## Set-Theoretic Completeness for Epistemic and Conditional Logic (1999)

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Venue: | In Fifth International Conference on AI and Mathematics. Proceedings |

Citations: | 13 - 3 self |

### BibTeX

@INPROCEEDINGS{Halpern99set-theoreticcompleteness,

author = {Joseph Halpern},

title = {Set-Theoretic Completeness for Epistemic and Conditional Logic},

booktitle = {In Fifth International Conference on AI and Mathematics. Proceedings},

year = {1999},

pages = {1--27}

}

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### Abstract

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. The work was supported in part by NSF under grant IRI-96-25901 and by the Air Force Office of Scientific Research under grant F49620-96-1-0323. 1 Introduction The standard approach to logic in the literature in philosophy and ...

### Citations

1497 | Reasoning about Knowledge - Fagin, Halpern, et al. - 1995 |

1184 | Automatic verification of finite-state concurrent systems using temporal logic specifications - Clarke, Emerson, et al. - 1986 |

540 | Nonmonotonic reasoning, preferential models and cumulative logics - Kraus, Lehmann, et al. |

432 |
Knowledge and Belief
- Hintikka
- 1962
(Show Context)
Citation Context ...es ¬, ∩, ∪. In this paper, I explore set-theoretic completeness proofs in the context of epistemic logics and conditional logics. Both of these logics were introduced in the philosophical literature [=-=Hintikka 1962-=-; Stalnaker 1968], but have been widely used in computer science and AI. Epistemic logic has been used as a tool for analyzing multi-agent systems [Fagin, 2Halpern, Moses, and Vardi 1995]; conditiona... |

283 |
Probability and conditionals
- Stalnaker
- 1981
(Show Context)
Citation Context ...this paper, I explore set-theoretic completeness proofs in the context of epistemic logics and conditional logics. Both of these logics were introduced in the philosophical literature [Hintikka 1962; =-=Stalnaker 1968-=-], but have been widely used in computer science and AI. Epistemic logic has been used as a tool for analyzing multi-agent systems [Fagin, 2Halpern, Moses, and Vardi 1995]; conditional logic has been... |

271 | An analysis of first-order logics of probability
- Halpern
- 1990
(Show Context)
Citation Context ...ut probability by working purely at a set-theoretic level. While some logics for reasoning about probability have been proposed, both propositional [Fagin, Halpern, and Megiddo 1990] and first-order [=-=Halpern 1990-=-], they certainly do not begin to capture all the subtleties of the reasoning we find in probability texts. For example, typical logics of probability cannot express notions such as expectation and va... |

268 |
Causation and Counterfactuals
- Lewis
- 2004
(Show Context)
Citation Context ...yzing multi-agent systems [Fagin, 2Halpern, Moses, and Vardi 1995]; conditional logic has been used as a framework for analyzing nonmonotonic reasoning [Boutilier 1994] and counterfactual reasoning [=-=Lewis 1973-=-]. It also has an important role to play in the analysis of causality [Lewis 1973], which is becoming an increasingly important issue in AI as well [Pearl 1995]. Set-theoretic completeness proofs for ... |

215 | A logic for reasoning about probabilities - Fagin, Halpern, et al. - 1990 |

117 | Model checking vs. theorem proving: A manifesto
- Halpern, Vardi
- 1991
(Show Context)
Citation Context ..., and Sistla 1986], typically works with one fixed model, the one generated by the program whose correctness we are trying to prove. Model checking has been advocated for epistemic reasoning as well [=-=Halpern and Vardi 1991-=-]. Perhaps when using the model-checking approach, it might make sense to work at the set-theoretic level. Probability provides another example. Probabilists start by defining a particular model—the p... |

99 |
Conditional Logics of Normality; A Modal Approach
- Boutilier
- 1994
(Show Context)
Citation Context ...istemic logic has been used as a tool for analyzing multi-agent systems [Fagin, 2Halpern, Moses, and Vardi 1995]; conditional logic has been used as a framework for analyzing nonmonotonic reasoning [=-=Boutilier 1994-=-] and counterfactual reasoning [Lewis 1973]. It also has an important role to play in the analysis of causality [Lewis 1973], which is becoming an increasingly important issue in AI as well [Pearl 199... |

79 | Plausibility measures and default reasoning
- Friedman, Halpern
- 1996
(Show Context)
Citation Context ...} ∩ H2 ❀ {w′} ⊆ H ❀ E. Thus, w ∈ H ❀ E. The opposite containment is obtained by a symmetric argument. Now we must deal with the case that C7 ′ ∈ C. The argument is similar in spirit to that given in [=-=Friedman and Halpern 1998-=-]. In this case, ≼w is not necessarily a total order. However, we can show that ≺w is modular: if w1 ≺w w2, then for all w3 ∈ Ww, either w3 ≺w w2 or w1 ≺w w3. To see this, suppose w1 ≺w w2 and it is n... |

45 |
completeness proofs for some logics of conditionals
- Burgess, Quick
- 1981
(Show Context)
Citation Context ...mpleteness result is well known. It says that in preferential structures, → satisfies C0, C1, C2, C5, C6, and RC1; moreover, P1, P2, P3, and P4 give us C3, C7, C4, and C8, respectively. Theorem 3.6: [=-=Burgess 1981-=-; Friedman and Halpern 1994; Lewis 1973] Let P be any (possibly empty) subset of {P1,P2,P3,P4} let C be the corresponding subset of {C3,C7,C4,C8}, and let W be a finite set of worlds. Let M P be the c... |

33 | On the complexity of conditional logics
- Friedman, Halpern
- 1994
(Show Context)
Citation Context ...ult is well known. It says that in preferential structures, → satisfies C0, C1, C2, C5, C6, and RC1; moreover, P1, P2, P3, and P4 give us C3, C7, C4, and C8, respectively. Theorem 3.6: [Burgess 1981; =-=Friedman and Halpern 1994-=-; Lewis 1973] Let P be any (possibly empty) subset of {P1,P2,P3,P4} let C be the corresponding subset of {C3,C7,C4,C8}, and let W be a finite set of worlds. Let M P be the class of preferential struct... |

29 | Defining relative likelihood in partially-ordered preferential structure
- Halpern
- 1997
(Show Context)
Citation Context ...er from it. Define w ′ ≤ w w ′′ either if w ′ ≺ w w ′′ or neither w ′ ≺ w w ′′ nor w ′′ ≺ w w ′ hold. It is a standard result (and not hard to show) that ≤ w is a total order if ≺ w is modular. (See [=-=Halpern 1997-=-, Lemma 2.6] for a proof.) Moreover, it is easy to see w ′ < w w ′′ iff w ′ ≺ w w ′′ . Thus, f≤ = f≼, so that if we use ≤ w to define the ternary relation, our previous argument shows that e → and ❀ s... |

19 |
An axiomatic characterization of common knowledge
- Milgrom
- 1981
(Show Context)
Citation Context ...ly important issue in AI as well [Pearl 1995]. Set-theoretic completeness proofs for logics of knowledge and common knowledge are standard in the economics literature (see, for example, [Aumann 1989; =-=Milgrom 1981-=-]). I compare them to the more familiar syntactic completeness proofs in the philosophical literature, and then do the same for conditional logic. For the logics considered here, every syntactic opera... |

18 | Causation, Action, and Counterfactuals
- Pearl
- 1996
(Show Context)
Citation Context ...lier 1994] and counterfactual reasoning [Lewis 1973]. It also has an important role to play in the analysis of causality [Lewis 1973], which is becoming an increasingly important issue in AI as well [=-=Pearl 1995-=-]. Set-theoretic completeness proofs for logics of knowledge and common knowledge are standard in the economics literature (see, for example, [Aumann 1989; Milgrom 1981]). I compare them to the more f... |

8 |
Notes on interactive epistemology. Cowles Foundation for Research in Economics working paper
- Aumann
- 1989
(Show Context)
Citation Context ...an increasingly important issue in AI as well [Pearl 1995]. Set-theoretic completeness proofs for logics of knowledge and common knowledge are standard in the economics literature (see, for example, [=-=Aumann 1989-=-; Milgrom 1981]). I compare them to the more familiar syntactic completeness proofs in the philosophical literature, and then do the same for conditional logic. For the logics considered here, every s... |

8 | Conditional first-order logic revisited - Friedman, Halpern, et al. - 1996 |

4 |
Models and Ultraproducts: An Introduction (third revised printing). Amsterdam/New
- Bell, Slomson
- 1974
(Show Context)
Citation Context ... Note that ❀ also does not satisfy C4 ′ since, for example, ({1, 2} ❀ {1}) ∪ ({1, 2} ❀ ¬{1}) = {1, 2} = W. Example 3.5: For this example, we need to review some material on filters and ultrafilters [=-=Bell and Slomson 1974-=-]. A filter on W is a nonempty set U of subsets of W that is closed under supersets (i.e., if E ∈ U and E ⊂ E ′ , then E ′ ∈ U) and finite intersections, and does not contain the empty set. An ultrafi... |

4 | Notes on conditional semantics - Stalnaker - 1992 |

2 | Also appears - Blackwell - 1981 |

1 | Also appears in Ifs - Blackwell - 1981 |

1 | Notes on conditional semantics - C - 1992 |