We introduce a general purpose typed ?-calculus ? ? which contains intuitionistic logic, is capab... more We introduce a general purpose typed ?-calculus ? ? which contains intuitionistic logic, is capable of internalizing its own derivations as ?-terms and yet enjoys strong normalization with respect to a natural reduction system. In particular, ?? subsumes the typed ?-calculus. The Curry-Howard isomorphism converting intuitionistic proofs into ?-terms is a simple instance of the internalization property of ??. The standard semantics of ?? is given by a proof system with proof checking capacities. The system ?? is a theoretical prototype of reflective extensions of a broad class of type-based systems in programming languages, provers, AI and knowledge representation, etc.
Imagine a database—a set of propositions $\varGamma =\{F_1,\ldots ,F_n\}$ with some kind of proba... more Imagine a database—a set of propositions $\varGamma =\{F_1,\ldots ,F_n\}$ with some kind of probability estimates and let a proposition $X$ logically follow from $\varGamma $. What is the best justified lower bound of the probability of $X$? The traditional approach, e.g. within Adams’ probability logic, computes the numeric lower bound for $X$ corresponding to the worst-case scenario. We suggest a more flexible parameterized approach by assuming probability events $u_1,u_2,\ldots ,u_n$ that support $\varGamma $ and calculating aggregated evidence$e(u_1,u_2,\ldots ,u_n)$ for $X$. The probability of $e$ provides a tight lower bound for any, not only a worst-case, situation. The problem is formalized in a version of justification logic and the conclusions are supported by corresponding completeness theorems. This approach can handle conflicting and inconsistent data and allows the gathering both positive and negative evidence for the same proposition.
Traditionally, Epistemic Logic represents epistemic scenarios using a single model. This, however... more Traditionally, Epistemic Logic represents epistemic scenarios using a single model. This, however, covers only complete descriptions that specify truth values of all assertions. Indeed, many—and perhaps most—epistemic descriptions are not complete. Syntactic Epistemic Logic, SEL, suggests viewing an epistemic situation as a set of syntactic conditions rather than as a model. This allows us to naturally capture incomplete descriptions; we discuss a case study in which our proposal is successful. In Epistemic Game Theory, this closes the conceptual and technical gap, identified by R. Aumann, between the syntactic character of game-descriptions and semantic representations of games.
We introduce a general purpose typed ?-calculus ? ? which contains intuitionistic logic, is capab... more We introduce a general purpose typed ?-calculus ? ? which contains intuitionistic logic, is capable of internalizing its own derivations as ?-terms and yet enjoys strong normalization with respect to a natural reduction system. In particular, ?? subsumes the typed ?-calculus. The Curry-Howard isomorphism converting intuitionistic proofs into ?-terms is a simple instance of the internalization property of ??. The standard semantics of ?? is given by a proof system with proof checking capacities. The system ?? is a theoretical prototype of reflective extensions of a broad class of type-based systems in programming languages, provers, AI and knowledge representation, etc.
Imagine a database—a set of propositions $\varGamma =\{F_1,\ldots ,F_n\}$ with some kind of proba... more Imagine a database—a set of propositions $\varGamma =\{F_1,\ldots ,F_n\}$ with some kind of probability estimates and let a proposition $X$ logically follow from $\varGamma $. What is the best justified lower bound of the probability of $X$? The traditional approach, e.g. within Adams’ probability logic, computes the numeric lower bound for $X$ corresponding to the worst-case scenario. We suggest a more flexible parameterized approach by assuming probability events $u_1,u_2,\ldots ,u_n$ that support $\varGamma $ and calculating aggregated evidence$e(u_1,u_2,\ldots ,u_n)$ for $X$. The probability of $e$ provides a tight lower bound for any, not only a worst-case, situation. The problem is formalized in a version of justification logic and the conclusions are supported by corresponding completeness theorems. This approach can handle conflicting and inconsistent data and allows the gathering both positive and negative evidence for the same proposition.
Traditionally, Epistemic Logic represents epistemic scenarios using a single model. This, however... more Traditionally, Epistemic Logic represents epistemic scenarios using a single model. This, however, covers only complete descriptions that specify truth values of all assertions. Indeed, many—and perhaps most—epistemic descriptions are not complete. Syntactic Epistemic Logic, SEL, suggests viewing an epistemic situation as a set of syntactic conditions rather than as a model. This allows us to naturally capture incomplete descriptions; we discuss a case study in which our proposal is successful. In Epistemic Game Theory, this closes the conceptual and technical gap, identified by R. Aumann, between the syntactic character of game-descriptions and semantic representations of games.
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Papers by Sergei Artemov