×

Logics of imprecise comparative probability. (English) Zbl 1520.03002

Summary: This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability and comparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

MSC:

03B48 Probability and inductive logic
60A05 Axioms; other general questions in probability
68T27 Logic in artificial intelligence
68T37 Reasoning under uncertainty in the context of artificial intelligence
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Alon, Shiri; Heifetz, Aviad, The logic of Knightian games, Econ. Theory Bull., 2, 2, 161-182 (2014)
[2] Alon, Shiri; Lehrer, Ehud, Subjective multi-prior probability: a representation of a partial likelihood relation, J. Econ. Theory, 151, 476-492 (2014) · Zbl 1296.91080
[3] Augustin, Thomas; Coolen, Frank P. A.; De Cooman, Gert; Troffaes, Matthias C. M., Introduction to Imprecise Probabilities (2014), John Wiley & Sons · Zbl 1290.62003
[4] Bareinboim, Elias; Correa, Juan D.; Ibeling, Duligur; Icard, Thomas, On Pearl’s hierarchy and the foundations of causal inference (2020), Causal AI Lab, Columbia University, Technical Report R-60
[5] van Benthem, Johan, Logical Dynamics of Information and Interaction (2011), Cambridge University Press: Cambridge University Press New York · Zbl 1251.03003
[6] van Benthem, Johan; Klein, Dominik, Logics for analyzing games, (Zalta, Edward N., The Stanford Encyclopedia of Philosophy (2019))
[7] van Benthem, Johan; Gerbrandy, Jelle; Kooi, Barteld, Dynamic update with probabilities, Stud. Log., 93, 1, 67-96 (2009) · Zbl 1183.03015
[8] Boole, George, An Investigation of the Laws of Thought (1854), Walton & Maberly · Zbl 1205.03003
[9] Bradley, Seamus, Imprecise probabilities, (Zalta, Edward N., The Stanford Encyclopedia of Philosophy (2019)) · Zbl 1325.60004
[10] Bradley, Seamus; Steele, Katie, Uncertainty, learning, and the “problem” of dilation, Erkenntnis, 79, 1287-1303 (2014) · Zbl 1329.03035
[11] Carnap, Rudolf, Testability and meaning (part I), Philos. Sci., 3, 4, 419-471 (1936)
[12] Couso, Inés; Moral, Serafín, Sets of desirable gambles: conditioning, representation, and precise probabilities, Int. J. Approx. Reason., 52, 7, 1034-1055 (2011) · Zbl 1234.68374
[13] Dekel, Eddie; Siniscalchi, Marciano, Epistemic game theory, (Handbook of Game Theory with Economic Applications, vol. 4 (2015)), 619-702
[14] Diaconis, Persi, Review of “A mathematical theory of evidence” (G. Shafer), J. Am. Stat. Assoc., 73, 363, 677-678 (1978)
[15] Diaconis, Persi; Zabell, Sandy L., Some alternatives to Bayes’s rule, (Grofman, B.; Owen, G., Information Pooling and Group Decision Making (1986), J.A.I. Press), 25-38
[16] Nicholas DiBella, Qualitative probability and infinitesimal probability, Draft of 9/7/18, 2018. · Zbl 1436.60004
[17] Ding, Yifeng; Harrison-Trainor, Matthew; Holliday, Wesley H., The logic of comparative cardinality, J. Symb. Log. (2021), forthcoming · Zbl 1485.03064
[18] van Ditmarsch, Hans; van der Hoek, Wiebe; Kooi, Barteld, Dynamic Epistemic Logic (2008), Springer: Springer Dordrecht · Zbl 1156.03015
[19] Domotor, Zoltan, Probabilistic relational structures and their applications (1969), California Institute for Mathematical Studies in the Social Sciences, Technical Report No. 144 Psychology Series
[20] Elliott, Edward, ‘Ramseyfying’ probabilistic comparativism, Philos. Sci., 87, 4, 727-754 (2020)
[21] Eva, Benjamin, Principles of indifference, J. Philos., 116, 7, 390-411 (2019)
[22] Fagin, R.; Halpern, J. Y.; Megiddo, N., A logic for reasoning about probabilities, Inf. Comput., 87, 78-128 (1990) · Zbl 0811.03014
[23] Fine, Terrence L., Theories of Probability (1973), Academic Press: Academic Press New York · Zbl 0275.60006
[24] Fine, Terrence L., An argument for comparative probability, (Butts, R. E.; Hintikka, J., Basic Problems in Methodology and Linguistics (1977), Springer), 105-119
[25] de Finetti, Bruno, La ‘logica del plausible’ secondo la concezione di Polya, Atti della XLII Riunione, Societa Italiana per il Progresso delle Scienze, 227-236 (1949)
[26] Fishburn, Peter C., The axioms of subjective probability, Stat. Sci., 1, 3, 335-358 (1986) · Zbl 0604.60004
[27] Brandon Fitelson, David McCarthy, Toward an epistemic foundation for comparative confidence, Draft of 1/19/14, 2014.
[28] Gärdenfors, Peter, Qualitative probability as an intensional logic, J. Philos. Log., 4, 2, 171-185 (1975) · Zbl 0317.02030
[29] Gardner, Marvin, Mathematical games, Sci. Am., 180-182 (1959), October issue
[30] Gardner, Marvin, Mathematical games, Sci. Am., 188 (1959), November issue
[31] Giarlotta, Alfio; Greco, Salvatore, Necessary and possible preference structures, J. Math. Econ., 49, 163-172 (2013) · Zbl 1271.91045
[32] Good, I. J., Subjective probability as the measure of a non-measurable set, (Nagel, Ernest; Suppes, Patrick; Tarski, Alfred, Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress (1962)), 319-329 · Zbl 0192.02104
[33] Halpern, J. Y., An analysis of first-order logics of probability, Artif. Intell., 46, 311-350 (1990) · Zbl 0723.03007
[34] Halpern, Joseph Y., Reasoning About Uncertainty (2003), MIT Press: MIT Press Cambridge, Mass · Zbl 1090.68105
[35] Halpern, Joseph Y.; Moses, Yoram, A guide to completeness and complexity for modal logics of knowledge and belief, Artif. Intell., 54, 3, 319-379 (1992) · Zbl 0762.68029
[36] Harrison-Trainor, Matthew; Holliday, Wesley H.; Icard, Thomas F., A note on cancellation axioms for comparative probability, Theory Decis., 80, 1, 159-166 (2016) · Zbl 1378.91066
[37] Harrison-Trainor, Matthew; Holliday, Wesley H.; Icard, Thomas F., Preferential structures for comparative probabilistic reasoning, (Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (2017)), 1135-1141
[38] Harrison-Trainor, Matthew; Holliday, Wesley H.; Icard, Thomas F., Inferring probability comparisons, Math. Soc. Sci., 91, 62-70 (2018) · Zbl 1396.91083
[39] Hawthorne, James, A logic of comparative support: qualitative conditional probability relations representable by Popper functions, (Hájek, Alan; Hitchcock, Christopher, Oxford Handbook of Probability and Philosophy (2016), Oxford University Press)
[40] Wiebe van der Hoek, Qualitative modalities, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 4, 1, 45-59 (1996) · Zbl 1232.03009
[41] Holliday, Wesley H.; Icard, Thomas F., Measure semantics and qualitative semantics for epistemic modals, (Snider, Todd, Proceedings of Semantics and Linguistic Theory (SALT) vol. 23 (2013), LSA and CLC Publications)
[42] Holliday, Wesley H.; Icard, Thomas F., Axiomatization in the meaning sciences, (Ball, Derek; Rabern, Brian, The Science of Meaning: Essays on the Metatheory of Natural Language Semantics (2018), Oxford University Press: Oxford University Press Oxford), 73-97
[43] Holliday, Wesley H.; Hoshi, Tomohiro; Icard, Thomas F., A uniform logic of information dynamics, (Bolander, Thomas; Braüner, Torben; Ghilardi, Silvio; Moss, Lawrence, Advances in Modal Logic, vol. 9 (2012), College Publications: College Publications London), 348-367 · Zbl 1291.03025
[44] Holliday, Wesley H.; Hoshi, Tomohiro; Icard, Thomas F., Information dynamics and uniform substitution, Synthese, 190, 1, 31-55 (2013) · Zbl 1310.03027
[45] Ibeling, Duligur; Icard, Thomas, Probabilistic reasoning across the causal hierarchy, (Proceedings of the Thirty-Fourth AAAI Conference on Artificial Intelligence (2020))
[46] Icard, Thomas F., Pragmatic considerations on comparative probability, Philos. Sci., 83, 3, 348-370 (2016)
[47] Joyce, James M., How probabilities reflect evidence, Philos. Perspect., 19, 153-178 (2005)
[48] Keynes, John Maynard, A Treatise on Probability (1921), Macmillan · Zbl 0925.01037
[49] Konek, Jason, Comparative probabilities, (Pettigrew, Richard; Weisberg, Jonathan, The Open Handbook of Formal Epistemology (2019), The PhilPapers Foundation), 267-348
[50] Konek, Jason, Epistemic conservativity and imprecise credence, Philos. Phenomenol. Res. (2021), forthcoming
[51] Koopman, Bernard O., The axioms and algebra of intuitive probability, Ann. Math., 41, 2, 269-292 (1940) · Zbl 0024.05001
[52] Kraft, Charles H.; Pratt, John W.; Seidenberg, A., Intuitive probability on finite sets, Ann. Math. Stat., 30, 2, 408-419 (1959) · Zbl 0173.19606
[53] Lassiter, Daniel, Gradable epistemic modals, probability, and scale structure, (Li, N.; Lutz, D., Semantics and Linguistic Theory (SALT), vol. 20 (2010), CLC (Cornell Linguistics Circle)), 1-18
[54] Lehrer, Ehud; Teper, Roee, Justifiable preferences, J. Econ. Theory, 146, 2, 762-774 (2011) · Zbl 1282.91106
[55] Levi, Isaac, On indeterminate probabilities, J. Philos., 71, 391-418 (1974)
[56] Duncan Luce, R., On the numerical representation of qualitative conditional probability, Ann. Math. Stat., 39, 2, 481-491 (1968) · Zbl 0164.46302
[57] Lutz, Carsten, Complexity and succinctness of public announcement logic, (Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems (2006), ACM), 137-143
[58] Mierzewski, Krzysztof, Probabilistic stability: dynamics, nonmonotonic logics, and stable revision (2018), Universiteit van Amsterdam, Master’s thesis
[59] Moss, Sarah, Probabilistic Knowledge (2018), Oxford University Press: Oxford University Press Oxford
[60] Moss, Sarah, Global constraints on imprecise credences: solving reflection violations, belief inertia, and other puzzles, Philos. Phenomenol. Res. (2020)
[61] Motzkin, T. S., Two consequences of the transposition theorem on linear inequalities, Econometrica, 19, 2, 184-185 (1951) · Zbl 0042.01201
[62] Pearl, Judea, Causality (2009), Cambridge University Press · Zbl 1188.68291
[63] Rinard, Susanna, Against radical credal imprecision, Thought: J. Philos., 2, 157-165 (2013)
[64] Ríos Insua, D., On the foundations of decision making under partial information, Theory Decis., 33, 1, 83-100 (1992) · Zbl 0756.90004
[65] Russell, Stuart, Unifying logic and probability, Commun. ACM, 58, 7, 88-97 (2015)
[66] Schoenfield, Miriam, Chilling out on epistemic rationality: a defense of imprecise credences (and other imprecise doxastic attitudes), Philos. Stud., 158, 197-219 (2012)
[67] Scott, Dana, Measurement structures and linear inequalities, J. Math. Psychol., 1, 233-247 (1964) · Zbl 0129.12102
[68] Segerberg, Krister, Qualitative probability in a modal setting, (Fenstad, E., Second Scandinavian Logic Symposium (1971), North-Holland: North-Holland Amsterdam), 341-352 · Zbl 0223.02013
[69] Seidenfeld, Teddy; Schervish, Mark J.; Kadane, Joseph B., Forecasting with imprecise probabilities, Int. J. Approx. Reason., 53, 8, 1248-1261 (2012) · Zbl 1284.60082
[70] Selvin, Steve, On the Monty Hall problem, Am. Stat., 29, 3, 134 (1975)
[71] Suppes, Patrick, The measurement of belief, J. R. Stat. Soc., Ser. B, 36, 2, 160-191 (1974) · Zbl 0287.60007
[72] Suppes, Patrick; Zanotti, Mario, Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualitative probability ordering, J. Philos. Log., 5, 3, 431-438 (1976) · Zbl 0351.60005
[73] Suppes, Patrick; Zanotti, Mario, Necessary and sufficient qualitative axioms for conditional probability, Z. Wahrscheinlichkeitstheor. Verw. Geb., 60, 163-169 (1982) · Zbl 0468.60002
[74] vos Savant, Marilyn, Marilyn vos Savant’s reply, Am. Stat., 45, 4, 347 (1991)
[75] Walley, Peter, Statistical Reasoning with Imprecise Probabilities (1991), Chapman and Hall · Zbl 0732.62004
[76] Walley, Peter, Towards a unified theory of imprecise probability, Int. J. Approx. Reason., 24, 2-3, 125-148 (2000) · Zbl 1007.28015
[77] Weatherson, Brian, The Bayesian and the dogmatist, Proc. Aristot. Soc., 107, 169-185 (2007)
[78] Yalcin, Seth, Context probabilism, (Logic, Language and Meaning - 18th Amsterdam Colloquium, Amsterdam, the Netherlands, December 19-21, 2011, Revised Selected Papers (2011)), 12-21 · Zbl 1366.03207
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.