scholarly journals Belief contraction in the context of the general theory of rational choice

1993 ◽  
Vol 58 (4) ◽  
pp. 1426-1450 ◽  
Author(s):  
Hans Rott
Keyword(s):  
General Theory ◽  
Model Theory ◽  
Revealed Preference ◽  
Representation Theorems ◽  
Belief Contraction ◽  
Elementary Model ◽  
Partial Meet Contraction ◽  
Finite Case ◽  
Open Question ◽  
Classic Paper

AbstractThis paper reorganizes and further develops the theory of partial meet contraction which was introduced in a classic paper by Alchourrón, Gärdenfors, and Makinson. Our purpose is threefold. First, we put the theory in a broader perspective by decomposing it into two layers which can respectively be treated by the general theory of choice and preference and elementary model theory. Second, we reprove the two main representation theorems of AGM and present two more representation results for the finite case that “lie between” the former, thereby partially answering an open question of AGM. Our method of proof is uniform insofar as it uses only one form of “revealed preference”, and it explains where and why the finiteness assumption is needed. Third, as an application, we explore the logic characterizing theory contractions in the finite case which are governed by the structure of simple and prioritized belief bases.

Revealed Preference and Model Theory

2016 ◽  
pp. 186-197 ◽  
Author(s):  
Christopher P. Chambers ◽  
Federico Echenique
Keyword(s):  
Model Theory ◽  
Revealed Preference

Wittgenstein and the Logic of Inference

Dialogue ◽  
1982 ◽  
Vol 21 (4) ◽  
pp. 671-692
Author(s):  
Jan Zwicky
Keyword(s):  
General Theory ◽  
Model Theory ◽  
Deductive System ◽  
Functional Interpretation ◽  
Logical Constants

TheTractatusfirst appeared in 1921, the same year that Post's “Introduction to a General Theory of Elementary Propositions” appeared in theAmerican Journal of Mathematics. As the latter is the first piece clearly to present and exploit the distinction between a deductive system and a truth-functional interpretation of such a system, we may conclude that Wittgenstein's views had been arrived at somewhat before a variety of logical concepts had received the clarification and refinement incipient on the now taken-for-granted distinction between proof and model theory. One such concept, of considerable interest to Wittgenstein, was that of inference. The following constitutes an attempt to explicate his notion. In particular, I shall attempt to show that his repudiation of “laws of inference” is closely tied to his rejection of logical constants; and that both can be seen as the product of what might be termed a “metaphysics of completeness”—before, of course, any (presently recognizable) notion of completeness had achieved a measure of precision or currency.

scholarly journals Applications of Kolmogorov complexity to computable model theory

2007 ◽  
Vol 72 (3) ◽  
pp. 1041-1054 ◽  
Author(s):  
Bakhadyr Khoussainov ◽  
Pavel Semukhin ◽  
Frank Stephan
Keyword(s):  
Model Theory ◽  
Kolmogorov Complexity ◽  
Isomorphism Type ◽  
Computable Model ◽  
Categorical Theory ◽  
Computable Model Theory ◽  
Open Question ◽  
Saturated Structure ◽  
Computable Isomorphism

AbstractIn this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-categorical saturated structure with a unique computable isomor-phism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ℵ1-categorical but not ℵ0-categorical saturated -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.

scholarly journals Some elementary results in intuitionistic model theory

1996 ◽  
Vol 61 (3) ◽  
pp. 745-767
Author(s):  
Wim Veldman ◽  
Frank Waaldijk
Keyword(s):  
Model Theory ◽  
Additive Group ◽  
Real Numbers ◽  
Elementary Model ◽  
The Real

AbstractWe establish constructive refinements of several well-known theorems in elementary model theory. The additive group of the real numbers may be embedded elementarily into the additive group of pairs of real numbers, constructively as well as classically.

scholarly journals Subgroups of elliptic elements of the Cremona group

2021 ◽  
Vol 2021 (770) ◽  
pp. 27-57
Author(s):  
Christian Urech
Keyword(s):  
Projective Plane ◽  
Model Theory ◽  
Complex Projective Plane ◽  
Elementary Model ◽  
Cremona Group ◽  
Birational Transformations ◽  
Derived Length ◽  
Tits Alternative ◽  
Torsion Subgroups ◽  
Complex Projective

Abstract The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an application, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.

Coalitional Expected Multi‐Utility Theory

Econometrica ◽  
2019 ◽  
Vol 87 (3) ◽  
pp. 933-980 ◽  
Author(s):  
Kazuhiro Hara ◽  
Efe A. Ok ◽  
Gil Riella
Keyword(s):  
Expected Utility ◽  
Utility Theory ◽  
Preference Relation ◽  
Revealed Preference ◽  
Portfolio Diversification ◽  
Independence Axiom ◽  
Representation Theorems ◽  
Utility Representation ◽  
Reversal Phenomenon ◽  
Preference Reversal Phenomenon

This paper begins by observing that any reflexive binary (preference) relation (over risky prospects) that satisfies the independence axiom admits a form of expected utility representation. We refer to this representation notion as the coalitional minmax expected utility representation. By adding the remaining properties of the expected utility theorem, namely, continuity, completeness, and transitivity, one by one, we find how this representation gets sharper and sharper, thereby deducing the versions of this classical theorem in which any combination of these properties is dropped from its statement. This approach also allows us to weaken transitivity in this theorem, rather than eliminate it entirely, say, to quasitransitivity or acyclicity. Apart from providing a unified dissection of the expected utility theorem, these results are relevant for the growing literature on boundedly rational choice in which revealed preference relations often lack the properties of completeness and/or transitivity (but often satisfy the independence axiom). They are also especially suitable for the (yet overlooked) case in which the decision‐maker is made up of distinct individuals and, consequently, transitivity is routinely violated. Finally, and perhaps more importantly, we show that our representation theorems allow us to answer many economic questions that are posed in terms of nontransitive/incomplete preferences, say, about the maximization of preferences, the existence of Nash equilibrium, the preference for portfolio diversification, and the possibility of the preference reversal phenomenon.

Finite Conformal Hypergraph Covers and Gaifman Cliques in Finite Structures

2003 ◽  
Vol 9 (3) ◽  
pp. 387-405 ◽  
Author(s):  
Ian Hodkinson ◽  
Martin Otto
Keyword(s):  
Model Theory ◽  
Extension Property ◽  
Alternative Proof ◽  
Finite Model ◽  
Finite Model Property ◽  
Finite Model Theory ◽  
Relational Structures ◽  
Finite Structures ◽  
Guarded Fragment ◽  
Open Question

AbstractWe provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques—thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In terms of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF.

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