Professor Jeff B. Paris
- Room 1-119, Alan Turing Building
- School of Mathematics,
- University of Manchester
- Oxford Road, Manchester M13 9PL, UK
- Jeff.Paris[at]manchester.ac.uk
- Tel: +44 (0) 161 275 5880
- Fax: +44 (0) 161 275 5819
Page contents:
Research interests
Mathematical Logic, in particular uncertain reasoning and inductive logic with an emphasis on rationality and common sense principles.
Writings
Books
- with A. Vencovsk\'{a}, Pure Inductive Logic, in the Association of Symbolic Logic Series Perspectives in Mathematical Logic , Cambridge University Press, April 2015, pp354. ISBN-10: 1107042305.
- The Uncertain Reasoner's Companion, - a mathematical perspective, Cambridge Tracts in Theoretical Computer Science 39, Cambridge University Press, 1994, pp212 ISBN 0-521-46089-1
- A Short Course in Predicate Logic, 2014. Available at http://bookboon.com/en/a-short-course-in-predicate-logic-ebook
Unpublished Papers
- with A. Vencovska. "An inequality concerning the expected values of row-sum and column-sum products in Boolean matrices"
- with E. Howarth. "A proof that FVP$_n$ does not imply FVP$_{n+1}$"
- with A. Vencovská. "Supplement to `An examination of the SEP Candidate Analogical Inference Rule within Pure Inductive Logic'"
- "A note on $*$-conditioning"
- with A. Vencovská. "Reichenbach's Axiom -- For the record"
- with A. Vencovská. "Postscript to "Symmetry's End?''"
- "A note on Gaifman's Condition"
- with O.W.Barnett "A proof of the base cases of the Emergence of Reason's Conjecture for $CM^{\infty}$", Manchester Centre for Pure Mathematics}, preprint number 2003/6.
- with A.Vencovská and M.Wafy. "Some limit theorems for $ME,MD$ and $CM^{\infty}$", Manchester Centre for Pure Mathematics, preprint number 2001/9, ISSN 1472-9210, 2001.
- with C.Dimitracopoulos, A.Vencovská and G.M.Wilmer. "A multivariate probability distribution based on the propositional calculus", Manchester Centre for Pure Mathematics}, preprint number 1999/6.
Published Papers
- with E.Howarth, The Theory of Spectrum Exchangeability, Review of Symbolic Logic, vol. 8(01) pp108-130, March 2015.
- with A.Vencovská, The Twin Continua of Inductive Methods'',, in Logic without Borders, eds. A.Hirvonen, J.Kontinen, R.Kossak, A.Villaveces, Ontos Mathematical Logic, 2015, pp355-366, 2015.
- with M.Kliess, Predicate Exchangeability and Language Invariance in Pure Inductive Logic, Logique et Analyse vol.228 pp413-540, December 2014.
- with M.S.Kliess, Second Order Inductive Logic and Wilmer's Principle, Journal of Applied Logic, vol.12(4) pp462-476, December 2014.
- with E.Howarth, Principles of Remembering and Forgetting, Logique et Analyse vol.228 pp489-511, December 2014. li>
- What you see is what you get, Entropy, vol.16, pp6186-6194, November 2014
- An Observation on Carnap's Continuum and Stochastic Independencies, Journal of Applied Logic, vol.11(4) pp 421-429, December 2013
- with A.Hill, The Counterpart Principle of Analogical Support by Structural Similarity, Erkenntnis, On line first, October 2013
- with A.Hill, An Analogy Principle in Inductive Logic, Annals of Pure and Applied Logic, vol.64, pp 1293-1321, 2013
- with A.Hill, Reasoning by Analogy in Inductive Logic, The Logica Yearbook 2011, eds. M.Pelis and V.Puncochar, College Publications, London, 2012.
- with Z.Adamovicz & L.A.Kolodziejczyk, Truth Definitions without Exponentiation and the \Sigma_1 Collection Scheme, Journal of Symbolic Logic, 77(2):649-655, June 2012
- with A.Vencovská, Symmetry in Polyadic Inductive Logic, Journal of Logic, Language and Information, 2012 21:189–216.
- with J.Landes & A.Vencovská, A survey of some recent results on Spectrum Exchangeability in Polyadic Inductive Logic, Synthese, 2011, 181 (Supplement 1) pp19-47
- Pure Inductive Logic, in The Continuum Companion to Philosophical Logic, Eds. L.Horsten & R.Pettigrew, Continuum International Publishing Group, London, 2011, pp428-449
- with A. Vencovská, A Note on Irrelevance in Inductive Logic, Journal of Philosophical Logic, 2011, 40(3):357-370.
- with A. Vencovská, Symmetry's End? Erkenntnis, 74(1):53-67, 2011.
- with A. Vencovská, A Note on Nathanial's Invariance Principle in Polyadic Inductive Logic, in Logic and its Applications, ICLA, Proceedings of the 4th Indian Logic Conference, Dehli, India, January 5-11, 2011, eds. M.Banerjee & A.Seth. pp137-146.
- with S.R.Rad, A note on the least informative model of a theory, in Programs, Proofs, Processes, CiE 2010, Eds. F. Ferreira, B. Löwe, E. Mayordomo, & L. Mendes Gomes, Sprnger LNCS 6158, pp342-351, 2010.
- with J. Landes and A. Vencovská, A Characterization of the Language Invariant Families satisfying Spectrum Exchangeability in Polyadic Inductive Logic, Annals of Pure and Applied Logic, 2010, 161:800-811. DOI: 10.1016/j.apal.2009.06.010
- with J. Landes and A. Vencovská, The Principle of Conformity and Spectrum Exchangeability, in Foundations of the Formal Sciences VI, Probabilistic Reasoning and Reasoning with Probabilities, Eds. B.Löwe, E.Pacuit, J-W.Romeijn, Studies in Logic Volume 16, College Publications, UK, 2009, pp81-90.
- with D. Picado-Muino and Mike Rosefield, Inconsistency as qualified truth: A probability logic approach, International Journal of Approximate Reasoning, 50:1151-1163, 2009.
- with A. Vencovská, A Generalization of Muirhead's Inequality, Journal of Mathematical Inequalities, 3(2), 2009, pp181-187.
- with A. Vencovská, A General Representation Theorem for Probability Functions Satisfying Spectrum Exchangeability, in CiE 2009, eds. K.Ambros-Spies, B.Lowe and W.Merkle. Springer LNCS 5635, pp379-388.
- with R. Simmonds, O is not enough, Review of Symbolic Logic, 2(02), 2009, pp298-309
- with P. Waterhouse, Atom Exchangeability and Instantial Relevance, Journal of Philosophical Logic,38(3), 2009, pp313-332.
- with J. Landes and A. Vencovská, Some aspects of polyadic inductive logic, Studia Logica, 90, 2008, pp3-16.
- with J.Landes and A. Vencovská, Instantial Relevance in Polyadic Inductive Logic, Proceedings of the 3nd India Logic Conference, ICLA 2009, eds. R.Ramanujam and Sundar Sarukkai, Chennai, India, 2009. Springer LNAI 5378, pp162-169.
- with S.R.Rad, Inference Processes for Quantified Predicate Knowledge, in Logic, Language, Information and Computation, WoLLIC, Edinburgh, 2008, Eds. W.Hodges and R. de Queiroz, Springer LNAI, 5110, pp249-259.
- with A. Sirokofskich, On $LP$-models of arithmetic, Journal of Symbolic Logic, 2008, 73(1), pp212-226.
- with D. Picado-Muino and M. Rosefield, Information from inconsistent knowledge: A Probability Logic approach, in Interval/Probabilistic Uncertainty and Non-Classical Logics, Advances in Soft Computing, 46,eds. Van-Nam Huynh et al, Springer, 2008, pp291-307.
- with O.W. Barnett, Maximum Entropy Inference with Quantified Knowledge, Logic Journal of the IGPL, 16(1), 2008, pp85-98.
- with J. Landes and A. Vencovská, Language Invariance and Spectrum Exchangeability in Inductive Logic, Symbolic and Quantitative Approaches to Reasoning with Uncertainty, Proceedings of the 9th European Conference, ECSQARU 2007, Hammamet, Tunisia. Springer LNAI 4724, pp151-160.
- with C.J. Nix, A note on binary inductive logic, Journal of Philosophical Logic, 36(6), 2007, pp735-771
- with N. Pathmanathan. A note on Priest's finite inconsistent arithmetics, Journal of Philosophical Logic, 35, 2006, pp529-537.
- with C.J. Nix. A continuum of inductive methods arising from a generalized principle of instantial relevance, Journal of Philosophical Logic, 35(1), 2006, pp83-115.
- Good news for prisoner A, in The Way Through Science and Philosophy: Essays in Honour of Stig Andur Pedersen, Eds. H.B.Andersen, F.V.Christiansen, K.F.Jorgensen & V.F.Hendricks, Printed by Lightning Source, Milton Keynes, UK. ISBN 1-904987-33-8, 2006, pp237-242.
- Formal Philosophy, Eds. V.F.Hendricks & J.Symons, Academic Press, 2005, pp147-152.
- On filling-in missing information in causal networks, International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems, 13(3), 2005, pp263-280
- with H.Hosni, Rationality as conformity, Synthese (Knowledge, Rationality and Action), 144 (3), 2005, pp 249-285.
- Deriving information from inconsistent knowledge bases: A completeness theorem, Logic Journal of the IGPL, 12, 2004, 345 - 353.
- with A. Vencovská. Why it is common sense to invent reasons, LOGICA Yearbook 2002, Institute of Philosophy of the Czech Academy of Sciences, Eds. T. Childers & O. Majer, 2003, pp.195-204.
- with A. Vencovská. The emergence of reasons conjecture, Journal of Applied Logic, 1, 2003, pp 167-195.
- with O.W. Barnett. A proof of the base case of the emergence of reason's conjecture for CM^{infinity}, Technical Report 2003/6 of the Manchester Centre for Pure Mathematics, ISSN 147-9210.
- with L. C. Hill. When maximising entropy gives the rational closure, Journal of Logic and Computation, 13, 2003, pp51-68.
- with Vencovská. There is a reason for everything (probably): On the application of maxent to induction, in the Proceedings of the Conditionals Information Inference Workshop, Fern Universitat, Hagen, Germany, 2002, pp211-229.
- with M. J. Hill & G. M. Wilmers. Some observations on induction in predicate probabilistic reasoning, Journal of Philosophical Logic, 31, 2002, pp43-75.
- with M. Wafy. On the emergence of reasons in inductive logic, Logic Journal of the IGPL, 2001; 9: 207 - 216.
- On the distribution of probability functions in the natural world, in Probability Theory: Philosophy, Recent History and Relations to Science, Eds. V.F.Hendricks, S.A.Pedersen \& K.F.J0gensen, Synthese Library Vol.297, 2001, pp125-145
- with A. Vencovská & M. Wafy. Some limit theorems for ME, MD and CM^{infinity}, Manchester Centre for Pure Mathematics, preprint number 2001/9.
- with A Vencovská. Common sense and stochastic independence, to appear in Foundations of Bayesianism, Eds. D. Corfield & J. Williamson, Kluwer Applied Logic Series, 2001.
- A note on the Dutch Book method. This is a slightly revised version of a paper of the same title which appeared in the Proceedings of the 2nd International Symposium on Imprecise Probabilities and their Applications, Ithaca, New York, 2001.
- Semantics for Fuzzy Logic supporting truth functionality, in Discovering the World with Fuzzy Logic, Eds. V.Novák & I.Perfilieva, Physica-Verlag Studies in Fuzziness and Soft Computing, 2000, pp82-104.
- with P. Hájek & J.C. Shepherdson. Rational Pavelka Predicate Logic is a conservative extension of Lukasiewicz Perdicate Logic, Journal of Symbolic Logic, 2000, vol 65, no.2, pp669-682.
- with P.N. Watton & G.M. Wilmers. On the structure of probability functions in the natural world, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2000, vol 8, no. 3, pp311.329
- with P. Hájek & J.C. Shepherdson. The Liar Paradox and Fuzzy Logic, Journal of Symbolic Logic, 2000, vol 65, no.1, pp339-346.
- with A.D.C. Bennett & A. Vencovská. A new criteria for comparing fuzzy logics for uncertain reasoning.Journal of Logic, Language, and Information, 2000, vol 9, pp 31-63.
- with P.N. Watton & G.M. Wilmers. On the distribution of natural probability functions. ISIPTA '99 Proc.First Int. Conf. on Imprecise Probabilities and their Applications, Eds G. de Cooman et al, Ghent, 1999, pp 302-310.
- with C. Dimitracopoulos, A. Vencovská & G. M. Wilmers. A multivariate natural prior probability distribution based on the propositional calculus. Technical Report of the Manchester Centre for Pure Mathematics,no. 1999/6, University of Manchester Manchester, M13 9PL, UK.
- Common sense and maxiumum entropy. Synthese, vol.117, pp75-93.
- with A.Vencovská. A proof theory for probabilistic uncertain reasoning. Journal of Symbolic Logic, 1998, vol. 63, no.3, pp1007-1039.
- with R. Booth. A note on the rational closure of knowledge bases with both positive and negative knowledge. Journal of Logic, Language and Information, 1998, vol.7, no.2, pp165-190.
- with A. Vencovská. In defence of the Maximum Entropy Inference Process. International Journal of Approximate Reasoning, 1997, vol.17, no.1, pp77-103.
- A semantics for Fuzzy Logic. Journal of Soft Computing,vol.1, no.3, pp143-147.
- A semantics for Fuzzy Logic, Journal of Soft Computing, vol.1, no.3, 1997, pp 143-147.
- with P.Hájek, A dialogue of Fuzzy Logic, Journal of Soft Computing, vol.1, no.1, 1997, pp3-5.
- with A.Vencovská, Belief formation by constructing models, in: L.Dorst et al Eds., Reasoning with Uncertainty in Robotics, Springer Lecture Notes in Artificial Intelligence, 1093, 1996, pp221-259.
- with A. Vencovská, Principles of Uncertain Reasoning, in: Philosophy and Cognitive Science, Ed. A.Clark et al, Kluwer Press, 1996, pp221-259.
- with A.Vencovská & G.M.Wilmers, A natural prior probability distribution derived from the propositional calculus, Annals of Pure and Applied Logic, 70, 1994, pp243-285.
- with A. Vencovská & G.M. Wilmers, A note on probability versus Truth-Functionality, Proceedings of the WUPES '94 Conference, Trest, Czech Republic, 1994, 12pp.
- with A. Vencovská, A Model of Belief, Artificial Intelligence, (1993), Vol. 64, pp.197-241.
- with R. Tavakol, Goodstein algorithm as a super-transient dynamical system, Physics Letters A, 180 (1993) pp.83-86.
- with A. Vencovská, A Method of Updating justifying Minimum Cross Entropy, International Journal of Approximate Reasoning, 1992, Vol 7, pp.1-18.
- with R. Kossak, Subsets of Models of Arithmetic, Arch. Math. Logic, (1992) 32, pp.65-73.
- with A. Vencovská & G.M. Wilmers, Natural Distributions and Random Sentences, Proceedings of WUPES, Prague, 1991.
- with A. Vencovská & G.M. Wilmers, A Note on Objective Inductive Inference; Proc. First World Conference on the Fundermentals of AI, Ed. M de Glas \& D. Gabbay, Ass. Francaise pour l'Intelligence Artificielle, Paris, (1991), pp.407-412.
- Combinatorial Statements Independent of Arithmetic, Mathematics of Ramsey Theory, Ed. J. Nesetril & V. Rodl, Springer-Verlag, (1991), pp. 232-245
- with A. Vencovská, Modelovani Domneni, Proc. Czechoslovak AI'90 Conf., Ed. V. Márik, Prague, (1990), pp.9-19.
- with I. Maung, A Note on the Infeasibility of some Inference & Processes, International Journal of Intelligent Systems, Vol.5, No.5, pp.595-604, (1990).
- with A. Vencovská, Modelling Belief, Proceedings of the Evolving Knowledge Conference, British Society for the Philosophy of Science, Reading University Pitman Press, (1990), pp.133-154.
- with A. Vencovská, A Note on the Inevitability of Maximum Entropy, International Journal of Approximate Reasoning, (1990) pp.183-224, Vol.4, No.3.
- with A. Vencovská, On the applicability of maximum entropy to inexact reasoning, International Journal of Approximate Reasoning>/em>, Vol 3(1) (1989), pp.1-34.
- with A. Vencovská, Maximum Entropy and inductive inference, in Maximum Entropy and Bayesian Methods, Ed. J. Skilling, Kluwer Academic Publishers (1989), pp.397-403.
- with A. Wilkie, On the existence of end extensions of models of bounded induction. Proceedings of the 8th International Congress of Logic, Methodology and Philosophy of Science, Moscow, pp.143-162, (1988).
- with A. Vencovská, Inexact and inductive reasoning. Proceedings of the 8th International Congress of Logic, Methodology and Philosophy of Science, Moscow, pp.111-120 (1988).
- with A.J. Wilkie & A.R. Woods, Provability of the pigeon hole principle and the existence of infinitely many primes, Journal of Symbolic Logic, 53, (1988), pp.1235-1244.
- with C. Dimitracopoulos & R. Kaye, On parameter free induction schemas, Journal of Symbolic Logic, 53, (1988), pp.1082-1097.
- C. Dimitracopoulos, A note on a theorem of H.Friedman, Zeitschr. fur math. Logik und Grundlagen der Math. Bd 34, (1988), pp 13-17.
- with A. Vencovská , The maximum entropy approach to inductive reasoning, ,em>Proceedings from the AI-workshop on inductive reasoning, Riso National Laboratory, Denmark, 1987.
- with A. Wilkie, On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, 35, (1987), pp 261-302.
- with A. Wilkie, Counting $\Delta_0$ sets, Fund. Math. 127, (1986), pp 67-76.
- with P. Hájek, Combinatorial principles concerning approximation of functions, Arch. Math. Logik 26 (1986/7), pp 13-28.
- with C. Dimitracopoulos, The pigeonhole principle and fragments of arithmetic, Zeit. Math. Logik, Bd. 32, (1986), pp.73-80.
- with W.G. Handley & A.J. Wilkie Characterizing some low arithmetic classes, Colloquia Mathematica Societatis János Bolyai, 44, (1984), pp353-364.
- with A.J. Wilkie, Counting problems in bounded arithmetic, Proceedings of the VI'th Latin American Logic Conference, Caracus 1983, Springer-Verlag Lecture Notes Series, No.1130, pp.317-340.
- with R. Kossak, Subsets of models of arithmetic. Proceedings of the Jadwisin Logic Conference, Poland 1981, Leeds University Press, pp.159-173.
- with A.J. Wilkie, Some results on bounded induction, Proc. 2nd Easter Conf. on Model Theory, Humbolt University, Berlin (1984), & pp.223-228.
- with A.J. Wilkie, $\Delta_0$ sets and induction, Proceedings of the Jadwisin Logic Conference, Poland, 1981, Leeds University Press, pp.237-248.
- O strukture modelu omezene $E_1$-indukce, Casopis pro Pestovani Matematiky (Journal of Czech. Academy of Science), 109, (1984), pp.372-379.
- with L. Csirmaz, A property of 2-sorted Peano models and program verification. Zeit Math. Logik, 30, (1984), pp.372-379.
- with G. Mills, Regularity in models of arithmetic, Journal of Symbolic Logic, 49, (1984), pp.412-420.
- with C. Dimitracopoulos, A note on the undefinability of cuts, Journal of Symbolic Logic, 48, (1983), pp.564-569.
- with L. Kirby, Accessible independence results for Peano Arithmetic, Bulletin London Mathematical Society, 14, (1982), pp.285-293.
- with C. Dimitracopoulos, Truth definitions for $\Delta_0$ formulae, Monographie No.30 de L'Enseignement Mathematique, Universite de Geneve (1982), pp.319-329.
- with A.J. Wilkie, Models of arithmetic and the rudimentary sets, Bulletin de la Societe Mathematique de Belgique, Vol.33, Ser.B, (1981), pp.157-169.
- Some conservation results for fragments of arithmetic, Proceedings of the Paris Logic Year 1979-80, Springer-Verlag Lecture Notes Series, No.890, pp.251-262.
- A hierarchy of cuts in models of arithmetic, Proceedings of the Logic Conference held in Karpacz, Poland, (1979). North Holland, pp.312-337.
- with G. Mills, Closure properties of countable non-standard integers, Fund. Math., CIII (1979), pp.205-215.
- Some independence results in Peano arithmetic, Journal of Symbolic Logic, 43, (1978), pp.725-731.
- with L. Kirby, $\Sigma_n$-collection schemes in arithmetic, Logic Colloquium '77, North Holland, (1978), pp.199-209.
- with L. Harrington, A mathematical incompleteness in Peano arithmetic, Handbook of mathematical logic, North Holland, (1977), pp.1133-1142.
- Note on an induction axiom, Journal of Symbolic Logic, 43, (1978), pp.113-117.
- Measure and minimal degrees, Annals Math. Logic, II, (1977), pp.203-216.
- with L. Kirby, Initial segments of models of Peano's axioms, Proceedings of the Bierutowice Conference, (1976), Springer & Verlag Lecture Notes series, No.619, pp.211-226.
- Sets satisfying $O^\sharp$, Bull. London Math.Soc., 8, (1976), pp.257-260.
- Models of arithmetic and the 1-3-1 lattice, Fund. Math., XCV, (1976), pp.195-199.
- Patterns of indiscernibles, Bull. London Math. Soc., (1974), pp.183-188.
- with K. Devlin, Certain sequences of ordinals, Colloquia Mathematica Societatis Janos Bolyai, Hungary, (1973), pp.333-359.
- with K. Devlin, More on the free subset problem, Annals Math. Logic, 5, (1973) pp.327-336.
- $ZF \vdash \Sigma^0_4$-determinateness, Journal of Symbolic Logic, 37, 1972, 4 pp.661-667.
- Minimal models of $ZF$, Proceedings of the Bertrand Russell Memorial Logic Conference, Denmark (1971), pp.327-331.
- with K. Kunen, Boolean extensions and measurable cardinals. Annals Math. Logic, 2 (1971), pp.359-378.
- On models of arithmetic. Proceedings of the Conference in Mathematical Logic - Bedford College, London, Ed. W.Hodges,(1970), pp.251-280.
Co-workers
- Alena Vencovská, George Wilmers, see below for some further publications of theirs.
Recent PhD Students
- Elizabeth Howarth, "New Rationality Principles in Pure Inductive Logic", Ph.D. dissertation, July 2015.
- Malte Kliess, "The Principle of Predicate Exchangeability in Pure Inductive Logic", Ph.D. dissertation, 2014.
- Alexandra Hill, "Reasoning by Analogy in Inductive Logic", Ph.D. dissertation, July 2013.
- Richard C.H. Simmonds, "On Horn Closure Conditions for Probabilistic Consequence Relations", Ph.D. dissertation, July 2010.
- Soroush Rafiee Rad, "Inference Processes for Probabilistic First Order Languages", Ph.D. dissertation, October 2009.
- Juergen Landes, "The Principle of Spectrum Exchangeability within Inductive Logic", Ph.D. dissertation, April 2009.
- David Picado Muino, "Deriving Information from Inconsistent Knowledge Bases: A Probabilistic Approach", Ph.D. dissertation, November 2008.
- Peter Waterhouse, "Probabilistic Relationships, Relevance and Irrelevance within the field of Uncertain Reasoning", Ph.D. dissertation, December 2007.
- Hykel Hosni, "Rationality-as-conformity", Ph.D. dissertation, June 2005.
- Christopher Nix, "Probabilistic Induction in the Predicate Calculus", Ph.D. dissertation, May 2005.
- Lee C. Hill "A New Relationship between Maximum Entropy and the Rational Closure of a Conditional Knowledge Base" , Ph.D. dissertation, December 2002.
- Kevin M Knight "A Theory of Inconsistency" , PhD thesis, July 2002
Teaching
- MATH33001/43001/63001 Predicate Logic
- MATH43032/63032 Nonstandard Logic
- MATH19821/19541, 0C1/1C1
Teaching Materials
- Progic Spring School Course Notes, 2015
- ISLA Winter School Slides, 2014
- Guangzhou Winter School course on Pure Inductive Logic, 2010
- Formal Epistemology Workshop slides on Pure Inductive Logic, Munich, 2012
- Some supporting notes to the Pure Inductive Logic slides from FEW, Munich, 2012
Professional affiliations
- British Academy
- British Logic Colloquium (President 2001-06)
- EPSRC (Panel)
- London Mathematical Society
Some further papers by George Wilmers
- Generalising the Maximum Entropy Inference Process to the Aggregation of Probabilistic Beliefs
- The Social Entropy Process: Axiomatising the Aggregation of Probabilistic Beliefs, in Probability, Uncertainty, and Rationality, 2010, eds. Hykel Kosni and Franco Montagna, vol. 10 of CRM series of the Scuola Normale Superiore, Pisa. Also available at http://eprints.ma.man.ac.uk/1395/01/covered/MIMS_ep2010_13.pdf