Set-theoretic completeness for epistemic and conditional logic

The standard approach to logic in the literature in philosophy and mathematics, which has also been adopted in computer science, is to define a language (the syntax), an appropriate class of models together with an interpretation of formulas in the language (the semantics), a collection of axioms and rules of inference characterizing reasoning (the proof theory), and then relate the proof theory to the semantics via soundness and completeness results. Here we consider an approach that is more common in the economics literature, which works purely at the semantic, set-theoretic level. We provide set-theoretic completeness results for a number of epistemic and conditional logics, and contrast the expressive power of the syntactic and set-theoretic approaches.

[1]  John P. Burgess,et al.  Quick completeness proofs for some logics of conditionals , 1981, Notre Dame J. Formal Log..

[2]  Paul R. Milgrom,et al.  AN AXIOMATIC CHARACTERIZATION OF COMMON KNOWLEDGE , 1981 .

[3]  Joseph Y. Halpern,et al.  First-Order Conditional Logic Revisited , 1996, AAAI/IAAI, Vol. 2.

[4]  Yoram Ofer Moses.  Theoretical aspects of reasoning about knowledge : proceedings of the fourth conference (TARK 1992), March 22-25, 1992, Monterey, California , 1992 .

[5]  Donald Nute,et al.  Counterfactuals , 1975, Notre Dame J. Formal Log..

[6]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[7]  Robert Stalnaker.  A Theory of Conditionals , 2019, Knowledge and Conditionals.

[8]  John L. Bell,et al.  Models and Ultraproducts: An Introduction. , 1969 .

[9]  Grigoris Antoniou,et al.  Nonmonotonic reasoning , 1997 .

[10]  E. Clarke,et al.  Automatic Veriication of Nite-state Concurrent Systems Using Temporal-logic Speciications. Acm , 1993 .

[11]  Franz von Kutschera,et al.  Causation , 1993, J. Philos. Log..

[12]  N. Malcolm.  On Knowledge and Belief , 1954 .

[13]  Joseph Y. Halpern.  Defining Relative Likelihood in Partially-Ordered Preferential Structures , 1996, UAI.

[14]  Ronald Fagin,et al.  A logic for reasoning about probabilities , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[15]  Judea Pearl,et al.  CAUSATION, ACTION, AND COUNTERFACTUALS , 2004 .

[16]  Joseph Y. Halpern,et al.  On the Complexity of Conditional Logics , 1994, KR.

[17]  Sarit Kraus,et al.  Nonmonotonic Reasoning, Preferential Models and Cumulative Logics , 1990, Artif. Intell..

[18]  Joseph Y. Halpern,et al.  Plausibility measures and default reasoning , 1996, JACM.

[19]  Craig Boutilier,et al.  Conditional Logics of Normality: A Modal Approach , 1994, Artif. Intell..

[20]  Joseph Y. Halpern,et al.  Model Checking vs. Theorem Proving: A Manifesto , 1991, KR.

[21]  Robert Stalnaker,et al.  Notes on Conditional Semantics , 1992, TARK.

[22]  Joseph Y. Halpern.  An Analysis of First-Order Logics of Probability , 1989, IJCAI.

[23]  KrausSarit,et al.  Nonmonotonic reasoning, preferential models and cumulative logics , 1990 .