An Analytic Logic of Aggregation

We present a modular approach to the logic of aggregated group preferences based on hybrid modal logic. The modularity of the system is twofold: 1) lifting preference relations between states to complex relations between propositions and 2) lifting individual preferences to group preferences. The preferences may be doxastic or proairetic, generating a logic of aggregated belief or aggregated desire, respectively, using a specific aggregation policy known as `lexicographic re-ordering'. Each agent and each group of agents has an associated modal operator representing their preferences between states. The addition of the existential modality and nominals allows us to produce, first, a Hilbert-style axiomatization of the logic and then a more thorough analysis of inference using a Gentzen-style sequent calculus, in which the role of each operator is revealed.

[1]  Dimiter G. Skordev.  Mathematical Logic and Its Applications , 2011 .

[2]  Helmut Schwichtenberg,et al.  Basic proof theory (2nd ed.) , 2000 .

[3]  I. Levi,et al.  Making It Explicit , 1994 .

[4]  K. Segerberg.  The Basic Dynamic Doxastic Logic of AGM , 2001 .

[5]  Jerry Seligman.  Internalization: The Case of Hybrid Logics , 2001, J. Log. Comput..

[6]  Henrik Lagerlund,et al.  Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg , 2006 .

[7]  K. Arrow.  Social Choice and Individual Values , 1951 .

[8]  Michael Dummett,et al.  The logical basis of metaphysics , 1991 .

[9]  R. Smullyan.  First-Order Logic , 1968 .

[10]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[11]  Mary-Anne Williams,et al.  Frontiers in Belief Revision , 2001 .

[12]  Frank Wolter,et al.  Handbook of Modal Logic , 2007, Studies in logic and practical reasoning.

[13]  Johan van Benthem,et al.  Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences , 2009, J. Philos. Log..

[14]  Leon van der Torre,et al.  Hidden Uncertainty in the Logical Representation of Desires , 2003, IJCAI.

[15]  Pierre-Yves Schobbens,et al.  Operators and Laws for Combining Preference Relations , 2002, J. Log. Comput..

[16]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[17]  B. T. Cate,et al.  Model theory for extended modal languages , 2005 .

[18]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[19]  Craig Boutilier,et al.  Toward a Logic for Qualitative Decision Theory , 1994, KR.

[20]  Olivier Roy,et al.  Preference logic, conditionals and solution concepts in games , 2005 .