, S., editors. Model Theoretic Logics. New York, NY: Springer-Verlag, pp. 579596.Google Scholar
Terwijn, S. A. (2005). Probabilistic logic and induction. Journal of Logic and Computation, 15, 507515.CrossRefGoogle Scholar
Terwijn, S. A. (2009). Decidability and undecidability in probability logic. In: Nerode, A., and Artemov, S., editors. Proceedings of Logical Foundations of Computer Science. Lecture Notes in Computer Science 5407. Berlin, Germany: Springer, pp. 441450.Google Scholar
Valiant, L. G. (2000). Robust logics. Artificial Intelligence, 117, 231253.CrossRefGoogle Scholar
Väänänen, J. (2008). The Craig interpolation theorem in abstract model theory. Synthese, 164, 401420.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 58 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 6th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

MODEL THEORY OF MEASURE SPACES AND PROBABILITY LOGIC
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

MODEL THEORY OF MEASURE SPACES AND PROBABILITY LOGIC
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

MODEL THEORY OF MEASURE SPACES AND PROBABILITY LOGIC
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *